Every executive is responsible for enforcing the policy all the way down the chain, recursively.
|
|
An empirical computation engine is an artificial system capable of recursively and intelligently searching a solution space.
|
|
Recursively, that means executives lower down the tree will do the same, because that itself is one of the values you enforce.
|
|
"It was based on that and I just extracted the probabilities to try to recursively build up the tree structure of the keyboard," he explains.
|
|
On November 11, it will release the recursively named "Nintendo Entertainment System: NES Classic Edition," a tiny little dedicated device packed with 30 of the greatest 8-bit hits.
|
|
In it, the character describes "recursively overloading" a memory, and I knew that it had something to do with going through a portal in the "installation" scene's red shift.
|
|
The recursively-named Nintendo Entertainment System: NES Classic Edition is a tiny $60 box that plugs into your TV with an HDMI cable and delivers '80s delights like Super Mario Bros.
|
|
When Homo erectus began using several symbols one after the other in a more predictable pattern (but not yet recursively), Mr Everett thinks he could be said to be using human language.
|
|
But these files had a natural limitation: Most Zip decompression routines max out at a compression ratio of 1032-to-one, which meant that "Zip bombs" could only reach their true compression potential recursively.
|
|
In an introductory essay he wrote for the collection Certain Noble Plays of Japan (1916), edited by Pound and Ernest Fenollosa, Yeats provides at least one clue, homing in on the way Japanese Noh dramatic verse recursively exploits a single recurring image or metaphor.
|
|
If there's anything that rises above bizarre to become actually objectionable where Bryant's storytelling is concerned, it's that—the way that his stories all loop recursively and blindly to the author and the one story he wants to tell, an advertisement for himself and his personal suite of ghosts.
|
|
The whole of it loops recursively, forever in the same year-spanning orbit, and if we turn our eyes to it at the right moment we will always see a Stanford quarterback being described as "cerebral" or a 22-year-old being dismissed for not "having that winner quality about him" or some bit of poker-faced and totally psychotic thumbnail psychologizing.
|
|
An example is the moving average (MA) filter, that can be implemented both recursively and non recursively.
|
|
A graph is weakly recursively simplicial if it has a simplicial vertex and the subgraph after removing a simplicial vertex and some edges (possibly none) between its neighbours is weakly recursively simplicial. A graph is moral if and only if it is weakly recursively simplicial. A chordal graph (a.k.a., recursive simplicial) is a special case of weakly recursively simplicial when no edge is removed during the elimination process.
|
|
A set B is called many-one complete, or simply m-complete, iff B is recursively enumerable and every recursively enumerable set A is m-reducible to B.
|
|
Shore's splitting theorem: Let A be \alpha recursively enumerable and regular. There exist \alpha recursively enumerable B_0,B_1 such that A=B_0 \cup B_1 \wedge B_0 \cap B_1 = \varnothing \wedge A ot\le_\alpha B_i (i<2). Shore's density theorem: Let A, C be α-regular recursively enumerable sets such that \scriptstyle A <_\alpha C then there exists a regular α-recursively enumerable set B such that \scriptstyle A <_\alpha B <_\alpha C.
|
|
The class of all recursively enumerable languages is called RE.
|
|
Therefore, there are finitely generated groups that cannot be recursively presented.
|
|
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context- free, context-sensitive and recursive languages are recursively enumerable.
|
|
The Measure phase recursively calls all elements and determines the size they will take. In the Arrange phase, the child elements are recursively arranged by their parents, invoking the layout algorithm of the layout module in use.
|
|
Later research dealt also with numberings of other classes like classes of recursively enumerable sets. Goncharov discovered for example a class of recursively enumerable sets for which the numberings fall into exactly two classes with respect to recursive isomorphisms.
|
|
After ten years, Kleene and Post showed in 1954 that there are intermediate Turing degrees between those of the computable sets and the halting problem, but they failed to show that any of these degrees contains a recursively enumerable set. Very soon after this, Friedberg and Muchnik independently solved Post's problem by establishing the existence of recursively enumerable sets of intermediate degree. This groundbreaking result opened a wide study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure. There are uncountably many sets that are not recursively enumerable, and the investigation of the Turing degrees of all sets is as central in recursion theory as the investigation of the recursively enumerable Turing degrees.
|
|
There are three equivalent definitions of a recursively enumerable language: # A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language. # A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) which will enumerate all valid strings of the language. Note that if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is "already" produced for a number which is less than n. If it already is produced, use the output for input n+1 instead (recursively), but again, test whether it is "new".
|
|
This algorithm computes hypergeometric solutions and reduces the order of the recurrence equation recursively.
|
|
In the latter case, you must recursively delete subkeys before deleing the parent key.
|
|
So, M does not halt if and only if eg\phi_M is true over all finite models. The set of machines that does not halt is not recursively enumerable, so the set of valid sentences over finite models is not recursively enumerable.
|
|
Any "axiomatizable" fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable. It is an open question to give supports for a "Church thesis" for fuzzy mathematics, the proposed notion of recursive enumerability for fuzzy subsets is the adequate one.
|
|
It is equivalent to the standard Turing machine and therefore accepts precisely the recursively enumerable languages.
|
|
In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same logician, William Craig.
|
|
An algebraic expression can be produced from a binary expression tree by recursively producing a parenthesized left expression, then printing out the operator at the root, and finally recursively producing a parenthesized right expression. This general strategy (left, node, right) is known as an in-order traversal. An alternate traversal strategy is to recursively print out the left subtree, the right subtree, and then the operator. This traversal strategy is generally known as post-order traversal.
|
|
If S is indexed by a set I consisting of all the natural numbers N or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) from the free group on S to the natural numbers, such that we can find algorithms that, given f(w), calculate w, and vice versa. We can then call a subset U of FS recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed as above and R recursively enumerable, then the presentation is a recursive presentation and the corresponding group is recursively presented. This usage may seem odd, but it is possible to prove that if a group has a presentation with R recursively enumerable then it has another one with R recursive.
|
|
The Ehrenfeucht–Mycielski sequence is a recursively defined sequence of binary digits with pseudorandom properties, defined by .
|
|
The Liouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.
|
|
The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory.
|
|
Hence the recursively indexed sequence for N = 49 with set S, is 10, 10, 10, 10, 9\.
|
|
For any finite unary function \theta on integers, let C(\theta) denote the 'frustum' of all partial-recursive functions that are defined, and agree with \theta, on \theta's domain. Equip the set of all partial-recursive functions with the topology generated by these frusta as base. Note that for every frustum C, Ix(C) is recursively enumerable. More generally it holds for every set A of partial-recursive functions: Ix(A) is recursively enumerable iff A is a recursively enumerable union of frusta.
|
|
Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non- isomorphic two generator groups.
|
|
This process continues recursively until the problems are of sufficiently small size to solve in a single processor.
|
|
Min/max kd-trees may be constructed recursively. Starting with the root node, the splitting plane orientation and position is evaluated. Then the children's splitting planes and min/max values are evaluated recursively. The min/max value of the current node is simply the minimum/maximum of its children's minima/maxima.
|
|
"Sierpinski Gasket by Trema Removal" This process of recursively removing triangles is an example of a finite subdivision rule.
|
|
The matrices can be constructed recursively, first in all even dimensions, = 2, and thence in odd ones, 2+1.
|
|
In computability theory, two disjoint sets of natural numbers are called recursively inseparable if they cannot be "separated" with a recursive set.Monk 1976, p. 100 These sets arise in the study of computability theory itself, particularly in relation to Π classes. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.
|
|
This process is repeated recursively, each time with atoms one bond farther from the stereocenter, until the tie is broken.
|
|
In computability theory, a Friedberg numbering is a numbering (enumeration) of the set of all uniformly recursively enumerable sets that has no repetitions: each recursively enumerable set appears exactly once in the enumeration (Vereščagin and Shen 2003:30). The existence of such numberings was established by Richard M. Friedberg in 1958 (Cutland 1980:78).
|
|
12, 211-218, 1993.Noy, M. and Ribó, A. "Recursively Constructible Families of Graphs." Adv. Appl. Math. 32, 350-363, 2004.
|
|
Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning.
|
|
In the pseudo code, samplesort is called recursively. Frazer and McKellar called samplesort just once and used quicksort in all following iterations.
|
|
That is, given such sets A and B, there is a total computable function f such that A = {x : f(x) ∈ B}. These sets are said to be many-one equivalent (or m-equivalent). Many-one reductions are "stronger" than Turing reductions: if a set A is many-one reducible to a set B, then A is Turing reducible to B, but the converse does not always hold. Although the natural examples of noncomputable sets are all many-one equivalent, it is possible to construct recursively enumerable sets A and B such that A is Turing reducible to B but not many-one reducible to B. It can be shown that every recursively enumerable set is many-one reducible to the halting problem, and thus the halting problem is the most complicated recursively enumerable set with respect to many-one reducibility and with respect to Turing reducibility. Post (1944) asked whether every recursively enumerable set is either computable or Turing equivalent to the halting problem, that is, whether there is no recursively enumerable set with a Turing degree intermediate between those two.
|
|
Agents may activate subagents dynamically and recursively. The development of Joyce formed the foundation of the language SuperPascal, also developed by Hansen around 1993.
|
|
If the element this node was a winner at its parent node, then the element and certificates at the parent must be recursively updated too.
|
|
Once the graph is partitioned into two parts, it can be furtherly recursively bisected on every partition until the necessary number of partitions is matched.
|
|
The objects of study in \alpha recursion are subsets of \alpha. A is said to be \alpha recursively enumerable if it is \Sigma_1 definable over L_\alpha. A is recursive if both A and \alpha \setminus A (its complement in \alpha) are \alpha recursively enumerable. Members of L_\alpha are called \alpha finite and play a similar role to the finite numbers in classical recursion theory.
|
|
The Floyd-Rivest algorithm is a divide and conquer algorithm, sharing many similarities with quickselect. It uses sampling to help partition the list into three sets. It then recursively selects the kth smallest element from the appropriate set. The general steps are: # Select a small random sample S from the list L. # From S, recursively select two elements, u and v, such that u < v.
|
|
262 (1961), pp. 455-475. On the other hand, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up to isomorphism) exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism).
|
|
The Traverser API makes it possible to analyze local data. Based on a number of nodes (local), neighboring nodes can be searched recursively (breadth-depth first).
|
|
A given probability distribution, including a heavy-tailed distribution, can be approximated by a hyperexponential distribution by fitting recursively to different time scales using Prony's method.
|
|
Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ; Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987.
|
|
A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.
|
|
A function may be recursively defined in terms of itself. A familiar example is the Fibonacci number sequence: F(n) = F(n − 1) + F(n − 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case F(0) = 0 and F(1) = 1. A famous recursive function is the Ackermann function, which, unlike the Fibonacci sequence, cannot be expressed without recursion.
|
|
He and Richard Friedberg independently introduced the priority method which gave an affirmative answer to Post's Problem regarding the existence of recursively enumerable Turing degrees between 0 and 0' . This result, now known as the Friedberg-Muchnik Theorem,Robert I. Soare, Recursively Enumberable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer-Verlag, 1999, ; p. 118Nikolai Vereshchagin, Alexander Shen, Computable functions. American Mathematical Society, 2003, ; p.
|
|
The set of reachable configurations is recognizable for lossy channel machines and machines capable of insertions of errors. It is recursively enumerable for machine capable of duplication error.
|
|
Fortunetellers divide a set of 50 yarrow stalks into piles and use modular arithmetic recursively to generate two bits of random information that have a non-uniform distribution.
|
|
This implicitly gives all modular partitions of V. It is in this sense that the modular decomposition tree "subsumes" all other ways of recursively decomposing G into quotients.
|
|
Pre-order, in-order, and post-order traversal visit each node in a tree by recursively visiting each node in the left and right subtrees of the root.
|
|
In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.
|
|
The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but according to the Cantor's theorem, there are uncountably many sets of natural numbers. Although the halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerable and semidecidable). Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.
|
|
Once the binary search tree has been created, its elements can be retrieved in-order by recursively traversing the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed. As with all binary trees, one may conduct a pre-order traversal or a post-order traversal, but neither are likely to be useful for binary search trees. An in-order traversal of a binary search tree will always result in a sorted list of node items (numbers, strings or other comparable items). The code for in-order traversal in Python is given below.
|
|
When securely deleting files recursively, srm 1.2.11 is unable to determine device boundaries on Windows. Therefore, the `-x` option, which restricts srm to one file system, is not supported.
|
|
If not, the computer computes the solution recursively and forwards the solution to the computer whose authority it falls under. This is what causes a lot of communication overhead.
|
|
Composite sentences are recursively built from simpler sentences through coordination, subordination, quantification, and negation. Note that ACE composite sentences overlap with what linguists call compound sentences and complex sentences.
|
|
Though this causes more iterations, it reduces cache misses and can make the algorithm run faster overall. In the case where the number of bins is at least the number of elements, spreadsort degenerates to bucket sort and the sort completes. Otherwise, each bin is sorted recursively. The algorithm uses heuristic tests to determine whether each bin would be more efficiently sorted by spreadsort or some other classical sort algorithm, then recursively sorts the bin.
|
|
The backtracking algorithm traverses this search tree recursively, from the root down, in depth-first order. At each node c, the algorithm checks whether c can be completed to a valid solution. If it cannot, the whole sub-tree rooted at c is skipped (pruned). Otherwise, the algorithm (1) checks whether c itself is a valid solution, and if so reports it to the user; and (2) recursively enumerates all sub-trees of c.
|
|
In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956..
|
|
Many related models have been considered and also the learning of classes of recursively enumerable sets from positive data is a topic studied from Gold's pioneering paper in 1967 onwards.
|
|
A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.
|
|
IETF protocols can be encapsulated recursively, as demonstrated by tunneling protocols such as Generic Routing Encapsulation (GRE). GRE uses the same mechanism that OSI uses for tunneling at the network layer.
|
|
By default, it uses the standard input/output for archive and listing operations, but this can be overwridden with the "tar-style" option that specifies the archive file. Pax differs from cpio by recursively considering the content of a directory; to disable this behavior, POSIX pax has an option to disable it. The command is a mish-mash of and features. Like , processes directory entries recursively, a feature that can be disabled with for cpio-style behavior.
|
|
This is done recursively for the smaller transforms. Illustration of row- and column-major order More generally, Cooley–Tukey algorithms recursively re-express a DFT of a composite size N = N1N2 as:Duhamel, P., and M. Vetterli, "Fast Fourier transforms: a tutorial review and a state of the art," Signal Processing 19, 259–299 (1990) # Perform N1 DFTs of size N2. # Multiply by complex roots of unity (often called the twiddle factors). # Perform N2 DFTs of size N1.
|
|
If an operation is to be performed on the whole structure, each node calls the operation on its children (recursively). This is an implementation of the composite pattern, which is a collection of nodes. The node is an abstract base class, and derivatives can either be leaves (singular), or collections of other nodes (which in turn can contain leaves or collection-nodes). When an operation is performed on the parent, that operation is recursively passed down the hierarchy.
|
|
If it is, swap the two. It is therefore ensured that Q1 < Q2 and that the root node of the merged heap will contain Q1. We then recursively merge Q2 with Q1.
|
|
Joint Conf. on Artificial Intelligence (Q334 .I571 1993), pp. 1022-1027 which uses mutual information to recursively define the best bins, CAIM, CACC, Ameva, and many othersDougherty, J.; Kohavi, R. ; Sahami, M. (1995).
|
|
As intermediate results, Post defined natural types of recursively enumerable sets like the simple, hypersimple and hyperhypersimple sets. Post showed that these sets are strictly between the computable sets and the halting problem with respect to many-one reducibility. Post also showed that some of them are strictly intermediate under other reducibility notions stronger than Turing reducibility. But Post left open the main problem of the existence of recursively enumerable sets of intermediate Turing degree; this problem became known as Post's problem.
|
|
According to the Church–Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is recursively enumerable if and only if there is some algorithm which yields an enumeration of S. This cannot be taken as a formal definition, however, because the Church–Turing thesis is an informal conjecture rather than a formal axiom. The definition of a recursively enumerable set as the domain of a partial function, rather than the range of a total recursive function, is common in contemporary texts. This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory, the definition corresponding to domains has been found to be more natural. Other texts use the definition in terms of enumerations, which is equivalent for recursively enumerable sets.
|
|
An enumerator is a Turing machine that lists, possibly with repetitions, elements of some set S, which it is said to enumerate. A set enumerated by some enumerator is said to be recursively enumerable.
|
|
The iterated extended Kalman filter improves the linearization of the extended Kalman filter by recursively modifying the centre point of the Taylor expansion. This reduces the linearization error at the cost of increased computational requirements.
|
|
This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.
|
|
A move can be associated with the position it leaves the next player in. Doing so allows positions to be defined recursively. For example, consider the following game of Nim played by Alice and Bob.
|
|
In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable.
|
|
Dynamic programming is a systematic technique in which a complex problem is decomposed recursively into smaller, overlapping subproblems for solution. Dynamic programming stores the results of the overlapping sub- problems locally using an optimization technique called memoization.
|
|
Since a heightfield occupies a box volume itself, recursively subdividing this box into eight subboxes (hence the 'oct' in octree) until individual heightfield elements are reached is efficient and natural. A quadtree is simply a 2D octree.
|
|
The halting problem is easy to solve, however, if we allow that the Turing machine that decides it may run forever when given input which is a representation of a Turing machine that does not itself halt. The halting language is therefore recursively enumerable. It is possible to construct languages which are not even recursively enumerable, however. A simple example of such a language is the complement of the halting language; that is the language consisting of all Turing machines paired with input strings where the Turing machines do not halt on their input.
|
|
Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic--including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion ' is interesting in set theory (and in other sufficiently expressive axiomatic systems).
|
|
A child of a vertex v is a vertex of which v is the parent. An ascendant of a vertex v is any vertex which is either the parent of v or is (recursively) the ascendant of the parent of v. A descendant of a vertex v is any vertex which is either the child of v or is (recursively) the descendant of any of the children of v. A sibling to a vertex v is any other vertex on the tree which has the same parent as v.
|
|
In logic, finite model theory, and computability theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences over finite models is not recursively enumerable (though it is co-recursively enumerable). Trakhtenbrot's theorem implies that Gödel's completeness theorem (that is fundamental to first-order logic) does not hold in the finite case. Also it seems counter-intuitive that being valid over all structures is 'easier' than over just the finite ones.
|
|
Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood). Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories.
|
|
An ordinal that is both admissible and a limit of admissibles, or equivalently such that \alpha is the \alpha-th admissible ordinal, is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals. For example, we can define recursively Mahlo ordinals: these are the \alpha such that every \alpha-recursive closed unbounded subset of \alpha contains an admissible ordinal (a recursive analog of the definition of a Mahlo cardinal). But note that we are still talking about possibly countable ordinals here.
|
|
The root node may have zero or more subtrees. The k-th subtree is recursively built of all elements b such that d(a,b) = k. BK-trees can be used for approximate string matching in a dictionary.
|
|
Another is to call the ziggurat algorithm recursively and add x1 to the result. For a normal distribution, Marsaglia suggests a compact algorithm: # Let x = −ln(U1)/x1. # Let y = −ln(U2). # If 2y > x2, return x + x1.
|
|
Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines. Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not.
|
|
The divide and conquer technique decomposes complex problems recursively into smaller sub-problems. Each sub-problem is then solved and these partial solutions are recombined to determine the overall solution. This technique is often used for searching and sorting.
|
|
In this section, the same code is used with the addition of #include guards. The C preprocessor preprocesses the header files, including and further preprocessing them recursively. This will result in a correct source file, as we will see.
|
|
We have previously shown, however, that the halting problem is undecidable. We have a contradiction, and we have thus shown that our assumption that M exists is incorrect. The complement of the halting language is therefore not recursively enumerable.
|
|
A set S of natural numbers is called recursively enumerable if there is a partial recursive function whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.
|
|
Applying the standard technique of proof by cases to recursively defined sets or functions, as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical logic and computer science.
|
|
The semantics of these is that they provide details of how to download a stringified IOR (or, recursively, download another URL that will eventually provide a stringified IOR). Some ORBs do deliver additional formats which are proprietary for that ORB.
|
|
Series and parallel composition operations for series-parallel graphs. In graph theory, series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.
|
|
A hierarchical watershed transformation converts the result into a graph display (i.e. the neighbor relationships of the segmented regions are determined) and applies further watershed transformations recursively. See Laurent Najman, Michel Schmitt. Geodesic Saliency of Watershed Contours and Hierarchical Segmentation.
|
|
The concept is primarily invoked in the context of discussions of recursively self-improving artificial agents that rapidly explode in intelligence, on the grounds that this hypothetical technology would have a large, rapid, and difficult-to-control impact on human society.
|
|
The decision problem of whether a given string s can be generated by a given unrestricted grammar is equivalent to the problem of whether it can be accepted by the Turing machine equivalent to the grammar. The latter problem is called the Halting problem and is undecidable. Recursively enumerable languages are closed under Kleene star, concatenation, union, and intersection, but not under set difference; see Recursively enumerable language#Closure properties. The equivalence of unrestricted grammars to Turing machines implies the existence of a universal unrestricted grammar, a grammar capable of accepting any other unrestricted grammar's language given a description of the language.
|
|
Suppose\left\vert P\right\vert = \left\vert P_0 \right\vert + \left\vert P_1 \right\vert , we can recursively compute the number of sets in a ZDD, enabling us to get the 34th set out a 54-member family. Random access is fast, and any operation possible for an array of sets can be done with efficiency on a ZDD. According to Minato, the above operations for ZDDs can be executed recursively like original BDDs. To describe the algorithms simply, we define the procedure `Getnode(top, P0, P1)` that returns a node for a variable top and two subgraphs P0 and P1.
|
|
Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent.
|
|
In addition to being constructed from primitives by functionals, a function may be defined recursively by an equation, the simplest kind being: f ≡ Ef where Ef is an expression built from primitives, other defined functions, and the function symbol f itself, using functionals.
|
|
Consequently, each bucket's size is also a power of two, and the procedure can be applied recursively. This approach can accelerate the scatter phase, since we only need to examine a prefix of the bit representation of each element to determine its bucket.
|
|
When Post defined the notion of a simple set as an r.e. set with an infinite complement not containing any infinite r.e. set, he started to study the structure of the recursively enumerable sets under inclusion. This lattice became a well-studied structure.
|
|
Otherwise, the remaining elements should be compared to u first and only to v if they are greater than u. # Based on the value of k, apply the algorithm recursively to the appropriate set to select the kth smallest element in L.
|
|
To traverse any tree with depth-first search, perform the following operations recursively at each node: # Perform pre-order operation. # For each i from 1 to the number of children do: ## Visit i-th, if present. ## Perform in-order operation. # Perform post-order operation.
|
|
They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
|
|
It will be complete whenever the set is recursively enumerable. #. proved that all creative sets are RE- complete. #The uniform word problem for groups or semigroups. (Indeed, the word problem for some individual groups is RE-complete.) #Deciding membership in a general unrestricted formal grammar.
|
|
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.
|
|
If that distance is less than then use the algorithm recursively to search the subtree of the node that contains the points closer to the vantage point than the threshold ; otherwise recurse to the subtree of the node that contains the points that are farther than the vantage point than the threshold . If the recursive use of the algorithm finds a neighboring point with distance to that is less than then it cannot help to search the other subtree of this node; the discovered node is returned. Otherwise, the other subtree also needs to be searched recursively. A similar approach works for finding the nearest neighbors of a point .
|
|
Traversals are the key to the power of applying operations to scene graphs. A traversal generally consists of starting at some arbitrary node (often the root of the scene graph), applying the operation(s) (often the updating and rendering operations are applied one after the other), and recursively moving down the scene graph (tree) to the child nodes, until a leaf node is reached. At this point, many scene graph engines then traverse back up the tree, applying a similar operation. For example, consider a render operation that takes transformations into account: while recursively traversing down the scene graph hierarchy, a pre-render operation is called.
|
|
Usually, on most filesystems, deleting a file requires write permission on the parent directory (and execute permission, in order to enter the directory in the first place). (Note that, confusingly for beginners, permissions on the file itself are irrelevant. However, GNU `rm` asks for confirmation if a write-protected file is to be deleted, unless the -f option is used.) To delete a directory (with `rm -r`), one must delete all of its contents recursively. This requires that one must have read and write and execute permission to that directory (if it's not empty) and all non-empty subdirectories recursively (if there are any).
|
|
RST relations are applied recursively in a text, until all units in that text are constituents in an RST relation. The result of such analyses is that RST structure are typically represented as trees, with one top level relation that encompasses other relations at lower levels.
|
|
Simplifying a piecewise linear curve with the Douglas–Peucker algorithm. The starting curve is an ordered set of points or lines and the distance dimension ε > 0\. The algorithm recursively divides the line. Initially it is given all the points between the first and last point.
|
|
Who Invented Backpropagation? when Linnainmaa introduced the reverse mode of automatic differentiation (AD), in order to efficiently compute the derivative of a differentiable composite function that can be represented as a graph, by recursively applying the chain rule to the building blocks of the function.Griewank, Andreas (2012).
|
|
Bayer is also known for his recursively defined matrix used in ordered dithering. Alternatives to the Bayer filter include both various modifications of colors and arrangement and completely different technologies, such as color co-site sampling, the Foveon X3 sensor, the dichroic mirrors or a transparent diffractive-filter array.
|
|
This decomposition is performed recursively when N is a power of two. The base cases of the recursion are N=1, where the DFT is just a copy X_0 = x_0, and N=2, where the DFT is an addition X_0 = x_0 + x_1 and a subtraction X_1 = x_0 - x_1.
|
|
Unfortunately, this technique does not scale well when multiple reflective objects are present. A unique dynamic environment map is usually required for each reflective object. Also, further complications are added if reflective objects can reflect each other - dynamic cube maps can be recursively generated approximating the effects normally generated using raytracing.
|
|
In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with :a_1 = 1. Then for n>1, a_n is the smallest integer such that every pairwise sum :a_i + a_j is distinct, for all i and j less than or equal to n.
|
|
The tromino can be recursively dissected into unit trominoes, and a dissection of the quarter-board with one square removed follows by the induction hypothesis. In contrast, when a chessboard of this size has one square removed, it is not always possible to cover the remaining squares by I-trominoes..
|
|
Mathscinet searches for the titles like "computably enumerable" and "c.e." show that many papers have been published with this terminology as well as with the other one. These researchers also use terminology such as partial computable function and computably enumerable (c.e.) set instead of partial recursive function and recursively enumerable (r.
|
|
Every first-order formula is logically equivalent (in classical logic) to some formula in prenex normal form.Hinman, P. (2005), p. 111 There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connectives appear in the formula.
|
|
Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as , require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined.
|
|
A polygon that is already convex has no pockets. One can form a hierarchical description of any given polygon by constructing its hull and its pockets in this way and then recursively forming a hierarchy of the same type for each pocket. This structure, called a convex differences tree, can be constructed efficiently.
|
|
In 1970, William A. Woods introduced the augmented transition network (ATN) to represent natural language input.Woods, William A (1970). "Transition Network Grammars for Natural Language Analysis". Communications of the ACM 13 (10): 591–606 Instead of phrase structure rules ATNs used an equivalent set of finite state automata that were called recursively.
|
|
Type-0 grammars include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine. These languages are also known as the recursively enumerable or Turing-recognizable languages. Note that this is different from the recursive languages, which can be decided by an always- halting Turing machine.
|
|
Recursively partitioning is method that creates a decision tree using qualitative data. Understanding the way rules break classes up with a low error of misclassification while repeating each step until no sensible splits can be found. However, recursive partitioning can have poor prediction ability potentially creating fine models at the same rate.
|
|
Doo–Sabin surfaces are defined recursively. Each refinement iteration replaces the current mesh with a smoother, more refined mesh, following the procedure described in. After many iterations, the surface will gradually converge onto a smooth limit surface. The figure below show the effect of two refinement iterations on a T-shaped quadrilateral mesh.
|
|
The Earley parser executes in cubic time in the general case {O}(n^3), where n is the length of the parsed string, quadratic time for unambiguous grammars {O}(n^2), p.145 and linear time for all deterministic context-free grammars. It performs particularly well when the rules are written left-recursively.
|
|
The Estonian diminutive suffix can be used recursively - it can be attached to a word more than once. Forms such as "pisikesekesekene", having three diminutive suffixes, are grammatically legitimate. As is demonstrated by the example, in recursive usage all but the last diminutive "-ne" suffix become "-se" as in forms inflected by case.
|
|
NegaScout calls the zero-window searches recursively. MTD(f) calls the zero-window searches from the root of the tree. Implementations of the MTD(f) algorithm have been shown to be more efficient (search fewer nodes) in practice than other search algorithms (e.g. NegaScout) in games such as chess , checkers, and Othello.
|
|
Recursive acronyms typically form backwardly: either an existing ordinary acronym is given a new explanation of what the letters stand for, or a name is turned into an acronym by giving the letters an explanation of what they stand for, in each case with the first letter standing recursively for the whole acronym.
|
|
A recursively enumerable set can be characterized as one for which there exists an algorithm that will ultimately halt when a member of the set is provided as input, but may continue indefinitely when the input is a non-member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intuitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable. This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation.
|
|
The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the converse is true: > Every recursively enumerable set is Diophantine. This result is variously known as Matiyasevich's theorem (because he provided the crucial step that completed the proof) and the MRDP theorem (for Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam). Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence. In fact, more can be said: there is a polynomial :p(a,x_1,\ldots,x_n) with integer coefficients such that the set of values of a for which the equation :p(a,x_1,\ldots,x_n)=0 has solutions in natural numbers is not computable.
|
|
These type of sets can be classified using the arithmetical hierarchy. For example, the index set FIN of class of all finite sets is on the level Σ2, the index set REC of the class of all recursive sets is on the level Σ3, the index set COFIN of all cofinite sets is also on the level Σ3 and the index set COMP of the class of all Turing-complete sets Σ4. These hierarchy levels are defined inductively, Σn+1 contains just all sets which are recursively enumerable relative to Σn; Σ1 contains the recursively enumerable sets. The index sets given here are even complete for their levels, that is, all the sets in these levels can be many- one reduced to the given index sets.
|
|
The second half recursively introduces John Gerard (a thinly disguised Jean Giraud) and family into the world of Arzach. A brief essay on the publishing history of Arzach concludes the book. Arzach was one of Panzer Dragoon's major artistic influences. Jean Giraud even contributed in the creative process of Team Andromeda's game with original artwork.
|
|
Third iteration Jerusalem cube A Jerusalem cube is a fractal object described by Eric Baird in 2011. It is created by recursively drilling Greek cross-shaped holes into a cube., published in Magazine Tangente 150, "l'art fractal" (2013), p. 45. The name comes from a face of the cube resembling a Jerusalem cross pattern.
|
|
Thus, the name may confuse some people into thinking it only provides the MD5 algorithm when the package supports many more. md5deep can be invoked in several different ways. Typically users operate it recursively, where md5deep walks through one directory at a time giving digests of each file found, and recursing into any subdirectories within.
|
|
G. Japaridze, "Decidable and enumerable predicate logics of provability". Studia Logica 49 (1990), pages 7-21. In the same paper he showed that, on the condition of the 1-completeness of the underlying arithmetical theory, predicate provability logic with non-iterated modalities is recursively enumerable. InG. Japaridze, "Predicate provability logic with non-modalized quantifiers".
|
|
The scattering amplitude is evaluated recursively through a set of Dyson-Schwinger equations. The computational cost of this algorithm grows asymptotically as 3n, where n is the number of particles involved in the process, compared to n! in the traditional Feynman graphs approach. Unitary gauge is used and mass effects are available as well.
|
|
Summarizing, GenVoca values are nested tuples of program artifacts, and features are nested delta tuples, where + recursively composes them by vector addition. This is the essence of AHEAD. The ideas presented above concretely expose two FOSD principles. The Principle of Uniformity states that all program artifacts are treated and modified in the same way.
|
|
A decision problem A is decidable or effectively solvable if A is a recursive set. A problem is partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set. Problems that are not decidable are undecidable. For those it is not possible to create an algorithm, efficient or otherwise, that solves them.
|
|
Debugging faulty actors include recursively performing coarse-grain replay on actors in the data-flow,Wenchao Zhou, Qiong Fei, Arjun Narayan, Andreas Haeberlen, Boon Thau Loo, and Micah Sherr. Secure network provenance. In Proceedings of 23rd ACM Symposium on Operating System Principles (SOSP), December 2011. which can be expensive in resources for long dataflows.
|
|
When curvature is specified, the triangle is decomposed recursively into four sub- triangles. The recursion must be executed five levels deep, so that the original curved triangle is ultimately replaced by 1024 flat triangles. These 1024 triangles are generated "on the fly" and stored temporarily only while layers intersecting that triangle are being processed for manufacturing.
|
|
Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.See and .
|
|
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Graham Higman from the 1960s. Graham Higman, Subgroups of finitely presented groups. Proceedings of the Royal Society. Series A. Mathematical and Physical Sciences. vol.
|
|
Sintzoff, M. "Existence of van Wijngaarden syntax for every recursively enumerable set", Annales de la Société Scientifique de Bruxelles 2 (1967), 115-118. Two-level grammar can also refer to a formal grammar for a two-level formal language, which is a formal language specified at two levels, for example, the levels of words and sentences.
|
|
If n1, n2, ..., nr is a strictly decreasing sequence of natural numbers, then an S-dévissage in dimensions n1, n2, ..., nr is defined recursively as: # An S-dévissage in dimension n1. Denote the cokernel of α by P1. # An S-dévissage in dimensions n2, ..., nr of P1. The dévissage is said to lie between dimensions n1 and nr.
|
|
But if one of the two shares is structured recursively, the efficiency of visual cryptography can be increased to 100%. Some antecedents of visual cryptography are in patents from the 1960s.Cook, Richard C. (1960) Cryptographic process and enciphered product, United States patent 4,682,954.Carlson, Carl O. (1961) Information encoding and decoding method, United States patent 3,279,095.
|
|
Like ports they are associated with a protocol. But other than ports they don't have to (and even cannot) be bound explicitly. Rather, an actor is bound to a concrete service by a layer connection and this binding of a service is propagated recursively to all sub actors of this actor. This concept is very similar to dependency injection.
|
|
Such factorization steps can be performed recursively. After steps, we obtain the factorization :, where has only two spikes. The reduced system will then be solved via :. The block LU factorization technique in the two- partition case can be used to handle the solving steps involving , ..., and for they essentially solve multiple independent systems of generalized two- partition forms.
|
|
Prescription cascade is the process whereby the side effects of drugs are misdiagnosed as symptoms of another problem, resulting in further prescriptions and further side effects and unanticipated drug interactions, which itself may lead recursively to further misdiagnoses and further symptoms. This is a pharmacological example of a feedback loop. Such cascades can be reversed through deprescribing.
|
|
The following statements hold. :# For any computable enumeration operator Φ there is a recursively enumerable set F such that Φ(F) = F and F is the smallest set with this property. :# For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property.
|
|
In computer science, average memory access time (AMAT) is a common metric to analyze memory system performance. AMAT uses hit time, miss penalty, and miss rate to measure memory performance. It accounts for the fact that hits and misses affect memory system performance differently. In addition, AMAT can be extended recursively to multiple layers of the memory hierarchy.
|
|
These procedures start at the last node in the tree (the lowest point). Following recursively upwards, they determine the relevance of each individual node. If the relevance for the classification is not given, the node is dropped or replaced by a leaf. The advantage is that no relevant sub-trees can be lost with this method.
|
|
Most English compound nouns are noun phrases (i.e. nominal phrases) that include a noun modified by adjectives or noun adjuncts. Due to the English tendency towards conversion, the two classes are not always easily distinguished. Most English compound nouns that consist of more than two words can be constructed recursively by combining two words at a time.
|
|
Searching is similar to searching a binary search tree. Starting at the root, the tree is recursively traversed from top to bottom. At each level, the search reduces its field of view to the child pointer (subtree) whose range includes the search value. A subtree's range is defined by the values, or keys, contained in its parent node.
|
|
Searching in a binary search tree for a specific key can be programmed recursively or iteratively. We begin by examining the root node. If the tree is null, the key we are searching for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and we return the node.
|
|
In automata theory, the class of unrestricted grammars (also called semi-Thue, type-0 or phrase structure grammars) is the most general class of grammars in the Chomsky hierarchy. No restrictions are made on the productions of an unrestricted grammar, other than each of their left-hand sides being non- empty. This grammar class can generate arbitrary recursively enumerable languages.
|
|
Telomerase restores short bits of DNA known as telomeres, which are otherwise shortened when a cell divides via mitosis. In normal circumstances, where telomerase is absent, if a cell divides recursively, at some point the progeny reach their Hayflick limit, which is believed to be between 50–70 cell divisions. At the limit the cells become senescent and cell division stops.Siegel, L (2013).
|
|
If every connected component of a graph has a vertex valued 3, then we can make the Jacobi diagram into a Chord diagram using the STU relation recursively. If we restrict ourselves only to chord diagrams, then the above four relations are reduced to the following two relations: :(The four term relation) 90px − 90px + 90px − 90px = 0. :(The FI relation) 90px = 0.
|
|
These hyperlinks are added to the frontier and will visit those new web pages based on the policies of the crawler frontier. This process continues recursively until all URLs in the crawl frontier are visited. The policies used to determine what pages to visit are commonly based on a score. This score is typically computed from a number of different attributes.
|
|
Given a binary tree, with this node structure: class node { node left node right } One may implement a tree size procedure recursively: function tree_size(node) { return 1 + tree_size(node.left) + tree_size(node.right) } Since the child nodes may not exist, one must modify the procedure by adding non-existence or null checks: function tree_size(node) { set sum = 1 if node.left exists { sum = sum + tree_size(node.
|
|
If it is found non-uniform (not homogeneous), then it is split into four child squares (the splitting process), and so on. If, in contrast, four child squares are homogeneous, they are merged as several connected components (the merging process). The node in the tree is a segmented node. This process continues recursively until no further splits or merges are possible.
|
|
In mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variables and function symbols.
|
|
Samplesort is a generalization of quicksort. Where quicksort partitions its input into two parts at each step, based on a single value called the pivot, samplesort instead takes a larger sample from its input and divides its data into buckets accordingly. Like quicksort, it then recursively sorts the buckets. To devise a samplesort implementation, one needs to decide on the number of buckets .
|
|
This algorithm is a combination of radix sort and quicksort. Pick an element from the array (the pivot) and consider the first character (key) of the string (multikey). Partition the remaining elements into three sets: those whose corresponding character is less than, equal to, and greater than the pivot's character. Recursively sort the "less than" and "greater than" partitions on the same character.
|
|
As originally developed by Strachey and Scott, denotational semantics provided the meaning of a computer program as a function that mapped input into output.Dana Scott and Christopher Strachey. Toward a mathematical semantics for computer languages Oxford Programming Research Group Technical Monograph. PRG-6. 1971. To give meanings to recursively defined programs, Scott proposed working with continuous functions between domains, specifically complete partial orders.
|
|
27–29 suggests that interactive computation can help mathematics form a more appropriate framework (empirical) than can be founded with rationalism alone. Related to this argument is that the function (even recursively related ad infinitum) is too simple a construct to handle the reality of entities that resolve (via computation or some type of analog) n-dimensional (general sense of the word) systems.
|
|
Each pass is based on a single digit (e.g. 4-bits per digit in the case of 16-radix), starting from the most significant digit. Each bin is then processed recursively using the next digit, until all digits have been used for sorting. Neither in-place binary-radix sort nor n-bit-radix sort, discussed in paragraphs above, are stable algorithms.
|
|
6 steps of a Sierpinski carpet. The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is one generalization of the Cantor set to two dimensions; another is the Cantor dust. The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing recursively can be extended to other shapes.
|
|
Once a process finishes, DTS dynamically reassigns the processors to other processes as to keep the efficiency to a maximum through good load-balancing, especially in irregular trees. Once a process finishes searching, it recursively sends and merges a resulting signal to its parent-process, until all the different sub-answers have been merged and the entire problem has been solved.
|
|
Concretely it can be defined as follows.John L. Hennessy and David A. Patterson, Computer Architecture a Quantitative Approach Fifth Edition, 2012, pp.B9-B19 AMAT = H + MR \cdot AMP It can also be defined recursively as, AMAT = H_1 + MR_1 \cdot AMP_1 where AMP_1 = H_2 + MR_2 \cdot AMP_2 In this manner, this recursive definition can be extended throughout all layers of the memory hierarchy.
|
|
The choice rule in Line 1 "generates" all subsets of the set of edges. The three constraints "weed out" the subsets that are not Hamiltonian cycles. The last of them uses the auxiliary predicate r(x) ("x is reachable from 0") to prohibit the vertices that do not satisfy this condition. This predicate is defined recursively in Lines 4 and 5.
|
|
An unordered tree is well-founded if the strict partial order is a well-founded relation. In particular, every finite tree is well-founded. Assuming the axiom of dependent choice a tree is well-founded if and only if it has no infinite branch. Well-founded trees can be defined recursively - by forming trees from a disjoint union of smaller trees.
|
|
This recursively exploits the nested dimensions group structure of , as follows. Generate a uniform angle and construct a rotation matrix. To step from to , generate a vector uniformly distributed on the -sphere , embed the matrix in the next larger size with last column , and rotate the larger matrix so the last column becomes . As usual, we have special alternatives for the case.
|
|
The tool created output streams based on interpreting data provided via multiple input sources. It was originally created by Univac for the creation of Operating System (OS) updates. It was later adopted by the general user community for the creation of complex batch and real-time computer processes. The sources could recursively reference additional sources, providing wide flexibility in input parsing.
|
|
It includes ranks and binomial nomenclature. The nomenclature of botanical organisms is codified in the International Code of Nomenclature for algae, fungi, and plants (ICN) and administered by the International Botanical Congress. Kingdom Plantae belongs to Domain Eukarya and is broken down recursively until each species is separately classified. The order is: Kingdom; Phylum (or Division); Class; Order; Family; Genus (plural genera); Species.
|
|
The result was the truth predicate is well arithmetically, it is even \Delta^0_2. So far down in the arithmetic hierarchy, and that goes for any recursively axiomatized (countable, consistent) theories. Even if you are true in all the natural numbers \Pi^0_1 formulas to the axioms. This classic proof is a very early, original application of the arithmetic hierarchy theory to a general-logical problem.
|
|
Unfoldment of coherences of recursively packed concepts by the repulsive "carapace" forces of like concepts and coalescence by the attraction of unlike concepts is a further feature. Pask's approach involves a psychodynamic and panpsychic element. He achieved this by placing co-ordinates on a participant rather than claiming non-participant observer status. Stafford Beer similarly regarded his Viable System Model as a model of the observer.
|
|
Many other of Benglis's earlier solo films are highly technically manipulated, edited, and re-taped, thus blending present and past video sequences and selves to enhance the feeling of artifice."Lynda Benglis: Biography" , Electronic Arts Intermix, Retrieved 15 April 2014. For instance, in Now (1973) the artist's face is recursively featured, but this time the self-evidencing frame of the television is cropped out.
|
|
The operation that searches for the successor of an element x in a vEB tree proceeds as follows: If then the search is complete, and the answer is . If then the next element does not exist, return M. Otherwise, let . If then the value being searched for is contained in so the search proceeds recursively in . Otherwise, we search for the value i in .
|
|
Thus if the theory is ω-consistent, is not provable. We have sketched a proof showing that: For any formal, recursively enumerable (i.e. effectively generated) theory of Peano Arithmetic, : if it is consistent, then there exists an unprovable formula (in the language of that theory). : if it is ω-consistent, then there exists a formula such that both it and its negation are unprovable.
|
|
The small list of initial prime numbers constitute complete parameters for the algorithm to generate the remainder of the list. These generators are referred to as wheels. While each wheel may generate an infinite list of numbers, past a certain point the numbers cease to be mostly prime. The method may further be applied recursively as a prime number wheel sieve to generate more accurate wheels.
|
|
For example, the function returns its arguments as a list, so the expression (list 1 2 (quote foo)) evaluates to the list . The "quote" before the in the preceding example is a "special operator" which returns its argument without evaluating it. Any unquoted expressions are recursively evaluated before the enclosing expression is evaluated. For example, (list 1 2 (list 3 4)) evaluates to the list .
|
|
Median cut is an algorithm to sort data of an arbitrary number of dimensions into series of sets by recursively cutting each set of data at the median point along the longest dimension. Median cut is typically used for color quantization. For example, to reduce a 64k-colour image to 256 colours, median cut is used to find 256 colours that match the original data well.
|
|
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let be a nonempty set of real numbers, and suppose that has an upper bound . Since is nonempty, there exists a real number that is not an upper bound for . Define sequences and recursively as follows: # Check whether is an upper bound for .
|
|
A hyperlink to a page counts as a vote of support. The PageRank of a page is defined recursively and depends on the number and PageRank metric of all pages that link to it ("incoming links"). A page that is linked to by many pages with high PageRank receives a high rank itself. Numerous academic papers concerning PageRank have been published since Page and Brin's original paper.
|
|
We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property).
|
|
In a general constraint satisfaction problem, every variable can take a value in a domain. A backtracking algorithm therefore iteratively chooses a variable and tests each of its possible values; for each value the algorithm is recursively run. Look ahead is used to check the effects of choosing a given variable to evaluate or to decide the order of values to give to it.
|
|
The visualization strategy is to recursively drop out the tail parts until the head parts are clear or visible enough.Jiang, Bin (2015). "Head/tail breaks for visualization of city structure and dynamics", Cities, 43, 69 - 77. In addition, it helps delineate cities or natural cities to be more precise from various geographic information such as street networks, social media geolocation data, and nighttime images.
|
|
One can recursively define an addition operator on the natural numbers by setting and for all , . Here, should be read as "successor". This turns the natural numbers into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property, and can be embedded in a group (in the group theory sense of the word).
|
|
If a column is not found the program returns to the last good state and then tries a different column. As an alternative to backtracking, solutions can be counted by recursively enumerating valid partial solutions, one row at a time. Rather than constructing entire board positions, blocked diagonals and columns are tracked with bitwise operations. This does not allow the recovery of individual solutions.
|
|
Nimber multiplication (nim-multiplication) is defined recursively by :. Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal is itself.
|
|
Interpolation Tag Sort is a variant of Interpolation Sort. Applying the bucket sorting and dividing method, the array data is distributed into a limited number of buckets by mathematical interpolation formula, and the bucket then recursively the original processing program until the sorting is completed. Interpolation tag sort is a recursive sorting method for interpolation sorting. To avoid stacking overflow caused by recursion, the memory crashes.
|
|
It depicts a delusional adolescent boy who is treated with projective psychotherapy. In this case the works of fiction are the previously published novels in the "World of Tiers" series. Characters and locations are recursively introduced in the mind of the protagonist. He travels into the World of Tiers although it is never certain if he is delusional or has found a gateway to an alternative universe.
|
|
WordPerfect for DOS stood out for its macros, in which sequences of keystrokes, including function codes, were recorded as the user typed them. These macros could then be assigned to any key desired. This enabled any sequence of keystrokes to be recorded, saved, and recalled. Macros could examine system data, make decisions, be chained together, and operate recursively until a defined "stop" condition occurred.
|
|
The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of 2^n numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to prove the next scale, which leads to 2^n-1 differences and a final sum.
|
|
In 2000, he published, in the style of Dr. Seuss, a proof of Turing's theorem that the Halting Problem is recursively unsolvable.Pullum, Geoffrey K. (2000) "Scooping the loop snooper: An elementary proof of the undecidability of the halting problem". Mathematics Magazine 73.4 (October 2000), 319–320. A corrected version appears on the author's website as "Scooping the loop snooper: A proof that the Halting Problem is undecidable".
|
|
A Domain Name System server translates a human-readable domain name (such as `example.com`) into a numerical IP address that is used to route communications between nodes. Normally if the server does not know a requested translation it will ask another server, and the process continues recursively. To increase performance, a server will typically remember (cache) these translations for a certain amount of time.
|
|
There are uncountably many of these sets and also some recursively enumerable but noncomputable sets of this type. Later, Degtev established a hierarchy of recursively enumerable sets that are (1, n + 1)-recursive but not (1, n)-recursive. After a long phase of research by Russian scientists, this subject became repopularized in the west by Beigel's thesis on bounded queries, which linked frequency computation to the above-mentioned bounded reducibilities and other related notions. One of the major results was Kummer's Cardinality Theory which states that a set A is computable if and only if there is an n such that some algorithm enumerates for each tuple of n different numbers up to n many possible choices of the cardinality of this set of n numbers intersected with A; these choices must contain the true cardinality but leave out at least one false one.
|
|
Dividing f by p gives the other factor q(x) = x^3 - x + 2, so that f = pq. Now one can test recursively to find factors of p and q. It turns out they both are irreducible over the integers, so that the irreducible factorization of f is Van der Waerden, Sections 5.4 and 5.6 :f(x) = p(x)q(x) = (x^2 + x + 1)(x^3 - x + 2).
|
|
Objects can be composed recursively, and their type is then called recursive type. Examples includes various kinds of trees, DAGs, and graphs. Each node in a tree may be a branch or leaf; in other words, each node is a tree at the same time when it belongs to another tree. In UML, recursive composition is depicted with an association, aggregation or composition of a class with itself.
|
|
Konqueror supports tabbed document interface and Split views, wherein a window can contain multiple documents in tabs. Multiple document interfaces are not supported, however it is possible to recursively divide a window to view multiple documents simultaneously, or simply open another window. Konqueror's user interface is somewhat reminiscent of Microsoft's Internet Explorer, though it is more customizable. It works extensively with "panels", which can be rearranged or added.
|
|
Fractal construction of an Osgood curve by recursively removing wedges from triangles. As the wedges narrow, the fraction of area removed decreases exponentially, so the area remaining in the final curve is nonzero. In mathematics, an Osgood curve is a non-self-intersecting curve (either a Jordan curve or a Jordan arc) of positive area. More formally, these are curves in the Euclidean plane with positive two-dimensional Lebesgue measure.
|
|
A sphere world is a space whose boundary is a sphere of the same dimension as the space. A star world is any world whose boundary can be mapped onto the boundary of a sphere world. Since a forest of stars is the union of a number of star worlds, the forest can be recursively mapped onto a single sphere world, and then navigation techniques for sphere worlds can be used.
|
|
Stanley Tennenbaum (April 11, 1927 – May 4, 2005) was an American mathematician who contributed to the field of logic. In 1959, he published Tennenbaum's theorem, which states that no countable nonstandard model of Peano arithmetic (PA) can be recursive, i.e. the operations + and × of a nonstandard model of PA are not recursively definable in the + and × operations of the standard model. He was a Professor at Yeshiva University in the 1960s.
|
|
Systemd mounts variables used by Unified Extensible Firmware Interface on Linux system's sysfs as writable by the root user of a system. As a result, it is possible for the root user of a system to completely brick a system with a non-conforming UEFI implementation (specifically some MSi laptops) by using the `rm` command to delete the `/sys/firmware/efi/efivars/` directory, or recursively delete the root directory.
|
|
It also involves considerable autoboxing and unboxing. What may not be obvious is that, at the end of the loop, the program has constructed a linked list of 11 objects and that all of the actual additions involved in computing the result are done in response to the call to `a.eval()` on the final line of code. This call recursively traverses the list to perform the necessary additions.
|
|
The `diff` command is invoked from the command line, passing it the names of two files: `diff original new`. The output of the command represents the changes required to transform the original file into the new file. If original and new are directories, then will be run on each file that exists in both directories. An option, `-r`, will recursively descend any matching subdirectories to compare files between directories.
|
|
In a product line of parsers, for example, a base parser f is defined by its grammar gf, Java source sf, and documentation df. Parser f is modeled by the tuple f=[gf, sf, df]. Each program representation may have subrepresentations, and they too may have subrepresentations, recursively. In general, a GenVoca value is a tuple of nested tuples that define a hierarchy of representations for a particular program.
|
|
HEVC specifies four transform units (TUs) sizes of 4x4, 8x8, 16x16, and 32x32 to code the prediction residual. A CTB may be recursively partitioned into 4 or more TUs. TUs use integer basis functions based on the discrete cosine transform (DCT). In addition, 4x4 luma transform blocks that belong to an intra coded region are transformed using an integer transform that is derived from discrete sine transform (DST).
|
|
The unique feature of Magnus was that it provided facilities for doing calculations in and about infinite groups. Almost all symbolic algebra systems are oriented toward finite computations that are guaranteed to produce answers, given enough time and resources. By contrast, Magnus was concerned with experiments and computations on infinite groups which in some cases are known to terminate, while in others are known to be generally recursively unsolvable.
|
|
A recursive neural network is created by applying the same set of weights recursively over a differentiable graph-like structure by traversing the structure in topological order. Such networks are typically also trained by the reverse mode of automatic differentiation. They can process distributed representations of structure, such as logical terms. A special case of recursive neural networks is the RNN whose structure corresponds to a linear chain.
|
|
Let S be a set that can be recursively enumerated by a Turing machine. Then there is a Turing machine T that for every n in S, T halts when given n as an input. This can be formalized by the first-order arithmetical formula presented above. The members of S are the numbers n satisfying the following formula: \exists n_1:\varphi(n,n_1) This formula is in \Sigma^0_1.
|
|
The most common approach to finding a MDS is divide-and-conquer. A typical algorithm in this approach looks like the following: # Divide the given set of shapes into two or more subsets, such that the shapes in each subset cannot overlap the shapes in other subsets because of geometric considerations. # Recursively find the MDS in each subset separately. # Return the union of the MDSs from all subsets.
|
|
Recently it is also being used in the domain of Bioinformatics. Forward algorithm can also be applied to perform Weather speculations. We can have a HMM describing the weather and its relation to the state of observations for few consecutive days (some examples could be dry, damp, soggy, sunny, cloudy, rainy etc.). We can consider calculating the probability of observing any sequence of observations recursively given the HMM.
|
|
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively approximable are also used. One can think of limit computable functions as those admitting an eventually correct computable guessing procedure at their true value. A set is limit computable just when its characteristic function is limit computable.
|
|
Recursive sets can be defined in this structure by the basic result that a set is recursive if and only if the set and its complement are both recursively enumerable. Infinite r.e. sets have always infinite recursive subsets; but on the other hand, simple sets exist but do not have a coinfinite recursive superset. Post (1944) introduced already hypersimple and hyperhypersimple sets; later maximal sets were constructed which are r.e.
|
|
One special type of operand is the parenthesis group. An expression enclosed in parentheses is typically recursively evaluated to be treated as a single operand on the next evaluation level. Each operator is given a position, precedence, and an associativity. The operator precedence is a number (from high to low or vice versa) that defines which operator that takes an operand surrounded by two operators of different precedence (or priority).
|
|
In this case, there is no obvious candidate for a new axiom that resolves the issue. The theory of first order Peano arithmetic seems to be consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete.
|
|
Let σ be a relational vocabulary with one at least binary relation symbol. :The set of σ-sentences valid in all finite structures is not recursively enumerable. Remarks # This implies that Gödel's completeness theorem fails in the finite since completeness implies recursive enumerability. # It follows that there is no recursive function f such that: if φ has a finite model, then it has a model of size at most f(φ).
|
|
To insert an object, the tree is traversed recursively from the root node. At each step, all rectangles in the current directory node are examined, and a candidate is chosen using a heuristic such as choosing the rectangle which requires least enlargement. The search then descends into this page, until reaching a leaf node. If the leaf node is full, it must be split before the insertion is made.
|
|
Louise Hay (June 14, 1935 – October 28, 1989) was a French-born American mathematician. Her work focused on recursively enumerable sets and computational complexity theory, which was influential with both Soviet and US mathematicians in the 1970s. When she was appointed head of the mathematics department at the University of Illinois at Chicago, she was the only woman to head a math department at a major research university in her era.
|
|
Furthermore, (unlike in the literature example), the third-level nested quote must be escaped in order not to conflict with either the first- or second-level quote delimiters. This is true regardless of alternating-symbol encapsulation. Every level after the third level must be recursively escaped for all the levels of quotes in which it is contained. This includes the escape character itself, the backslash (“\”), which is escaped by itself (“\\\”).
|
|
Otherwise, consists of a complete binary tree of height covering PEs , a recursively constructed tree covering PEs , and a root at PE whose children are the roots of the left and the right subtree. : There are two ways to construct . With shifting, is first constructed like , except that it contains an additional processor. Then is shifted by one position to the left and the leftmost leaf is removed.
|
|
An n-flake, polyflake, or Sierpinski n-gon, is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n-gons must touch yet not overlap.
|
|
It is possible that one or more of the partitions still does not fit into the available memory, in which case the algorithm is recursively applied: an additional orthogonal hash function is chosen to hash the large partition into sub-partitions, which are then processed as before. Since this is expensive, the algorithm tries to reduce the chance that it will occur by forming the smallest partitions possible during the initial partitioning phase.
|
|
A first-order theory of a particular signature is a set of axioms, which are sentences consisting of symbols from that signature. The set of axioms is often finite or recursively enumerable, in which case the theory is called effective. Some authors require theories to also include all logical consequences of the axioms. The axioms are considered to hold within the theory and from them other sentences that hold within the theory can be derived.
|
|
Multigrid methods may be used to accelerate the methods. One can first compute an approximation on a coarser grid – usually the double spacing 2h – and use that solution with interpolated values for the other grid points as the initial assignment. This can then also be done recursively for the coarser computation.William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000), A Multigrid Tutorial (2nd ed.), Philadelphia: Society for Industrial and Applied Mathematics, .
|
|
Practical geopolitics describes the actual practice of geopolitical strategy (i.e. foreign policy). Studies of practical geopolitics focus both on geopolitical action and geopolitical reasoning, and the ways in which these are linked recursively to both 'formal' and 'popular' geopolitical discourse. Because critical geopolitics is concerned with geopolitics as discourse, studies of practical geopolitics pay attention both to geopolitical actions (for example, military deployment), but also to the discursive strategies used to narrativize these actions.
|
|
The crossover operation involves swapping random parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Mutation involves substitution of some random part of a program with some other random part of a program. Some programs not selected for reproduction are copied from the current generation to the new generation. Then the selection and other operations are recursively applied to the new generation of programs.
|
|
Von Neumann was a founding figure in computing. Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. Von Neumann wrote the 23 pages long sorting program for the EDVAC in ink. On the first page, traces of the phrase "TOP SECRET", which was written in pencil and later erased, can still be seen.
|
|
The notion of enumeration algorithms is also used in the field of computability theory to define some high complexity classes such as RE, the class of all recursively enumerable problems. This is the class of sets for which there exist an enumeration algorithm that will produce all elements of the set: the algorithm may run forever if the set is infinite, but each solution must be produced by the algorithm after a finite time.
|
|
A drawback of this method is that DNS caches hide the end user's IP address. Both redirection methods, HTTP and DNS based, can be performed in the CDNI, either iteratively or recursively. The recursive redirection is more transparent for the end user because it involves only one UE redirection, but it has other dependencies on the interconnection realisation. A single UE redirection may be preferable if the number of interconnected CDNs exceeds two.
|
|
A decision problem A is called decidable or effectively solvable if A is a recursive set. A problem is called partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set. This means that there exists an algorithm that halts eventually when the answer is yes but may run for ever if the answer is no. Partially decidable problems and any other problems that are not decidable are called undecidable.
|
|
Some of these problem sets required the use of the Oracle object-relational database management system behind Web pages. Others were basic computer science problems such as computing a Fibonacci series recursively using the Tcl programming language. Approximately 180 Ars Digita employees were hired at the company's peak, but with the crash of the dot com economy, many of ArsDigita's clients went out of business. Others cut back heavily on their technology initiatives.
|
|
Systems with a known topology can be initialized in a system specific manner without affecting interoperability. The RapidIO system initialization specification supports system initialization when system topology is unknown or dynamic. System initialization algorithms support the presence of redundant hosts, so system initialization need not have a single point of failure. Each system host recursively enumerates the RapidIO fabric, seizing ownership of devices, allocating device IDs to endpoints and updating switch routing tables.
|
|
The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: # divide the line segment into three segments of equal length. # draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. # remove the line segment that is the base of the triangle from step 2. The first iteration of this process produces the outline of a hexagram.
|
|
A third strategy is to print out the operator first and then recursively print out the left and right subtree known as pre-order traversal. These three standard depth-first traversals are representations of the three different expression formats: infix, postfix, and prefix. An infix expression is produced by the inorder traversal, a postfix expression is produced by the post-order traversal, and a prefix expression is produced by the pre-order traversal.
|
|
More specifically, to compute the static slice for (x,v), we first find all statements that can directly affect the value of v before statement x is encountered. Recursively, for each statement y which can affect the value of v in statement x, we compute the slices for all variables z in y that affect the value of v. The union of all those slices is the static slice for (x,v).
|
|
For example one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym for "recursive language" used is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of . All recursive languages are also recursively enumerable.
|
|
The shuffle sortA revolutionary new sort from John Cohen Nov 26, 1997 is a variant of bucket sort that begins by removing the first 1/8 of the n items to be sorted, sorts them recursively, and puts them in an array. This creates n/8 "buckets" to which the remaining 7/8 of the items are distributed. Each "bucket" is then sorted, and the "buckets" are concatenated into a sorted array.
|
|
When inserting a node into an AVL tree, you initially follow the same process as inserting into a Binary Search Tree. If the tree is empty, then the node is inserted as the root of the tree. In case the tree has not been empty then we go down the root, and recursively go down the tree searching for the location to insert the new node. This traversal is guided by the comparison function.
|
|
Three G-structures (light blue) divide up the five-element base set between them; then, an F-structure (red) is built to connect the G-structures. center These last two operations may be illustrated by the example of trees. First, define X to be the species "singleton" whose generating series is X(x) = x. Then the species Ar of rooted trees (from the French "arborescence") is defined recursively by Ar = X · E(Ar).
|
|
The construction of the Sierpiński carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position.
|
|
We can recursively find solutions for these two sets because of optimal sub- structure. As we don't know , we can try each of the activities. This approach leads to an O(n^3) solution. This can be optimized further considering that for each set of activities in (i, j), we can find the optimal solution if we had known the solution for (i, t), where is the last non-overlapping interval with in (i, j).
|
|
A Web crawler starts with a list of URLs to visit, called the seeds. As the crawler visits these URLs, it identifies all the hyperlinks in the pages and adds them to the list of URLs to visit, called the crawl frontier. URLs from the frontier are recursively visited according to a set of policies. If the crawler is performing archiving of websites, it copies and saves the information as it goes.
|
|
Complete inline expansion is not always possible, due to recursion: recursively inline expanding the calls will not terminate. There are various solutions, such as expanding a bounded amount, or analyzing the call graph and breaking loops at certain nodes (i.e., not expanding some edge in a recursive loop). An identical problem occurs in macro expansion, as recursive expansion does not terminate, and is typically resolved by forbidding recursive macros (as in C and C++).
|
|
Such a subgraph is called a Highly Connected Subgraph (HCS). Single vertices are not considered clusters and are grouped into a singletons set S. Given a similarity graph G(V,E), HCS clustering algorithm will check if it is already highly connected, if yes, returns G, otherwise uses the minimum cut of G to partition G into two subgraphs H and H', and recursively run HCS clustering algorithm on H and H'.
|
|
85 opened study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure. He also has a significant contribution in the subject of mass problems where he introduced the generalisation of Turing degrees, called "Muchnik degrees" in his work On Strong and Weak Reducibilities of Algorithmic Problems published in 1963.A. A. Muchnik, On strong and weak reducibility of algorithmic problems.
|
|
If a name server cannot answer a query because it does not contain an entry for the host in its DNS cache, it may recursively query name servers higher up in the hierarchy. This is known as a recursive query or recursive lookup. A server providing recursive queries is known as a recursive name server or recursive DNS, sometimes abbreviated as recdns. In principle, authoritative name servers suffice for the operation of the Internet.
|
|
Apply the optimal algorithm recursively to this graph. The runtime of all steps in the algorithm is O(m), except for the step of using the decision trees. The runtime of this step is unknown, but it has been proved that it is optimal - no algorithm can do better than the optimal decision tree. Thus, this algorithm has the peculiar property that it is provably optimal although its runtime complexity is unknown.
|
|
The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using Turing machines, λ calculus, and other systems. More advanced results concern the structure of the Turing degrees and the lattice of recursively enumerable sets. Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such as hyperarithmetical theory and α-recursion theory.
|
|
The latter also focus on complexity and interworking parts as the effect needing explanation, whereas the Fifth Way takes as its starting point any regularity. (E.g., that an eye has a complicated function therefore a design therefore a designer) but an argument from final cause (e.g., that the pattern that things exist with a purpose itself allows us to recursively arrive at God as the ultimate source of purpose without being constrained by any external purpose).
|
|
The merge is done recursively by merging B with A's right subtree. This might change the S-value of A's right subtree. To maintain the leftist tree property, after each merge is done, we check if the S-value of right subtree became bigger than the S-value of left subtree during the recursive merge calls. If so, we swap the right and left subtrees (If one child is missing, it should be the right one).
|
|
A curved triangle patch. Normals at vertices are used to recursively subdivide the triangle into four sub-triangles In order to improve geometric fidelity, the format allows curving the triangle patches. By default, all triangles are assumed to be flat and all triangle edges are assumed to be straight lines connecting their two vertices. However, curved triangles and curved edges can optionally be specified in order to reduce the number of mesh elements required to describe a curved surface.
|
|
2 Standard headers, p. 181, footnote 182: "A header is not necessarily a source file, nor are the `<` and `>` delimited sequences in header names necessarily valid source file names. Inclusion continues recursively on these included contents, up to an implementation-defined nesting limit. Headers need not have names corresponding to files: in C++ standard headers are typically identified with words, like "vector", hence `#include ` while in C standard headers have identifiers in the form of filenames with a ".
|
|
While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize discussion of the economy as a whole through a theory of general competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption side. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium.
|
|
Thus, any set of items can be extended by recursively adding all the appropriate items until all nonterminals preceded by dots are accounted for. The minimal extension is called the closure of an item set and written as clos(I) where I is an item set. It is these closed item sets that are taken as the states of the parser, although only the ones that are actually reachable from the begin state will be included in the tables.
|
|
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
|
|
The cardinal B-splines are defined recursively starting from the B-spline of order 1, namely N_1(x), which takes the value 1 in the interval [0, 1) and 0 elsewhere. Computer algebra systems may have to be employed to obtain concrete expressions for higher order cardinal B-splines. The concrete expressions for cardinal B-splines of all orders up to 6 are given below. The graphs of cardinal B-splines of orders up to 4 are also exhibited.
|
|
The '$' sign is used to denote 'end of input' is expected, as is the case for the starting rule. This is not the complete item set 0, though. Each item set must be 'closed', which means all production rules for each nonterminal following a '•' have to be recursively included into the item set until all of those nonterminals are dealt with. The resulting item set is called the closure of the item set we began with.
|
|
The second proof is more reminiscent of R.D. Laing:R.D. Laing (1970) Your concept of your concept is not my concept of your concept—a reproduced concept is not the same as the original concept. Pask defined concepts as persisting, countably infinite, recursively packed spin processes (like many cored cable, or skins of an onion) in any medium (stars, liquids, gases, solids, machines and, of course, brains) that produce relations. Here we prove A(T) ≠ B(T).
|
|
For more than two factors, a 2k factorial experiment can usually be recursively designed from a 2k−1 factorial experiment by replicating the 2k−1 experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to, e.g., designing three replicates for three level factors, etc. A factorial experiment allows for estimation of experimental error in two ways.
|
|
The phenotype, however, is the same as in Koza-style GP: a tree-like structure that is evaluated recursively. This model is more in line with how genetics work in nature, where there is a separation between an organism's genotype and the final expression of phenotype in proteins, etc. Separating genotype and phenotype allows a modular approach. In particular, the search portion of the GE paradigm needn't be carried out by any one particular algorithm or method.
|
|
The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to fuzzy set theory is a crucial one. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable.
|
|
ECOsystem was content aware, with selection of compression solution based on the type of data being processed. This went beyond file-extension filtering. ECOsystem recursively decomposed compound files, until elemental text, media, or binary components are identified. At the heart of the optimizer software is a context-weighted neural net that will apply the most effective compression solution based on the nature of the elemental file component identified, and will efficiently remember optimal settings based on similar files processed.
|
|
After the array has been partitioned, the two partitions can be sorted recursively in parallel. Assuming an ideal choice of pivots, parallel quicksort sorts an array of size in work in time using additional space. Quicksort has some disadvantages when compared to alternative sorting algorithms, like merge sort, which complicate its efficient parallelization. The depth of quicksort's divide-and-conquer tree directly impacts the algorithm's scalability, and this depth is highly dependent on the algorithm's choice of pivot.
|
|
Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductive work.
|
|
Most often, the makefile directs Make on how to compile and link a program. A makefile works upon the principle that files only need recreating if their dependencies are newer than the file being created/recreated. The makefile is recursively carried out (with dependency prepared before each target depending upon them) until everything has been updated (that requires updating) and the primary/ultimate target is complete. These instructions with their dependencies are specified in a makefile.
|
|
The ideas of the factor method and binary method can be combined into Brauer's m-ary method by choosing any number m (regardless of whether it divides n), recursively constructing a chain for \lfloor n/m\rfloor, concatenating a chain for m (modified in the same way as above) to obtain m\lfloor n/m\rfloor, and then adding the remainder. Additional refinements of these ideas lead to a family of methods called sliding window methods.
|
|
Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics.
|
|
RE-complete is the set of decision problems that are complete for RE. In a sense, these are the "hardest" recursively enumerable problems. Generally, no constraint is placed on the reductions used except that they must be many-one reductions. Examples of RE-complete problems: #Halting problem: Whether a program given a finite input finishes running or will run forever. #By Rice's Theorem, deciding membership in any nontrivial subset of the set of recursive functions is RE-hard.
|
|
Each of the bins are recursively processed, as is done for the in-place MSD radix sort. After the sort by the last digit has been completed, the output buffer is checked to see if it is the original input array, and if it's not, then a single copy is performed. If the digit size is chosen such that the key size divided by the digit size is an even number, the copy at the end is avoided.
|
|
If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then A ∩ B, A ∪ B and the image of A × B under the Cantor pairing function are recursive sets. A set A is a recursive set if and only if A and the complement of A are both recursively enumerable sets. The preimage of a recursive set under a total computable function is a recursive set.
|
|
It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that is suitably flexible, and is typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one.Barbara Partee and Mats Rooth. 1983.
|
|
A zipper is a technique of representing an aggregate data structure so that it is convenient for writing programs that traverse the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997. It includes and generalizes the gap buffer technique sometimes used with arrays. The zipper technique is general in the sense that it can be adapted to lists, trees, and other recursively defined data structures.
|
|
An Apollonian network The Goldner–Harary graph, a non-Hamiltonian Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
|
|
When number (generally large number) is represented in a finite alphabet set, and it cannot be represented by just one member of the set, recursive indexing is used. Recursive indexing itself is a method to write the successive differences of the number after extracting the maximum value of the alphabet set from the number, and continuing recursively till the difference falls in the range of the set. Recursive indexing with a 2-letter alphabet is called unary code.
|
|
In such a case we use the same test recursively on the large factors of , until all of the primes are below a reasonable threshold. In our example, we can say with certainty that 2 and 3 are prime, and thus we have proved our result. The primality certificate is the list of (p, a_p)pairs, which can be quickly checked in the corollary. If our example had included large prime factors, the certificate would be more complicated.
|
|
Indeed, one can enumerate all the primitive recursive functions and define a function en such that for all n, m: en(n,m) = fn(m), where fn is the n-th primitive recursive function (for k-ary functions, this will be set to fn(m,m...m)). Now, g(n) = en(n,n)+1 is provably total but not primitive recursive, by a diagonalization argument: had there been a j such that g = fj, we would have got g(j) = en(j,j)+1 = fj (j)+1= g(j)+1, a contradiction. (The Gödel numbers of all primitive recursive functions can be enumerated by a primitive recursive function, though the primitive recursive functions' values cannot.) One such function, which is provable total but not primitive recursive, is the Ackermann function: since it is recursively defined, it is indeed easy to prove its computability (However, a similar diagonalization argument can also be built for all functions defined by recursive definition; thus, there are provable total functions that cannot be defined recursively ).
|
|
In this algorithmic process entropy is transferred reversibly to specific qubits (named reset spins) that are coupled with the environment much more strongly than others. After a sequence of reversible steps that let the entropy of these reset qubits increase they become hotter than the environment. Then the strong coupling results in a heat transfer (irreversibly) from these reset spins to the environment. The entire process may be repeated and may be applied recursively to reach low temperatures for some qubits.
|
|
Hal Burch and William Cheswick propose a controlled flooding of links to determine how this flooding affects the attack stream. Flooding a link will cause all packets, including packets from the attacker, to be dropped with the same probability. We can conclude from this that if a given link were flooded, and packets from the attacker slowed, then this link must be part of the attack path. Then recursively upstream routers are “coerced” into performing this test until the attack path is discovered.
|
|
Conway proposed the following problem: given a constant finite language L, is the greatest solution of the equation LX=XL always regular? This problem was studied by Karhumäki and Petre who gave an affirmative answer in a special case. A strongly negative answer to Conway's problem was given by Kunc who constructed a finite language L such that the greatest solution of this equation is not recursively enumerable. Kunc also proved that the greatest solution of inequality LX \subseteq XL is always regular.
|
|
Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability. The Church–Turing thesis states that the two notions coincide: any number- theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical proof.
|
|
Roles Report - recursively list the members of the role and the members of groups in order to easily determine which members actually have access via each role Similar Aggregations - allows viewing a report that lists any aggregations which are very similar to each other. Smart Diff - compares versions of a SSAS, SSIS, and SSRS files. BIDS Helper pre-processes XML files so that the diff versus source control is more meaningful. Show Extra Properties - exposes hidden properties on several Analysis Services objects.
|
|
"Perhaps the most challenging idea incorporated in the theory of autopoiesis is that social systems should not be defined in terms of human agency or norms, but of communications. Communication is in turn the unity of utterance, information and understanding and constitutes social systems by recursively reproducing communication. This sociologically radical thesis, which raises the fear of a dehumanised theory of law and society, attempts to highlight the fact that social systems are constituted by communication."Banakar and Max Travers 2005: 28.
|
|
The solution is shown in the right half: a virtual ancestor (the dashed circle) is created. Fortunately, in this case it can be shown that there are at most two possible candidate ancestors, and recursive three-way merge constructs a virtual ancestor by merging the non- unique ancestors first. This merge can itself suffer the same problem, so the algorithm recursively merges them. Since there is a finite number of versions in the history, the process is guaranteed to eventually terminate.
|
|
It is advantageous not to discard the part of the histogram that exceeds the clip limit but to redistribute it equally among all histogram bins. 300 px The redistribution will push some bins over the clip limit again (region shaded green in the figure), resulting in an effective clip limit that is larger than the prescribed limit and the exact value of which depends on the image. If this is undesirable, the redistribution procedure can be repeated recursively until the excess is negligible.
|
|
Finally, Alon applies an observation of , that selecting alternating subsets of edges in an Euler tour of the graph partitions it into two regular subgraphs, to split the edge coloring problem into two smaller subproblems, and his algorithm solves the two subproblems recursively. The total time for his algorithm is . For planar graphs with maximum degree , the optimal number of colors is again exactly . With the stronger assumption that , it is possible to find an optimal edge coloring in linear time .
|
|
It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined.
|
|
Since sub-arrays of sorted / identical elements crop up a lot towards the end of a sorting procedure on a large set, versions of the quicksort algorithm that choose the pivot as the middle element run much more quickly than the algorithm described in this diagram on large sets of numbers. Quicksort is a divide and conquer algorithm. It first divides the input array into two smaller sub-arrays: the low elements and the high elements. It then recursively sorts the sub-arrays.
|
|
Recursively sort the "equal to" partition by the next character (key). Given we sort using bytes or words of length bits, the best case is and the worst case or at least as for standard quicksort, given for unique keys , and is a hidden constant in all standard comparison sort algorithms including quicksort. This is a kind of three-way quicksort in which the middle partition represents a (trivially) sorted subarray of elements that are exactly equal to the pivot.
|
|
Therefore, S is in \Sigma^0_1. Thus every recursively enumerable set is in \Sigma^0_1. The converse is true as well: for every formula \varphi(n) in \Sigma^0_1 with k existential quantifiers, we may enumerate the k-tuples of natural numbers and run a Turing machine that goes through all of them until it finds the formula is satisfied. This Turing machine halts on precisely the set of natural numbers satisfying \varphi(n), and thus enumerates its corresponding set.
|
|
The simplest such procedure is termed the "k-d Construction Algorithm", by analogy with the process used to construct k-d trees. This is an off-line algorithm, that is, an algorithm that operates on the entire data set at once. The tree is built top-down by recursively splitting the data points into two sets. Splits are chosen along the single dimension with the greatest spread of points, with the sets partitioned by the median value of all points along that dimension.
|
|
One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has been pushed a long way. The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.
|
|
In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, it is a function of one variable which is the composition of a finite number of arithmetic operations , exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives. The logarithm function does not need to be explicitly included since it is the integral of 1/x.
|
|
Several fast tests exist that tell if a segment of the real line or a region of the complex plane contains no roots. By bounding the modulus of the roots and recursively subdividing the initial region indicated by these bounds, one can isolate small regions that may contain roots and then apply other methods to locate them exactly. All these methods involve finding the coefficients of shifted and scaled versions of the polynomial. For large degrees, FFT-based accelerated methods become viable.
|
|
In a "flat" design, only primitives are instanced. Hierarchical designs can be recursively "exploded" ("flattened") by creating a new copy (with a new name) of each definition each time it is used. If the design is highly folded, expanding it like this will result in a much larger netlist database, but preserves the hierarchy dependencies. Given a hierarchical netlist, the list of instance names in a path from the root definition to a primitive instance specifies the single unique path to that primitive.
|
|
Much recent research on Turing degrees has focused on the overall structure of the set of Turing degrees and the set of Turing degrees containing recursively enumerable sets. A deep theorem of Shore and Slaman (1999) states that the function mapping a degree x to the degree of its Turing jump is definable in the partial order of the Turing degrees. A recent survey by Ambos-Spies and Fejer (2006) gives an overview of this research and its historical progression.
|
|
Equivalently, RE is the class of decision problems for which a Turing machine can list all the 'yes' instances, one by one (this is what 'enumerable' means). Each member of RE is a recursively enumerable set and therefore a Diophantine set. Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever.
|
|
The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.
|
|
First, we obtain an algorithm that moves each node exactly once, which may not be optimal. Do this recursively: consider any leaf of the smallest tree in the graph containing both the initial and desired sets. If a leaf of this tree is in both, remove it and recurse down. If a leaf is in the initial set only, find a path from it to a vertex in the desired set that does not pass through any other vertices in the desired set.
|
|
Modern alternative classification systems generally start with the three-domain system: Archaea (originally Archaebacteria); Bacteria (originally Eubacteria); Eukaryota (including protists, fungi, plants, and animals) These domains reflect whether the cells have nuclei or not, as well as differences in the chemical composition of the cell exteriors. Further, each kingdom is broken down recursively until each species is separately classified. The order is: Domain; kingdom; phylum; class; order; family; genus; species. The scientific name of an organism is generated from its genus and species.
|
|
Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also allows preserving the curve definition under all linear or perspective 2D and 3D transforms and projections. Font engines, like FreeType, draw the font's curves (and lines) on a pixellated surface using a process known as font rasterization.
|
|
The objects that an application made available through its AEOM support were arranged in a hierarchy. At the top was the application itself, referenced via a null object descriptor. Other objects were referenced by (recursively) specifying their parent object, together with other information identifying it as a child of that parent, all collected in an AERecord. An iterator was provided by parents to enumerate their children, or children of a certain class, allowing applications to address a set of elements.
|
|
Barry Wellman and Bernie Hogan, with Kristen Berg, Jeffrey Boase, Juan-Antonio Carrasco, Rochelle Côté, Jennifer Kayahara, Tracy L.M. Kennedy and Phouc Tran. "Connected Lives: The Project" Pp. 157-211 in Networked Neighbourhoods: The Online Community in Context, edited by Patrick Purcell. Guildford, UK: Springer, 2006. More focused research (with Jennifer Kayahara) has shown how the onetime two-step flow of communication has become more recursively multi-step as the result of the Internet's facilitation of information seeking and communication.
|
|
The completeness theorem is a central property of first-order logic that does not hold for all logics. Second-order logic, for example, does not have a completeness theorem for its standard semantics (but does have the completeness property for Henkin semantics), and the set of logically-valid formulas in second-order logic is not recursively enumerable. The same is true of all higher-order logics. It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.
|
|
There are several approaches to solving the all maximal scoring subsequences problem. A natural approach is to use existing, linear time algorithms to find the maximum subsequence (see maximum subarray problem) and then recursively find the maximal subsequences to the left and right of the maximum subsequence. The analysis of this algorithm is similar to that of Quicksort: The maximum subsequence could be small in comparison to the rest of sequence, leading to a running time of O(n^2) in the worst case.
|
|
Further, a curried function at a fixed point is (trivially), a partial application. For further evidence, note that, given any function f(x,y), a function g(y,x) may be defined such that g(y,x) = f(x,y). Thus, any partial application may be reduced to a single curry operation. As such, curry is more suitably defined as an operation which, in many theoretical cases, is often applied recursively, but which is theoretically indistinguishable (when considered as an operation) from a partial application.
|
|
The method has been rediscovered by the parallel processing community under the name "Domain Decomposition".Lai C. H. (1994) "Diakoptics, Domain Decomposition and Parallel Computing", The Computer Journal, Vol 37, No 10, pp. 840–846 According to Keith Bowden, "Kron was undoubtedly searching for an ontology of engineering".K. Bowden (1998) "Physical computation and parallelism (constructive postmodern physics)", International Journal of General Systems 27(1–3):93–103 Bowden also described "a multilevel hierarchical version of the method, in which the subsystems are recursively torn into subsubsystems".
|
|
This is implemented through a model called structural regular expressions, which can recursively apply regular-expression matching to obtain other (sub)selections within a given selection. In this way, sam's command set can be applied to substrings that are identified by arbitrarily complex context. Sam extends its basic text-editing command set to handling of multiple files, providing similar pattern-based conditional and loop commands for filename specification. Any sequence of text-editing commands may be applied as a unit to each such specification.
|
|
This amounts to saying that n is the value of f(m). The game is now brought down to the elementary n=f(m), which is won by the machine if and only if n is indeed the value of f(m). Let p be a unary predicate. Then ⊓x(p(x)⊔¬p(x)) expresses the problem of deciding p, ⊓x(p(x)&ᐁ¬p(x)) expresses the problem of semideciding p, and ⊓x(p(x)⩛¬p(x)) the problem of recursively approximating p.
|
|
The key insight to the algorithm is a random sampling step which partitions a graph into two subgraphs by randomly selecting edges to include in each subgraph. The algorithm recursively finds the minimum spanning forest of the first subproblem and uses the solution in conjunction with a linear time verification algorithm to discard edges in the graph that cannot be in the minimum spanning tree. A procedure taken from Borůvka's algorithm is also used to reduce the size of the graph at each recursion.
|
|
This is a recursive method that decomposes a given tree into paths. This stages starts off by finding the longest root-to- leaf path in the tree. It then removes this path by breaking its ties from the tree which will break the remaining of the tree into sub-trees and then it recursively processes each sub-tree. Every time a path is decomposed, an array is created in association with the path that contains the elements on the path from the root to the leaf.
|
|
Texts, plots and cinematography are discussed and the delusions approached tangentially. This use of fiction to decrease the malleability of a delusion was employed in a joint project by science-fiction author Philip Jose Farmer and Yale psychiatrist A. James Giannini. They wrote the novel Red Orc's Rage, which, recursively, deals with delusional adolescents who are treated with a form of projective therapy. In this novel's fictional setting other novels written by Farmer are discussed and the characters are symbolically integrated into the delusions of fictional patients.
|
|
A function with three fixed pointsIn mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f(c) = c. This means f(f(...f(c)...)) = f n(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
|
|
Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics). Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable propositions. For example, the continuum hypothesis is undecidable in the Zermelo-Fraenkel set theory as shown by Cohen.
|
|
Loop Subdivision of an icosahedron (top) after one and after two refinement steps In computer graphics, Loop subdivision surface is an approximating subdivision scheme developed by Charles Loop in 1987 for triangular meshes. Loop subdivision surfaces are defined recursively, dividing each triangle into four smaller ones. The method is based on a quartic box spline, which generate C2 continuous limit surfaces everywhere except at extraordinary vertices where they are C1 continuous. Geologists have also applied Loop Subdivision Surfaces to erosion on mountain faces, specifically in the Appalachians.
|
|
A range tree on a set of points in d-dimensions is a recursively defined multi-level binary search tree. Each level of the data structure is a binary search tree on one of the d-dimensions. The first level is a binary search tree on the first of the d-coordinates. Each vertex v of this tree contains an associated structure that is a (d−1)-dimensional range tree on the last (d−1)-coordinates of the points stored in the subtree of v.
|
|
From the top-level synchronized XooML fragment, Planz builds a top-level Plan. Step two is to recursively retrieve and process additional XooML fragments as needed. In Planz, for example, the subfolders and folder shortcuts of a folder appear within the folder's Plan as document-like heading associations. For each of these headings that were last shown as "expanded," Planz retrieves folder content information and an associated XooML fragment and then uses the results of their synchronization to determine the display of a sub-Plan.
|
|
Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called runs in this example) into sublists until sublist size is 1, then merges those sublists to produce a sorted list. The copy back step is avoided with alternating the direction of the merge with each level of recursion (except for an initial one time copy). To help understand this, consider an array with 2 elements. The elements are copied to B[], then merged back to A[].
|
|
A divide and conquer algorithm for triangulations in two dimensions was developed by Lee and Schachter and improved by Guibas and Stolfi and later by Dwyer. In this algorithm, one recursively draws a line to split the vertices into two sets. The Delaunay triangulation is computed for each set, and then the two sets are merged along the splitting line. Using some clever tricks, the merge operation can be done in time O(n), so the total running time is O(n log n).
|
|
In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor's result as showing that the real numbers (collectively) are not recursively enumerable. Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable. They are, therefore exactly countable.
|
|
Diving further into the details, a common technique is to apply rules in a cyclical fashion (recursively, as computer scientists would say). After applying the suffix substitution rule in this example scenario, a second pass is made to identify matching rules on the term friendly, where the ly stripping rule is likely identified and accepted. In summary, friendlies becomes (via substitution) friendly which becomes (via stripping) friend. This example also helps illustrate the difference between a rule-based approach and a brute force approach.
|
|
R. Deriche (1987) Using Canny's criteria to derive an optimal edge detector recursively implemented, Int. J. Computer Vision, vol 1, pages 167–187. The differential edge detector described below can be seen as a reformulation of Canny's method from the viewpoint of differential invariants computed from a scale space representation leading to a number of advantages in terms of both theoretical analysis and sub-pixel implementation. In that aspect, Log Gabor filter have been shown to be a good choice to extract boundaries in natural scenes.
|
|
Recursive least squares (RLS) is an adaptive filter algorithm that recursively finds the coefficients that minimize a weighted linear least squares cost function relating to the input signals. This approach is in contrast to other algorithms such as the least mean squares (LMS) that aim to reduce the mean square error. In the derivation of the RLS, the input signals are considered deterministic, while for the LMS and similar algorithm they are considered stochastic. Compared to most of its competitors, the RLS exhibits extremely fast convergence.
|
|
TrueType fonts use composite Bézier curves composed of quadratic Bézier curves. Other languages and imaging tools (such as PostScript, Asymptote, Metafont, and SVG) use composite Béziers composed of cubic Bézier curves for drawing curved shapes. OpenType fonts can use either kind, depending on the flavor of the font. The internal rendering of all Bézier curves in font or vector graphics renderers will split them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments.
|
|
One then wishes to find a recursive factorization of z^N-1 into polynomials of few terms and smaller and smaller degree. To compute the DFT, one takes x(z) modulo each level of this factorization in turn, recursively, until one arrives at the monomials and the final result. If each level of the factorization splits every polynomial into an O(1) (constant-bounded) number of smaller polynomials, each with an O(1) number of nonzero coefficients, then the modulo operations for that level take O(N) time; since there will be a logarithmic number of levels, the overall complexity is O (N log N). More explicitly, suppose for example that z^N-1 = F_1(z) F_2(z) F_3(z), and that F_k(z) = F_{k,1}(z) F_{k,2}(z), and so on. The corresponding FFT algorithm would consist of first computing xk(z) = x(z) mod Fk(z), then computing xk,j(z) = xk(z) mod Fk,j(z), and so on, recursively creating more and more remainder polynomials of smaller and smaller degree until one arrives at the final degree-0 results.
|
|
Tree rotation A binary tree is a structure consisting of a set of nodes, one of which is designated as the root node, in which each remaining node is either the left child or right child of some other node, its parent, and in which following the parent links from any node eventually leads to the root node. (In some sources, the nodes described here are called "internal nodes", there exists another set of nodes called "external nodes", each internal node is required to have exactly two children, and each external node is required to have zero children. The version described here can be obtained by removing all the external nodes from such a tree.) For any node in the tree, there is a subtree of the same form, rooted at and consisting of all the nodes that can reach by following parent links. Each binary tree has a left-to-right ordering of its nodes, its inorder traversal, obtained by recursively traversing the left subtree (the subtree at the left child of the root, if such a child exists), then listing the root itself, and then recursively traversing the right subtree.
|
|
If they don't, the Turing machine will go back to step 1. It is easy to see that this Turing machine will generate all and only the sentential forms of G on its second tape after the last step is executed an arbitrary number of times, thus the language L(G) must be recursively enumerable. The reverse construction is also possible. Given some Turing machine, it is possible to create an equivalent unrestricted grammar which even uses only productions with one or more non- terminal symbols on their left-hand sides.
|
|
The Davis–Putnam algorithm was developed by Martin Davis and Hilary Putnam for checking the validity of a first-order logic formula using a resolution-based decision procedure for propositional logic. Since the set of valid first-order formulas is recursively enumerable but not recursive, there exists no general algorithm to solve this problem. Therefore, the Davis–Putnam algorithm only terminates on valid formulas. Today, the term "Davis–Putnam algorithm" is often used synonymously with the resolution-based propositional decision procedure that is actually only one of the steps of the original algorithm.
|
|
In computer programming languages, a recursive data type (also known as a recursively-defined, inductively-defined or inductive data type) is a data type for values that may contain other values of the same type. Data of recursive types are usually viewed as directed graphs. An important application of recursion in computer science is in defining dynamic data structures such as Lists and Trees. Recursive data structures can dynamically grow to an arbitrarily large size in response to runtime requirements; in contrast, a static array's size requirements must be set at compile time.
|
|
A third hypothesis is that if the knowledge to select or apply an operator is incomplete or uncertain, an impasse arises and the architecture automatically creates a substate. In the substate, the same process of problem solving is recursively used, but with the goal to retrieve or discover knowledge so that decision making can continue. This can lead to a stack of substates, where traditional problem methods, such as planning or hierarchical task decomposition, naturally arise. When results created in the substate resolve the impasse, the substate and its associated structures are removed.
|
|
In game theory, perfect play is the behavior or strategy of a player that leads to the best possible outcome for that player regardless of the response by the opponent. Perfect play for a game is known when the game is solved. Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). By backward reasoning, one can recursively evaluate a non-final position as identical to the position that is one move away and best valued for the player whose move it is.
|
|
Context-free grammars arise in linguistics where they are used to describe the structure of sentences and words in a natural language, and they were in fact invented by the linguist Noam Chomsky for this purpose. By contrast, in computer science, as the use of recursively-defined concepts increased, they were used more and more. In an early application, grammars are used to describe the structure of programming languages. In a newer application, they are used in an essential part of the Extensible Markup Language (XML) called the Document Type Definition.
|
|
Since the time of Pāṇini, at least, linguists have described the grammars of languages in terms of their block structure, and described how sentences are recursively built up from smaller phrases, and eventually individual words or word elements. An essential property of these block structures is that logical units never overlap. For example, the sentence: : John, whose blue car was in the garage, walked to the grocery store. can be logically parenthesized (with the logical metasymbols [ ]) as follows: : [John[, [whose [blue car [was [in [the garage ], [walked [to [the [grocery store .
|
|
In propositional calculus a literal is simply a propositional variable or its negation. In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbol applied to some terms, P(t_1,\ldots,t_n) with the terms recursively defined starting from constant symbols, variable symbols, and function symbols. For example, eg Q(f(g(x), y, 2), x) is a negative literal with the constant symbol 2, the variable symbols x, y, the function symbols f, g, and the predicate symbol Q.
|
|
With this reduction, the algorithm is O(nL)-time and O(nL)-space. However, the original paper, "A fast algorithm for optimal length-limited Huffman codes", shows how this can be improved to O(nL)-time and O(n)-space. The idea is to run the algorithm a first time, only keeping enough data to be able to determine two equivalent subproblems that sum to half the size of the original problem. This is done recursively, resulting in an algorithm that takes about twice as long but requires only linear space.
|
|
Recursion is the definition of a function using the function itself. Lambda calculus cannot express this as directly as some other notations: all functions are anonymous in lambda calculus, so we can't refer to a value which is yet to be defined, inside the lambda term defining that same value. However, recursion can still be achieved by arranging for a lambda expression to receive itself as its argument value, for example in `(λx.x x) E`. Consider the factorial function `F(n)` recursively defined by : `F(n) = 1, if n = 0; else n × F(n − 1)`.
|
|
Not considering suffixes, the longest Turkish dictionary words have 20 letters: These are "kuyruksallayangiller" (the biological genus Motacillidae), "ademimerkeziyetçilik" (decentralization) and "elektroensefalografi" (electroencephalography). In comparison, the word "muvaffakiyet" has 12 letters, so it should be possible to use various other suffixes to make an even longer word from these ones. There is no principled grammatical reason for not being able to make a Turkish word indefinitely long, as there are suffixes that can act recursively on a word stem. In practice, however, such words would become unintelligible after a few cycles of recursion.
|
|
For instance, :q(f(x_1,\dots,x_3)) \to g(a,q'(x_1),h(q(x_3))) is a rule – one customarily writes q(x_i) instead of the pair (q,x_i) – and its intuitive semantics is that, under the action of q, a tree with f at the root and three children is transformed into :g(a,q'(x_1),h(q(x_3))) where, recursively, q'(x_1) and q(x_3) are replaced, respectively, with the application of q' on the first child and with the application of q on the third.
|
|
Parallel-move is used to extend an i-cell to (i+1)-cell. In other words, if A and B are two i-cells and A is a parallel-move of B, then {A,B} is an (i+1)-cell. Therefore, k-cells can be defined recursively. Basically, a connected set of grid points M can be viewed as a digital k-manifold if: (1) any two k-cells are (k-1)-connected, (2) every (k-1)-cell has only one or two parallel-moves, and (3) M does not contain any (k+1)-cells.
|
|
Histogram-based methods are very efficient compared to other image segmentation methods because they typically require only one pass through the pixels. In this technique, a histogram is computed from all of the pixels in the image, and the peaks and valleys in the histogram are used to locate the clusters in the image. Color or intensity can be used as the measure. A refinement of this technique is to recursively apply the histogram-seeking method to clusters in the image in order to divide them into smaller clusters.
|
|
A naive use of this idea would increase the storage space to O(n²). In the same fashion as in the slab decomposition, the similarity between consecutive data structures can be exploited in order to reduce the storage space to O(n log n), but the query time increases to O(log² n). In d-dimensional space, point location can be solved by recursively projecting the faces into a (d-1)-dimensional space. While the query time is O(log n), the storage space can be as high as O(n^{2^d}).
|
|
A proof procedure for a logic is complete if it produces a proof for each provable statement. The theorems of logical systems are typically recursively enumerable, which implies the existence of a complete but extremely inefficient proof procedure; however, a proof procedure is only of interest if it is reasonably efficient. Faced with an unprovable statement, a complete proof procedure may sometimes succeed in detecting and signalling its unprovability. In the general case, where provability is a semidecidable property, this is not possible, and instead the procedure will diverge (not terminate).
|
|
In a modal logic, a model comprises a set of possible worlds, each one associated to a truth evaluation; an accessibility relation tells when a world is accessible from another one. A modal formula may specify not only conditions over a possible world, but also on the ones that are accessible from it. As an example, \Box A is true in a world if A is true in all worlds that are accessible from it. As for propositional logic, tableaux for modal logics are based on recursively breaking formulae into its basic components.
|
|
A language which is accepted by such a Turing machine is called a recursively enumerable language. The Turing machine, it turns out, is an exceedingly powerful model of automata. Attempts to amend the definition of a Turing machine to produce a more powerful machine have surprisingly met with failure. For example, adding an extra tape to the Turing machine, giving it a two-dimensional (or three- or any-dimensional) infinite surface to work with can all be simulated by a Turing machine with the basic one-dimensional tape.
|
|
In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms). Note that the definition of inductive reasoning described here differs from mathematical induction, which, in fact, is a form of deductive reasoning. Mathematical induction is used to provide strict proofs of the properties of recursively defined sets. The deductive nature of mathematical induction derives from its basis in a non-finite number of cases, in contrast with the finite number of cases involved in an enumerative induction procedure like proof by exhaustion.
|
|
Ray casting is the most basic of many computer graphics rendering algorithms that use the geometric algorithm of ray tracing. Ray tracing-based rendering algorithms operate in image order to render three-dimensional scenes to two-dimensional images. Geometric rays are traced from the eye of the observer to sample the light (radiance) travelling toward the observer from the ray direction. The speed and simplicity of ray casting comes from computing the color of the light without recursively tracing additional rays that sample the radiance incident on the point that the ray hit.
|
|
The MEA technique is a strategy to control search in problem-solving. Given a current state and a goal state, an action is chosen which will reduce the difference between the two. The action is performed on the current state to produce a new state, and the process is recursively applied to this new state and the goal state. Note that, in order for MEA to be effective, the goal-seeking system must have a means of associating to any kind of detectable difference those actions that are relevant to reducing that difference.
|
|
Yongge Wang (born 1967) is a computer science professor at the University of North Carolina at Charlotte specialized in algorithmic complexity and cryptography. He is the inventor of IEEE P1363 cryptographic standards SRP5 and WANG-KE and has contributed to the mathematical theory of algorithmic randomness. He co-authored a paper demonstrating that a recursively enumerable real number is an algorithmically random sequence if and only if it is a Chaitin's constant for some encoding of programs. He also showed the separation of Schnorr randomness from recursive randomness.
|
|
However, the first may require more computation. For example, if the constraint store contains the constraint `X<-2`, the interpreter recursively evaluates `B(X)` in the first case; if it succeeds, it then finds out that the constraint store is inconsistent when adding `X>0`. In the second case, when evaluating that clause, the interpreter first adds `X>0` to the constraint store and then possibly evaluates `B(X)`. Since the constraint store after the addition of `X>0` turns out to be inconsistent, the recursive evaluation of `B(X)` is not performed at all.
|
|
Because optimal vertex orderings are hard to find, heuristics have been used that attempt to reduce the number of colors while not guaranteeing an optimal number of colors. A commonly used ordering for greedy coloring is to choose a vertex of minimum degree, order the subgraph with removed recursively, and then place last in the ordering. The largest degree of a removed vertex that this algorithm encounters is called the degeneracy of the graph, denoted . In the context of greedy coloring, the same ordering strategy is also called the smallest last ordering.
|
|
The boldface Π01 classes are exactly the same as the closed sets of 2ω and thus the same as the boldface Π01 subsets of 2ω in the Borel hierarchy. Lightface Π01 classes in 2ω (that is, Π01 classes whose tree is computable with no oracle) correspond to effectively closed sets. A subset B of 2ω is effectively closed if there is a recursively enumerable sequence ⟨σi : i ∈ ω⟩ of elements of 2< ω such that each g ∈ 2ω is in B if and only if σi is an initial segment of B.
|
|
Wget can optionally work like a web crawler by extracting resources linked from HTML pages and downloading them in sequence, repeating the process recursively until all the pages have been downloaded or a maximum recursion depth specified by the user has been reached. The downloaded pages are saved in a directory structure resembling that on the remote server. This "recursive download" enables partial or complete mirroring of web sites via HTTP. Links in downloaded HTML pages can be adjusted to point to locally downloaded material for offline viewing.
|
|
For example, in 2D cases, scene graphs typically render themselves by starting at the tree's root node and then recursively draw the child nodes. The tree's leaves represent the most foreground objects. Since drawing proceeds from back to front with closer objects simply overwriting farther ones, the process is known as employing the Painter's algorithm. In 3D systems, which often employ depth buffers, it is more efficient to draw the closest objects first, since farther objects often need only be depth-tested instead of actually rendered, because they are occluded by nearer objects.
|
|
Randomized depth-first search on a hexagonal grid The depth-first search algorithm of maze generation is frequently implemented using backtracking. This can be described with a following recursive routine: # Given a current cell as a parameter, # Mark the current cell as visited # While the current cell has any unvisited neighbour cells ## Choose one of the unvisited neighbours ## Remove the wall between the current cell and the chosen cell ## Invoke the routine recursively for a chosen cell which is invoked once for any initial cell in the area.
|
|
Automated Retroactive Minimal Moderation (ARMM) was a program developed by Richard Depew in 1993 to aid in the control of Usenet abuse. Concerned by abusive posts emanating from certain anonymous-posting sites, Depew developed ARMM to allow news administrators to automatically issue cancel messages for such posts. This was a controversial act, as many news administrators and users were concerned about censorship of the netnews medium. An early version of ARMM contained a bug which caused it to post follow-ups to its own messages, recursively sending posts to the news.admin.
|
|
Church-encoded data and operations on them are typable in system F, but Scott-encoded data and operations are not obviously typable in system F. Universal as well as recursive types appear to be required,See the note "Types for the Scott numerals" by Martín Abadi, Luca Cardelli and Gordon Plotkin (February 18, 1993).. As strong normalization does not hold for unrestricted recursive types , establishing termination of programs manipulating Scott-encoded data by determining well-typedness requires the type system provide additional restrictions on the formation of recursively typed terms.
|
|
In linear algebra, a mapping that preserves a specified property is called a transformation, and that is the sense in which Harris introduced the term into linguistics. Harris's transformational analysis refined the word classes found in the 1946 "From Morpheme to Utterance" grammar of expansions. By recursively defining semantically more and more specific subclasses according to the combinatorial privileges of words, one may progressively approximate a grammar of individual word combinations. One form in which this is exemplified is in the lexicon- grammar work of Maurice Gross and his colleagues e.g.
|
|
Fractal landscape Not only do animated images form part of computer-generated imagery, natural looking landscapes (such as fractal landscapes) are also generated via computer algorithms. A simple way to generate fractal surfaces is to use an extension of the triangular mesh method, relying on the construction of some special case of a de Rham curve, e.g. midpoint displacement. For instance, the algorithm may start with a large triangle, then recursively zoom in by dividing it into four smaller Sierpinski triangles, then interpolate the height of each point from its nearest neighbors.
|
|
Given a function (or, similarly, a set), one may be interested not only if it is computable, but also whether this can be proven in a particular proof system (usually first order Peano arithmetic). A function that can be proven to be computable is called provably total. The set of provably total functions is recursively enumerable: one can enumerate all the provably total functions by enumerating all their corresponding proofs, that prove their computability. This can be done by enumerating all the proofs of the proof system and ignoring irrelevant ones.
|
|
Finally, F is taken to be \bigcup F_k. The remainder of the proof consists of a verification that F is recursively enumerable and is the least fixed point of Φ. The sequence Fk used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem. The second part of the first recursion theorem follows from the first part. The assumption that Φ is a recursive operator is used to show that the fixed point of Φ is the graph of a partial function.
|
|
Further reducibilities (positive, disjunctive, conjunctive, linear and their weak and bounded versions) are discussed in the article Reduction (recursion theory). The major research on strong reducibilities has been to compare their theories, both for the class of all recursively enumerable sets as well as for the class of all subsets of the natural numbers. Furthermore, the relations between the reducibilities has been studied. For example, it is known that every Turing degree is either a truth-table degree or is the union of infinitely many truth-table degrees.
|
|
There are close relationships between the Turing degree of a set of natural numbers and the difficulty (in terms of the arithmetical hierarchy) of defining that set using a first-order formula. One such relationship is made precise by Post's theorem. A weaker relationship was demonstrated by Kurt Gödel in the proofs of his completeness theorem and incompleteness theorems. Gödel's proofs show that the set of logical consequences of an effective first-order theory is a recursively enumerable set, and that if the theory is strong enough this set will be uncomputable.
|
|
Some intelligence technologies, like "seed AI", may also have the potential to not just make themselves faster, but also more efficient, by modifying their source code. These improvements would make further improvements possible, which would make further improvements possible, and so on. The mechanism for a recursively self-improving set of algorithms differs from an increase in raw computation speed in two ways. First, it does not require external influence: machines designing faster hardware would still require humans to create the improved hardware, or to program factories appropriately.
|
|
A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2005, p. 112). This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC). The theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic.
|
|
Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980). Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).
|
|
Structure charts are used in structured analysis to specify the high-level design, or architecture, of a computer program. As a design tool, they aid the programmer in dividing and conquering a large software problem, that is, recursively breaking a problem down into parts that are small enough to be understood by a human brain. The process is called top- down design, or functional decomposition. Programmers use a structure chart to build a program in a manner similar to how an architect uses a blueprint to build a house.
|
|
In a variation of the standard chain- of-responsibility model, some handlers may act as dispatchers, capable of sending commands out in a variety of directions, forming a tree of responsibility. In some cases, this can occur recursively, with processing objects calling higher-up processing objects with commands that attempt to solve some smaller part of the problem; in this case recursion continues until the command is processed, or the entire tree has been explored. An XML interpreter might work in this manner. This pattern promotes the idea of loose coupling.
|
|
An illustration of the fork–join paradigm, in which three regions of the program permit parallel execution of the variously colored blocks. Sequential execution is displayed on the top, while its equivalent fork–join execution is on the bottom. In parallel computing, the fork–join model is a way of setting up and executing parallel programs, such that execution branches off in parallel at designated points in the program, to "join" (merge) at a subsequent point and resume sequential execution. Parallel sections may fork recursively until a certain task granularity is reached.
|
|
In finance, bootstrapping is a method for constructing a (zero-coupon) fixed- income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps. A bootstrapped curve, correspondingly, is one where the prices of the instruments used as an input to the curve, will be an exact output, when these same instruments are valued using this curve. Here, the term structure of spot returns is recovered from the bond yields by solving for them recursively, by forward substitution: this iterative process is called the bootstrap method.
|
|
In 1970, she married fellow mathematician Richard Larson, was diagnosed with breast cancer in 1974 and in 1975 was promoted to full professor. She published prolifically throughout the 1970s on recursively enumerable sets and introduced the concept of the "weak jump," a generalization of the Halting problem distinct from the usual notion of the Turing jump. She also proved analogues of Rice and Rice-Shapiro theorems, as well as working on theories of computational complexity theory. Her work was influential with both Soviet and US mathematicians of the period.
|
|
In formal language theory, a cone is a set of formal languages that has some desirable closure properties enjoyed by some well-known sets of languages, in particular by the families of regular languages, context-free languages and the recursively enumerable languages. The concept of a cone is a more abstract notion that subsumes all of these families. A similar notion is the faithful cone, having somewhat relaxed conditions. For example, the context-sensitive languages do not form a cone, but still have the required properties to form a faithful cone.
|
|
Within these two subsequences, the path can be constructed recursively by the same rule, linking the two subsequences at the ends of the subsequences at which the second bit is 0. Thus, e.g., in the Fibonacci cube of order 4, the sequence constructed in this way is (0100-0101-0001-0000-0010)-(1010-1000-1001), where the parentheses demark the subsequences within the two subgraphs of the partition. Fibonacci cubes with an even number of nodes greater than two have a Hamiltonian cycle.. investigate the radius and independence number of Fibonacci cubes.
|
|
First, a connectionist knowledge representation is created as a semantic network consisting of concepts and their relations to serve as the basis for the representation of meaning.Johannes Fähndrich est First Search Planning of Service Composition Using Incrementally Redefined Context-Dependent Heuristics. In the German Conference Multiagent System Technologies, pages 404-407, Springer Berlin Heidelberg, 2013 This graph is built out of different knowledge sources like WordNet, Wiktionary, and BabelNET. The graph is created by lexical decomposition that recursively breaks each concept semantically down into a set of semantic primes.
|
|
Fortunately, there exists such a recursive decomposition of a graph that implicitly represents all ways of decomposing it; this is the modular decomposition. It is itself a way of decomposing a graph recursively into quotients, but it subsumes all others. The decomposition depicted in the figure below is this special decomposition for the given graph. A graph, its quotient where "bags" of vertices of the graph correspond to the children of the root of the modular decomposition tree, and its full modular decomposition tree: series nodes are labeled "s", parallel nodes "//" and prime nodes "p".
|
|
However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a straight line to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0.5 and 0.5 ≤ t ≤ 1, and the same procedure is applied recursively to each half.
|
|
The divide and conquer algorithm computes the smaller multiplications recursively, using the scalar multiplication as its base case. The complexity of this algorithm as a function of is given by the recurrence :T(1) = \Theta(1); :T(n) = 8T(n/2) + \Theta(n^2), accounting for the eight recursive calls on matrices of size and to sum the four pairs of resulting matrices element-wise. Application of the master theorem for divide-and-conquer recurrences shows this recursion to have the solution , the same as the iterative algorithm.
|
|
For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. A linear arrangement formed by recursively partitioning each of these two subforests, placing the separating vertices between them, has logarithmic vertex searching number. The same technique, applied to a tree-decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the pathwidth of G is O(t log n)., Theorem 6, p.
|
|
This is necessary because, as a recursively implemented FIR filter, a CIC filter relies on exact cancellation of poles from the integrator sections by zeros from the comb sections. While the reasons are less than intuitive, an inherent characteristic of the CIC architecture is that if fixed bit length overflows occur in the integrators, they are corrected in the comb sections. The range of filter shapes and responses available from a CIC filter is somewhat limited. Larger amounts of stopband rejection can be achieved by increasing the number of poles.
|
|
For example, assuming that the value of 0 represents a parent node and 1 a leaf node, whenever the latter is encountered the tree building routine simply reads the next 8 bits to determine the character value of that particular leaf. The process continues recursively until the last leaf node is reached; at that point, the Huffman tree will thus be faithfully reconstructed. The overhead using such a method ranges from roughly 2 to 320 bytes (assuming an 8-bit alphabet). Many other techniques are possible as well.
|
|
GOAL's syntax resembles the Lisp dialect Scheme, though with many idiosyncratic object-oriented programming features such as classes, inheritance, and virtual functions. GOAL encourages an imperative programming style: programs tend to consist of a sequence of events to be executed rather than the functional programming style of functions to be evaluated recursively. This is a diversion from Scheme, which allows such side effects but does not encourage imperative style. GOAL does not run in an interpreter, but instead is compiled directly into PlayStation 2 machine code to execute.
|
|
Steps 1-2: Divide points in two subsets Under average circumstances the algorithm works quite well, but processing usually becomes slow in cases of high symmetry or points lying on the circumference of a circle. The algorithm can be broken down to the following steps: # Find the points with minimum and maximum x coordinates, as these will always be part of the convex hull. If many points with the same minimum/maximum x exist, use ones with minimum/maximum y correspondingly. # Use the line formed by the two points to divide the set in two subsets of points, which will be processed recursively.
|
|
For higher-dimensional regular polytopes, the Schläfli symbol is defined recursively as {p1, p2,...,pn − 1} if the facets have Schläfli symbol {p1,p2,...,pn − 2} and the vertex figures have Schläfli symbol {p2,p3,...,pn − 1}. A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytope are the same: {p2,p3,...,pn − 2}. There are only 3 regular polytopes in 5 dimensions and above: the simplex, {3,3,3,...,3}; the cross- polytope, {3,3, ..., 3,4}; and the hypercube, {4,3,3,...,3}. There are no non- convex regular polytopes above 4 dimensions.
|
|
The standard decimation-in-frequency (DIF) radix-r Cooley–Tukey algorithm corresponds closely to a recursive factorization. For example, radix-2 DIF Cooley–Tukey factors z^N-1 into F_1 = (z^{N/2}-1) and F_2 = (z^{N/2}+1). These modulo operations reduce the degree of x(z) by 2, which corresponds to dividing the problem size by 2. Instead of recursively factorizing F_2 directly, though, Cooley–Tukey instead first computes x2(z ωN), shifting all the roots (by a twiddle factor) so that it can apply the recursive factorization of F_1 to both subproblems.
|
|
Regardless of N, exactly n−1 additions are performed in total, the same as for naive summation, so if the recursion overhead is made negligible then pairwise summation has essentially the same computational cost as for naive summation. A variation on this idea is to break the sum into b blocks at each recursive stage, summing each block recursively, and then summing the results, which was dubbed a "superblock" algorithm by its proposers.Anthony M. Castaldo, R. Clint Whaley, and Anthony T. Chronopoulos, "Reducing floating-point error in dot product using the superblock family of algorithms," SIAM J. Sci. Comput., vol.
|
|
As a precomputation, we can take each physical body (represented by a set of triangles) and recursively decompose it into a binary tree, where each node N represents a set of triangles, and its two children represent L(N) and R(N). At each node in the tree, we can precompute the bounding sphere B(N). When the time comes for testing a pair of objects for collision, their bounding sphere tree can be used to eliminate many pairs of triangles. Many variants of the algorithms are obtained by choosing something other than a sphere for B(T).
|
|
Thus solving is reduced to the simpler problems of solving and . Conversely, the factor theorem asserts that, if is a root of , then may be factored as :P(x)=(x-r)Q(x), where is the quotient of Euclidean division of by . If the coefficients of are real or complex numbers, the fundamental theorem of algebra asserts that has a real or complex root. Using the factor theorem recursively, it results that :P(x)=a_0(x-r_1)\cdots (x-r_n), where r_1, \ldots, r_n are the real or complex roots of , with some of them possibly repeated.
|
|
This method can be used as a stop-the-world mechanism for parallel programs, and also with a concurrent reference counting collector. ; Not real-time: Naive implementations of reference counting do not in general provide real-time behavior, because any pointer assignment can potentially cause a number of objects bounded only by total allocated memory size to be recursively freed while the thread is unable to perform other work. It is possible to avoid this issue by delegating the freeing of objects whose reference count dropped to zero to other threads, at the cost of extra overhead.
|
|
Epub 2008 Jan 8. Furthermore, building from the Dehaene-Changeux Model, Werner (2007b) proposed that the application of the twin concepts of scaling and universality of the theory of non-equilibrium phase transitions can serve as an informative approach for elucidating the nature of underlying neural-mechanisms, with emphasis on the dynamics of recursively reentrant activity flow in intracortical and cortico-subcortical neuronal loops. Friston (2000) also claimed that "the nonlinear nature of asynchronous coupling enables the rich, context-sensitive interactions that characterize real brain dynamics, suggesting that it plays a role in functional integration that may be as important as synchronous interactions".
|
|
A conventional third-generation computer is recursively virtualizable if: # it is virtualizable and # a VMM without any timing dependencies can be constructed for it. Some architectures, like the non-hardware-assisted x86, do not meet these conditions, so they cannot be virtualized in the classic way. But architectures can still be fully virtualized (in the x86 case meaning at the CPU and MMU level) by using different techniques like binary translation, which replaces the sensitive instructions that do not generate traps, which are sometimes called critical instructions. This additional processing however makes the VMM less efficient in theory,Smith and Nair, p.
|
|
These domains reflect whether the cells have nuclei or not, as well as differences in the chemical composition of key biomolecules such as ribosomes. Further, each kingdom is broken down recursively until each species is separately classified. The order is: Domain; Kingdom; Phylum; Class; Order; Family; Genus; Species. Outside of these categories, there are obligate intracellular parasites that are "on the edge of life" in terms of metabolic activity, meaning that many scientists do not actually classify such structures as alive, due to their lack of at least one or more of the fundamental functions or characteristics that define life.
|
|
Higman's embedding theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely presented group with algorithmically undecidable word problem. Indeed, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem. Then any finitely presented group that contains this group as a subgroup will have undecidable word problem as well. The usual proof of the theorem uses a sequence of HNN extensions starting with R and ending with a group G which can be shown to have a finite presentation.
|
|
Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers, including their addition and multiplication, axiomatized by the first-order Peano axioms. This is a "first-order" theory: the quantifiers extend over natural numbers, but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence. Gentzen showed that the consistency of the first-order Peano axioms is provable over the base theory of primitive recursive arithmetic with the additional principle of quantifier- free transfinite induction up to the ordinal ε0.
|
|
A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations). The language of a propositional calculus consists of # a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and # a set of operator symbols, variously interpreted as logical operators or logical connectives. A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.
|
|
Like in a k-d tree, updates in a K-D-B-tree may result in the requirement for the splitting of several nodes recursively. This is incredibly inefficient and can result in sub-optimal memory utilization as it may result in many near-empty leaves. Lomet and Salzberg proposed a structure called the hB-tree (holey brick tree) to improve performance of K-D-B-trees by limiting the splits that occur after an insertion to only one root-to-leaf path. This was achieved by storing regions not only as rectangles, but as rectangles with a rectangle removed from the center.
|
|
Although a normal Voronoi cell is defined as the set of points closest to a single point in S, an nth-order Voronoi cell is defined as the set of points having a particular set of n points in S as its n nearest neighbors. Higher- order Voronoi diagrams also subdivide space. Higher-order Voronoi diagrams can be generated recursively. To generate the nth-order Voronoi diagram from set S, start with the (n − 1)th-order diagram and replace each cell generated by X = {x1, x2, ..., xn−1} with a Voronoi diagram generated on the set S − X.
|
|
The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions, and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k > 0, and there are examples to show that this containment is strict (Ck ⊊ Ck−1).
|
|
Since we have only one equation but n variables, infinitely many solutions exist (and are easy to find) in the complex plane; however, the problem becomes impossible if solutions are constrained to integer values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Gödel's Incompleteness Theorem. In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term. This result was later generalized by Rice's theorem.
|
|
As for the integers, the Euclidean division allows us to define Euclid's algorithm for computing GCDs. Starting from two polynomials a and b, Euclid's algorithm consists of recursively replacing the pair by (where "" denotes the remainder of the Euclidean division, computed by the algorithm of the preceding section), until b = 0. The GCD is the last non zero remainder. Euclid's algorithm may be formalized in the recursive programming style as: In the imperative programming style, the same algorithm becomes, giving a name to each intermediate remainder: The sequence of the degrees of the is strictly decreasing.
|
|
HNN-extensions play a key role in Higman's proof of the Higman embedding theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group. Most modern proofs of the Novikov–Boone theorem about the existence of a finitely presented group with algorithmically undecidable word problem also substantially use HNN-extensions. Both HNN- extensions and amalgamated free products are basic building blocks in the Bass–Serre theory of groups acting on trees. The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.
|
|
The fast Walsh–Hadamard transform applied to a vector of length 8 Example for the input vector (1, 0, 1, 0, 0, 1, 1, 0) In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHTh) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order n = 2^m would have a computational complexity of O(n^2). The FWHTh requires only n \log n additions or subtractions. The FWHTh is a divide and conquer algorithm that recursively breaks down a WHT of size n into two smaller WHTs of size n/2.
|
|
A general solution for this problem is that of searching the space of tableaux until a closed one is found (if any exists, that is, the set is unsatisfiable). In this approach, one starts with an empty tableau and then recursively applies every possible applicable rule. This procedure visits a (implicit) tree whose nodes are labeled with tableaux, and such that the tableau in a node is obtained from the tableau in its parent by applying one of the valid rules. Since each branch can be infinite, this tree has to be visited breadth-first rather than depth-first.
|
|
The representations of a program are computed recursively by nested vector addition. The representations for parser p2 (whose GenVoca expression is j+f) are: p2 = j + f -- GenVoca expression = [\Deltagj, \Deltasj, \Deltadj] + [gf, sf, df] -- substitution = [\Deltagj+gf, \Deltasj+sf, \Deltadj+df] -- compose tuples element-wise That is, the grammar of p2 is the base grammar composed with its extension (\Deltagj+gf), the source of p2 is the base source composed with its extension (\Deltasj+sf), and so on. As elements of delta tuples can themselves be delta tuples, composition recurses, e.g., \Deltasj+sf= [\Deltac1…\Deltacn]+[c1…cn]=[\Deltac1+c1…\Deltacn+cn].
|
|
Any n-vertex forest has tree- depth O(log n). For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. By recursively partitioning each of these two subforests, we can easily derive a logarithmic upper bound on the tree-depth. The same technique, applied to a tree decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the tree-depth of G is O(t log n).
|
|
For example, in one shows that the fuzzy Turing machines are not adequate for fuzzy language theory since there are natural fuzzy languages intuitively computable that cannot be recognized by a fuzzy Turing Machine. Then, they proposed the following definitions. Denote by Ü the set of rational numbers in [0,1]. Then a fuzzy subset s : S \rightarrow[0,1] of a set S is recursively enumerable if a recursive map h : S×N \rightarrowÜ exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n).
|
|
This clause states one condition under which the statement `A(X,Y)` holds: `X+Y` is greater than zero and both `B(X)` and `C(Y)` are true. As in regular logic programming, programs are queried about the provability of a goal, which may contain constraints in addition to literals. A proof for a goal is composed of clauses whose bodies are satisfiable constraints and literals that can in turn be proved using other clauses. Execution is performed by an interpreter, which starts from the goal and recursively scans the clauses trying to prove the goal.
|
|
This file is available for download on various websites across the Internet. In many anti-virus scanners, only a few layers of recursion are performed on archives to help prevent attacks that would cause a buffer overflow, an out-of-memory condition, or exceed an acceptable amount of program execution time. Zip bombs often (if not always) rely on repetition of identical files to achieve their extreme compression ratios. Dynamic programming methods can be employed to limit traversal of such files, so that only one file is followed recursively at each level, effectively converting their exponential growth to linear.
|
|
A 1-dimensional range tree on a set of n points is a binary search tree, which can be constructed in O(n \log n) time. Range trees in higher dimensions are constructed recursively by constructing a balanced binary search tree on the first coordinate of the points, and then, for each vertex v in this tree, constructing a (d−1)-dimensional range tree on the points contained in the subtree of v. Constructing a range tree this way would require O(n \log ^d n) time. This construction time can be improved for 2-dimensional range trees to O(n \log n).
|
|
Another benefit is that by providing reentrancy, recursion is automatically supported. When a function calls itself recursively, a return address needs to be stored for each activation of the function so that it can later be used to return from the function activation. Stack structures provide this capability automatically. Depending on the language, operating-system, and machine environment, a call stack may serve additional purposes, including for example: ; Local data storage : A subroutine frequently needs memory space for storing the values of local variables, the variables that are known only within the active subroutine and do not retain values after it returns.
|
|
Because primitive recursive functions use natural numbers rather than integers, and the natural numbers are not closed under subtraction, a truncated subtraction function (also called "proper subtraction") is studied in this context. This limited subtraction function sub(a, b) [or b ∸ a] returns b - a if this is nonnegative and returns 0 otherwise. The predecessor function acts as the opposite of the successor function and is recursively defined by the rules: :pred(0) = 0, :pred(n + 1) = n. These rules can be converted into a more formal definition by primitive recursion: :pred(0) = 0, :pred(S(n)) = P12(n, pred(n)).
|
|
Elements are distributed among bins Then, elements are sorted within each bin Bucket sort, or bin sort, is a sorting algorithm that works by distributing the elements of an array into a number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting algorithm. It is a distribution sort, a generalization of pigeonhole sort, and is a cousin of radix sort in the most- to-least significant digit flavor. Bucket sort can be implemented with comparisons and therefore can also be considered a comparison sort algorithm.
|
|
Systems in the special strict-feedback form have a recursive structure similar to the many-integrator system structure. Likewise, they are stabilized by stabilizing the smallest cascaded system and then backstepping to the next cascaded system and repeating the procedure. So it is critical to develop a single-step procedure; that procedure can be recursively applied to cover the many-step case. Fortunately, due to the requirements on the functions in the strict-feedback form, each single-step system can be rendered by feedback to a single- integrator system, and that single-integrator system can be stabilized using methods discussed above.
|
|
In computations with rounded arithmetic, e.g. with floating-point numbers, a divide-and- conquer algorithm may yield more accurate results than a superficially equivalent iterative method. For example, one can add N numbers either by a simple loop that adds each datum to a single variable, or by a D&C; algorithm called pairwise summation that breaks the data set into two halves, recursively computes the sum of each half, and then adds the two sums. While the second method performs the same number of additions as the first, and pays the overhead of the recursive calls, it is usually more accurate.
|
|
An example for merge sort The merge algorithm plays a critical role in the merge sort algorithm, a comparison-based sorting algorithm. Conceptually, merge sort algorithm consists of two steps: # Recursively divide the list into sublists of (roughly) equal length, until each sublist contains only one element, or in the case of iterative (bottom up) merge sort, consider a list of n elements as n sub-lists of size 1. A list containing a single element is, by definition, sorted. # Repeatedly merge sublists to create a new sorted sublist until the single list contains all elements.
|
|
Lower elementary recursive functions follow the definitions as above, except that bounded product is disallowed. That is, a lower elementary recursive function must be a zero, successor, or projection function, a composition of other lower elementary recursive functions, or the bounded sum of another lower elementary recursive function. Also known as Skolem elementary functions.Th. Skolem, "Proof of some theorems on recursively enumerable sets", Notre Dame Journal of Formal Logic, 1962, Volume 3, Number 2, pp 65-74, .S. A. Volkov, "On the class of Skolem elementary functions", Journal of Applied and Industrial Mathematics, 2010, Volume 4, Issue 4, pp 588-599, .
|
|
When combining a non-executable stack with mmap() base randomization, the difficulty in exploiting bugs protected against by PaX is greatly increased due to the forced use of return- to-libc attacks. On 32-bit systems, this amounts to 16 orders of magnitude; that is, the chances of success are recursively halved 16 times. Combined with stack randomization, the effect can be quite astounding; if every person in the world (assuming 6 billion total) attacks the system once, roughly 1 to 2 should succeed on a 32-bit system. 64-bit systems of course benefit from greater randomization.
|
|
Recursive transpiling (or recursive transcompiling) is the process of applying the notion of transpiling recursively, to create a pipeline of transformations (often starting from a single source of truth) which repeatedly turn one technology into another. By repeating this process, one can turn A → B → C → D → E → F and then back into A(v2). Some information will be preserved through this pipeline, from A → A(v2), and that information (at an abstract level) demonstrates what each of the components A–F agree on. In each of the different versions that the transcompiler pipeline produces, that information is preserved.
|
|
It is known to be undecidable when 9 pairs are used (however, Stephen Wolfram (2002) suggested that it is also undecidable with just 3 pairs). The undecidability of his Post correspondence problem turned out to be exactly what was needed to obtain undecidability results in the theory of formal languages. In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research.
|
|
The modification is performed directly on the new node, without using the modification box. (One of the new node's fields overwritten and its modification box stays empty.) Finally, this change is cascaded to the node's parent, just like path copying. (This may involve filling the parent's modification box, or making a copy of the parent recursively. If the node has no parent—it's the root—it is added the new root to a sorted array of roots.) With this algorithm, given any time t, at most one modification box exists in the data structure with time t.
|
|
Typically, a cache-oblivious algorithm works by a recursive divide and conquer algorithm, where the problem is divided into smaller and smaller subproblems. Eventually, one reaches a subproblem size that fits into cache, regardless of the cache size. For example, an optimal cache-oblivious matrix multiplication is obtained by recursively dividing each matrix into four sub-matrices to be multiplied, multiplying the submatrices in a depth-first fashion. In tuning for a specific machine, one may use a hybrid algorithm which uses blocking tuned for the specific cache sizes at the bottom level, but otherwise uses the cache-oblivious algorithm.
|
|
This process is repeated recursively, which is the source of the cascade name. After all blocks have been compared, Alice and Bob both reorder their keys in the same random way, and a new round begins. At the end of multiple rounds Alice and Bob have identical keys with high probability; however, Eve has additional information about the key from the parity information exchanged. However, from a coding theory point of view information reconciliation is essentially source coding with side information, in consequence any coding scheme that works for this problem can be used for information reconciliation.
|
|
Then x \in C \implies Gx = a and similarly, x otin C \implies Gx = b. By the Second Recursion Theorem, there is a term X which is equal to f applied to the Church numeral of its Gödel numbering, X. Then X \in C implies that X = G(X') = b so in fact X otin C. The reverse assumption X otin C gives X = G(X') = a so X \in C. Either way we arise at a contradiction, and so f cannot be a function which separates A and B. Hence A and B are recursively inseparable.
|
|
Robin Hanson expressed skepticism of human intelligence augmentation, writing that once the "low-hanging fruit" of easy methods for increasing human intelligence have been exhausted, further improvements will become increasingly difficult to find. Despite all of the speculated ways for amplifying human intelligence, non-human artificial intelligence (specifically seed AI) is the most popular option among the hypotheses that would advance the singularity. Whether or not an intelligence explosion occurs depends on three factors.David Chalmers John Locke Lecture, 10 May, Exam Schools, Oxford, Presenting a philosophical analysis of the possibility of a technological singularity or "intelligence explosion" resulting from recursively self-improving AI .
|
|
By structuring data in such cons-lists, these languages facilitate recursive means for building and processing data—for example, by recursively accessing the head and tail elements of lists of lists; e.g. "taking the car of the cdr of the cdr". By contrast, memory management based on pointer dereferencing in some approximation of an array of memory addresses facilitates treating variables as slots into which data can be assigned imperatively. When dealing with arrays, the critical lookup operation typically involves a stage called address calculation which involves constructing a pointer to the desired data element in the array.
|
|
A Turing machine is a general example of a central processing unit (CPU) that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More specifically, it is a machine (automaton) capable of enumerating some arbitrary subset of valid strings of an alphabet; these strings are part of a recursively enumerable set. A Turing machine has a tape of infinite length on which it can perform read and write operations. Assuming a black box, the Turing machine cannot know whether it will eventually enumerate any one specific string of the subset with a given program.
|
|
Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree. Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side.
|
|
Terminal symbols are literal symbols which may appear in the outputs of the production rules of a formal grammar and which cannot be changed using the rules of the grammar. Applying the rules recursively to a source string of symbols will usually terminate in a final output string consisting only of terminal symbols. Consider a grammar defined by two rules. Using pictoric marks interacting with each other: # The symbol `ר` can become `ди` # The symbol `ר` can become `д` Here `д` is a terminal symbol because no rule exists which would change it into something else.
|
|
While the O(n2) time taken by Knuth's algorithm is substantially better than the exponential time required for a brute-force search, it is still too slow to be practical when the number of elements in the tree is very large. In 1975, Kurt Mehlhorn published a paper proving that a much simpler algorithm could be used to closely approximate the statically optimal tree in only time. In this algorithm, the root of the tree is chosen so as to most closely balance the total weight (by probability) of the left and right subtrees. This strategy is then applied recursively on each subtree.
|
|
Elias ω coding or Elias omega coding is a universal code encoding the positive integers developed by Peter Elias. Like Elias gamma coding and Elias delta coding, it works by prefixing the integer with a representation of its order of magnitude in a universal code. Unlike those other two codes, however, Elias omega recursively encodes that prefix; thus, they are sometimes known as recursive Elias codes. Omega coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values.
|
|
For instance, the OSType code `inte` indicates that the data was a four-byte signed integer in big-endian format. Besides predefined type codes for various common simple types, there are two predefined structured descriptor types: an AERecord, which has data type `reco` (record), and AEList with type `list` (list or array). The internal structure of these contain recursively-nested AEDescs, while the AERecord also associates each element with a unique record field ID, which is an OSType. The Apple Event Manager provides API calls to construct these structures, as well as extract their contents and query the type of contents they hold.
|
|
Because of the large constant factors arising in the analysis of the AKS sorting network, parametric search using this network as the test algorithm is not practical. Instead, suggest using a parallel form of quicksort (an algorithm that repeatedly partitions the input into two subsets and then recursively sorts each subset). In this algorithm, the partition step is massively parallel (each input element should be compared to a chosen pivot element) and the two recursive calls can be performed in parallel with each other. The resulting parametric algorithm is slower in the worst case than an algorithm based on the AKS sorting network.
|
|
A. thaliana has been extensively studied as a model for flower development. The developing flower has four basic organs: sepals, petals, stamens, and carpels (which go on to form pistils). These organs are arranged in a series of whorls: four sepals on the outer whorl, followed by four petals inside this, six stamens, and a central carpel region. Homeotic mutations in A. thaliana result in the change of one organ to another—in the case of the agamous mutation, for example, stamens become petals and carpels are replaced with a new flower, resulting in a recursively repeated sepal-petal-petal pattern.
|
|
For factoring a multivariate polynomial over a field or over the integers, one may consider it as a univariate polynomial with coefficients in a polynomial ring with one less indeterminate. Then the factorization is reduced to factorizing separately the primitive part and the content. As the content has one less indeterminate, it may be factorized by applying the method recursively. For factorizing the primitive part, the standard method consists of substituting integers to the indeterminates of the coefficients in a way that does not change the degree in the remaining variable, factorizing the resulting univariate polynomial, and lifting the result to a factorization of the primitive part.
|
|
The unrestricted grammars characterize the recursively enumerable languages. This is the same as saying that for every unrestricted grammar G there exists some Turing machine capable of recognizing L(G) and vice versa. Given an unrestricted grammar, such a Turing machine is simple enough to construct, as a two-tape nondeterministic Turing machine. The first tape contains the input word w to be tested, and the second tape is used by the machine to generate sentential forms from G. The Turing machine then does the following: # Start at the left of the second tape and repeatedly choose to move right or select the current position on the tape.
|
|
Rather than telling the story of a few people, Romains' aim is "to paint a complete picture of our twentieth century civilization, in all its aspects, human and inhuman, social as well as psychological". Plot threads, too, are begun, recursively interrupted, and returned to later, sometimes several volumes on. This cross-volume inter- cutting presses the reader to take the work as a whole, since plots are not always resolved within single volumes, which therefore cannot be fully understood without reference to the others. In all, there are approximately 40 main characters but counts of the full cast vary widely, from 600 total through 1,000 to 1,600.
|
|
This algorithm first generates a new pair of public and secret keys for the homomorphic encryption scheme, and then uses these keys with the homomorphic scheme to encrypt the correct input wires, represented as the secret key of the garbled circuit. The produced ciphertexts represent the public encoding of the input (σx) that is given to the worker, while the secret key (τx) is kept private by the client. After that, the worker applies the computation steps of the Yao's protocol over the ciphertexts generated by the problem generation algorithm. This is done by recursively decrypting the gate ciphertexts until arriving to the final output wire values (σy).
|
|
In pseudocode, the pairwise summation algorithm for an array of length > 0 can be written: s = pairwise(x[1…n]) if n ≤ N base case: naive summation for a sufficiently small array s = x[1] for i = 2 to n s = s + x[i] else divide and conquer: recursively sum two halves of the array m = floor(n / 2) s = pairwise(x[1…m]) + pairwise(x[m+1…n]) end if The non-recursive version of the algorithm uses a stack to accumulate partial sums: partials = new Stack for i = 1 to n partials.push(x[i]) j = i while is_even(j) j = floor(j / 2) p = partials.pop() q = partials.pop() partials.
|
|
Although most space-filling curves are not Osgood curves (they have positive area but often include infinitely many self- intersections, failing to be Jordan curves) it is possible to modify the recursive construction of space-filling curves or other fractal curves to obtain an Osgood curve.; ; ). For instance, Knopp's construction involves recursively splitting triangles into pairs of smaller triangles, meeting at a shared vertex, by removing triangular wedges. When the removed wedges at each level of this construction cover the same fraction of the area of their triangles, the result is a Cesàro fractal such as the Koch snowflake, but removing wedges whose areas shrink more rapidly produces an Osgood curve.
|
|
Gerald Enoch Sacks (1933 – October 4, 2019) was a logician whose most important contributions were in recursion theory. Named after him is Sacks forcing, a forcing notion based on perfect sets. and the Sacks Density Theorem, which asserts that the partial order of the recursively enumerable Turing degrees is dense.. Sacks had a joint appointment as a professor at the Massachusetts Institute of Technology and at Harvard University starting in 1972 and became emeritus at M.I.T. in 2006 and at Harvard in 2012.Short CV, retrieved 2015-06-26..Chi Tat Chong, Yue Yang, "An interview with Gerald E. Sacks", Recursion Theory: Computational Aspects of Definability, , 2015, p.
|
|
TPS (later renamed to "Interleaf 5," up through "Interleaf 7") was an integrated, networked text-and-graphics document creation system initially designed for technical publishing departments. Versions after its first release in 1984 added instantaneous updating of page numbering and reference numbers through multi-chapter and multi-volumes sets, increased graphics capabilities, automatic index and table of content generation, hyphenation, equations, "microdocuments" that recursively allowed fully functional whole document elements to be embedded in any document, and the ability to program any element of a document (a capability the company called "Active Documents"). Interleaf software was available in many languages including Japanese text layout. TPS was a structured document editor.
|
|
This can be applied recursively, as done in the radix-2 FFT and the Fast Walsh–Hadamard transform. Splitting a known matrix into the Hadamard product of two smaller matrices is known as the "nearest Kronecker Product" problem, and can be solved exactly by using the SVD. To split a matrix into the Hadamard product of more than two matrices, in an optimal fashion, is a difficult problem and the subject of ongoing research; some authors cast it as a tensor decomposition problem. In conjunction with the least squares method, the Kronecker product can be used as an accurate solution to the hand eye calibration problem.
|
|
Nodes are placed on a stack in the order in which they are visited. When the depth-first search recursively visits a node `v` and its descendants, those nodes are not all necessarily popped from the stack when this recursive call returns. The crucial invariant property is that a node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node earlier on the stack. In other words it means that in the DFS a node would be only removed from the stack after all its connected paths have been traversed.
|
|
The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. It is not expected that Euler's number e and Euler-Mascheroni constant γ are periods. The periods can be extended to exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions.
|
|
Alternatively, the information may be provided indirectly, as is the case with Google's page rank algorithms which orders search results based on the number of pages that (recursively) point to them. In all of these cases, information that is produced by a group of people is used to provide or enhance the functioning of a system. Social computing is concerned with systems of this sort and the mechanisms and principles that underlie them. Social computing can be defined as follows: > "Social Computing" refers to systems that support the gathering, > representation, processing, use, and dissemination of information that is > distributed across social collectivities such as teams, communities, > organizations, and markets.
|
|
Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is the divide-and-conquer approach from computer science. An eigenvalue problem is divided into two problems of roughly half the size, each of these are solved recursively, and the eigenvalues of the original problem are computed from the results of these smaller problems. Here we present the simplest version of a divide-and-conquer algorithm, similar to the one originally proposed by Cuppen in 1981.
|
|
We obtain the finite rooted tree representing α by joining the roots of the trees representing \beta_1,\ldots,\beta_k to a new root. (This has the consequence that the number 0 is represented by a single root while the number 1=\omega^0 is represented by a tree containing a root and a single leaf.) An order on the set of finite rooted trees is defined recursively: we first order the subtrees joined to the root in decreasing order, and then use lexicographic order on these ordered sequences of subtrees. In this way the set of all finite rooted trees becomes a well-ordered set which is order- isomorphic to ε0.
|
|
A universal Turing machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church–Turing thesis, the problems solvable by a universal Turing machine are exactly those problems solvable by an algorithm or an effective method of computation, for any reasonable definition of those terms. For these reasons, a universal Turing machine serves as a standard against which to compare computational systems, and a system that can simulate a universal Turing machine is called Turing complete. An abstract version of the universal Turing machine is the universal function, a computable function which can be used to calculate any other computable function.
|
|
73–79 This method may be used to enumerate either free or fixed polyominoes. A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes (without requiring that all polyominoes of order n be stored in order to enumerate those of order n+1), but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each n-omino n times, once from starting from each of its n squares, or may be arranged to count each once only.
|
|
Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.Grünbaum and Shephard, section 9.4 The two tiling nonominoes not satisfying the Conway criterion. The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to order 9 form a patch of at least one tile satisfying it, with higher-order exceptions more frequent. Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane.
|
|
Bjorn H. Jernudd and Michael Shapiro, pp. 53-80. Berlin: Mouton de Gruyter. Among the Tewa, for example, the influence of theocratic institutions and ritualized linguistic forms in other domains of Tewa society have led to a strong resistance to the extensive borrowing and shift many of its neighboring language communities have experienced. According to Paul Kroskrity this is due to a "dominant language ideology" through which ceremonial Kiva speech is elevated to a linguistic ideal and the cultural preferences that it embodies, namely regulation by convention, indigenous purism, strict compartmentalization, and linguistic indexing of identity, are recursively projected onto the Tewa language as a whole.
|
|
When performing this kind of automatic mirroring of web sites, Wget supports the Robots Exclusion Standard (unless the option `-e robots=off` is used). Recursive download works with FTP as well, where Wget issues the `LIST` command to find which additional files to download, repeating this process for directories and files under the one specified in the top URL. Shell-like wildcards are supported when the download of FTP URLs is requested. When downloading recursively over either HTTP or FTP, Wget can be instructed to inspect the timestamps of local and remote files, and download only the remote files newer than the corresponding local ones.
|
|
Intuitively, addition can be recursively defined with the rules: :add(0,x)=x, :add(n+1,x)=add(n,x)+1 To fit this into a strict primitive recursive definition, define: :add(0,x)=P_1^1(x), :add(S(n),x)=S(P_2^3(n, add(n,x), x)) Here S(n) is "the successor of n" (i.e., n+1), P11 is the identity function, and P23 is the projection function that takes 3 arguments and returns the second one. Functions f and g required by the above definition of the primitive recursion operation are respectively played by P11 and the composition of S and P23.
|
|
A C program consists of units called source files (or preprocessing files), which, in addition to source code, includes directives for the C preprocessor. A translation unit is the output of the C preprocessor – a source file after it has been preprocessed. Preprocessing notably consists of expanding a source file to recursively replace all `#include` directives with the literal file declared in the directive (usually header files, but possibly other source files); the result of this step is a preprocessing translation unit. Further steps include macro expansion of `#define` directives, and conditional compilation of `#ifdef` directives, among others; this translates the preprocessing translation unit into a translation unit.
|
|
Their range reduction algorithm replaces each digit by a signature, which is a hashed value with bits such that different digit values have different signatures. If is sufficiently small, the numbers formed by this replacement process will be significantly smaller than the original keys, allowing the non-conservative packed sorting algorithm of to sort the replaced numbers in linear time. From the sorted list of replaced numbers, it is possible to form a compressed trie of the keys in linear time, and the children of each node in the trie may be sorted recursively using only keys of size , after which a tree traversal produces the sorted order of the items.
|
|
Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible. For the frequent case of propositional logic, the problem is decidable but co-NP- complete, and hence only exponential-time algorithms are believed to exist for general proof tasks. For a first order predicate calculus, Gödel's completeness theorem states that the theorems (provable statements) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven. However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized.
|
|
A language is called computable (synonyms: recursive, decidable) if there is a computable function such that for each word over the alphabet, if the word is in the language and if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language. A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function such that is defined if and only if the word is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers.
|
|
For example, a simple stub resolver running on a home router typically makes a recursive query to the DNS server run by the user's ISP. A recursive query is one for which the DNS server answers the query completely by querying other name servers as needed. In typical operation, a client issues a recursive query to a caching recursive DNS server, which subsequently issues non-recursive queries to determine the answer and send a single answer back to the client. The resolver, or another DNS server acting recursively on behalf of the resolver, negotiates use of recursive service using bits in the query headers.
|
|
Idea of "internal viewer" generates infinite regress of internal viewers. The homunculus argument is a fallacy whereby a concept is explained in terms of the concept itself, recursively, without first defining or explaining the original concept. This fallacy arises most commonly in the theory of vision. One may explain human vision by noting that light from the outside world forms an image on the retinas in the eyes and something (or someone) in the brain looks at these images as if they are images on a movie screen (this theory of vision is sometimes termed the theory of the Cartesian theater: it is most associated, nowadays, with the psychologist David Marr).
|
|
Another important question is the existence of automorphisms in recursion-theoretic structures. One of these structures is that one of recursively enumerable sets under inclusion modulo finite difference; in this structure, A is below B if and only if the set difference B − A is finite. Maximal sets (as defined in the previous paragraph) have the property that they cannot be automorphic to non-maximal sets, that is, if there is an automorphism of the recursive enumerable sets under the structure just mentioned, then every maximal set is mapped to another maximal set. Soare (1974) showed that also the converse holds, that is, every two maximal sets are automorphic.
|
|
Gitem began working at the stash in 2016 and started to help Walt Flanagan to create games for the podcast. His most famous episodes include one where he gets married to a fan on the air in Episode 300 "Gitem to the Chapel". Ming Chen - Ming is possibly the longest running guest on TESD having been a guest on episodes 8 and 9 of TESD: "Party in the USA I and II", recursively. Ming early on upset Walt and in an act of revenge Walt created a listener contest called the "Not So Superbowl", in which Ming was made to judge dozens of listener submitted pictures of feces covered toilet bowls.
|
|
The basic goal of the meld (also called merge) operation is to take two heaps (by taking each heaps root nodes), Q1 and Q2, and merges them, returning a single heap node as a result. This heap node is the root node of a heap containing all elements from the two subtrees rooted at Q1 and Q2. A nice feature of this meld operation is that it can be defined recursively. If either heaps are null, then the merge is taking place with an empty set and the method simply returns the root node of the non-empty heap. If both Q1 and Q2 are not nil, check if Q1 > Q2.
|
|
This design choice has a slight "cost" in that code `else if` branch is, effectively, adding an extra nesting level, complicating the job for some compilers (or its implementers), which has to analyse and implement arbitrarily long `else if` chains recursively. If all terms in the sequence of conditionals are testing the value of a single expression (e.g., `if x=0` ... `else if x=1` ... `else if x=2`...), then an alternative is the switch statement, also called case-statement or select- statement. Conversely, in languages that do not have a switch statement, these can be produced by a sequence of `else if` statements.
|
|
The entire directory structure is recorded in the metadata, so the data section purely contains data from files. The metadata describes the location of data in files with extents of blocks, which makes the metadata quite compact. When a metadata update occurs, the system looks at the block containing the metadata to be changed, and copies it to a newly allocated block from the metadata section, with the change made, then it recursively changes the metadata in the block that points to that block in the same way. This way, eventually the root block needs to be changed, which causes the atomic metadata update.
|
|
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of N1 smaller DFTs of sizes N2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT.
|
|
Circuit representation of a work-efficient 16-input parallel prefix sum A work-efficient parallel prefix sum can be computed by the following steps. #Compute the sums of consecutive pairs of items in which the first item of the pair has an even index: , , etc. #Recursively compute the prefix sum of the sequence #Express each term of the final sequence as the sum of up to two terms of these intermediate sequences: , , , , etc. After the first value, each successive number is either copied from a position half as far through the sequence, or is the previous value added to one value in the sequence.
|
|
To encode a number N, keep reducing the maximum element of this set (Smax) from N and output Smax for each such difference, stopping when the number lies in the half closed half open range [0 – Smax). Example: Let S = [0 1 2 3 4 … 10], be an 11-element set, and we have to recursively index the value N=49. According to this method, we need to keep removing 10 from 49, and keep proceeding till we reach a number in the 0–10 range. So the values are 10 (N = 49 – 10 = 39), 10 (N = 39 – 10 = 29), 10 (N = 29 – 10 = 19), 10 (N = 19 – 10 = 9), 9.
|
|
A recursive neural network is a kind of deep neural network created by applying the same set of weights recursively over a structured input, to produce a structured prediction over variable-size input structures, or a scalar prediction on it, by traversing a given structure in topological order. Recursive neural networks, sometimes abbreviated as RvNNs, have been successful, for instance, in learning sequence and tree structures in natural language processing, mainly phrase and sentence continuous representations based on word embedding. RvNNs have first been introduced to learn distributed representations of structure, such as logical terms. Models and general frameworks have been developed in further works since the 1990s.
|
|
A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature. More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transforms. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT.
|
|
In Unix-like and some other operating systems, `find` is a command-line utility that locates files based on some user-specified criteria and then applies some requested action on each matched object. It initiates a search from a desired starting location and then recursively traversing the nodes (directories) of a hierarchical structure (typically a tree). find can traverse and search through different file systems of partitions belonging to one or more storage devices mounted under the starting directory. The possible search criteria include a pattern to match against the filename or a time range to match against the modification time or access time of the file.
|
|
Most notably, this difference affects how the interpreter behaves when more than one clause is applicable: non-concurrent constraint logic programming recursively tries all clauses; concurrent constraint logic programming chooses only one. This is the most evident effect of an intended directionality of the interpreter, which never revise a choice it has previously taken. Other effects of this are the semantical possibility of having a goal that cannot be proved while the whole evaluation does not fail, and a particular way for equating a goal and a clause head. Constraint handling rules can be seen as a form of concurrent constraint logic programming,Frühwirth, Thom.
|
|
Convex hull of a simple polygon The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its convex differences tree. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates.
|
|
An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions. This comes in contrast with the direct meaning of the notion of semantic consequence, that quantifies over all structures in a particular language, which is clearly not a recursive definition. Also, it makes the concept of "provability," and thus of "theorem," a clear concept that only depends on the chosen system of axioms of the theory, and not on the choice of a proof system.
|
|
An explicit construction of a parametrix for second order partial differential operators based on power series developments was discovered by Jacques Hadamard. It can be applied to the Laplace operator, the wave equation and the heat equation. In the case of the heat equation or the wave equation, where there is a distinguished time parameter , Hadamard's method consists in taking the fundamental solution of the constant coefficient differential operator obtained freezing the coefficients at a fixed point and seeking a general solution as a product of this solution, as the point varies, by a formal power series in . The constant term is 1 and the higher coefficients are functions determined recursively as integrals in a single variable.
|
|
Minesweeper for versions of Windows protects the first square revealed; from Windows Vista onward, players may elect to replay a board, in which the game is played by revealing squares of the grid by clicking or otherwise indicating each square. If a square containing a mine is revealed, the player loses the game. If no mine is revealed, a digit is instead displayed in the square, indicating how many adjacent squares contain mines; if no mines are adjacent, the square becomes blank, and all adjacent squares will be recursively revealed. The player uses this information to deduce the contents of other squares, and may either safely reveal each square or mark the square as containing a mine.
|
|
FindRoot refers to finding the root of the represented tree that contains the node v. Since the access subroutine puts v on the preferred path, we first execute an access. Now the node v is on the same preferred path, and thus the same auxiliary tree as the root R. Since the auxiliary trees are keyed by depth, the root R will be the leftmost node of the auxiliary tree. So we simply choose the left child of v recursively until we can go no further, and this node is the root R. The root may be linearly deep (which is worst case for a splay tree), we therefore splay it so that the next access will be quick.
|
|
An exhaustive search algorithm can solve the problem in time 2knO(1), where k is the size of the vertex cover. Vertex cover is therefore fixed-parameter tractable, and if we are only interested in small k, we can solve the problem in polynomial time. One algorithmic technique that works here is called bounded search tree algorithm, and its idea is to repeatedly choose some vertex and recursively branch, with two cases at each step: place either the current vertex or all its neighbours into the vertex cover. The algorithm for solving vertex cover that achieves the best asymptotic dependence on the parameter runs in time O(1.2738^k+(k\cdot n)).
|
|
To set in further context Pask won a prize from Old Dominion University for his complementarity principle: "All processes produce products and all products are produced by processes". This can be written: Ap(ConZ(T)) => DZ (T) where => means produces and Ap means the "application of", D means "description of" and Z is the concept mesh or coherence of which T is part. This can also be written : Z (T)), DZ (T)>. Pask distinguishes Imperative (written &Ap; or IM) from Permissive Application (written Ap)Pask (1993) para 188 where information is transferred in the Petri net manner, the token appearing as a hole in a torus producing a Klein bottle containing recursively packed concepts.
|
|
Greebles implemented in computer graphics using bump mapping In 3D computer graphics, greebles can be created by specific software in order to avoid the time-consuming process of manually creating large numbers of precise, custom geometry. This is often tedious, repetitive work, and may be best suited to automatic, software-based procedural generation, particularly if a great degree of control is unnecessary or the greebles will be small on screen. Most greeble-generating software work by subdividing the surface to be greebled into smaller regions, adding some detail to each new surface, and then recursively continuing this process on each new surface to some specified level of detail. Similar algorithms are used in the creation of fractal surfaces.
|
|
Two incomparable families examined at length are WRB (languages generated by normal regular- based W-grammars) and WS (languages generated by simple W-grammars). Both properly contain the context-free languages and are properly contained in the family of quasirealtime languages. In addition, WRB is closed under nested iterate ... "An Infinite Hierarchy of Context-Free Languages," Journal of the ACM, Volume 16 Issue 1, January 1969 "A New Normal-Form Theorem for Context- Free Phrase Structure Grammars," JACM, Volume 12 Issue 1, January 1965 "The Unsolvability of the Recognition of Linear Context-Free Languages," JACM, Volume 13 Issue 4, October 1966 :The problem of whether a given context-free language is linear is shown to be recursively undecidable.
|
|
Local in time means that the updates take place continually (on-line) and depend only on the most recent time step rather than on multiple time steps within a given time horizon as in BPTT. Biological neural networks appear to be local with respect to both time and space. For recursively computing the partial derivatives, RTRL has a time-complexity of O(number of hidden x number of weights) per time step for computing the Jacobian matrices, while BPTT only takes O(number of weights) per time step, at the cost of storing all forward activations within the given time horizon. An online hybrid between BPTT and RTRL with intermediate complexity exists, along with variants for continuous time.
|
|
Most definitions of ordinal collapsing functions found in the recent literature differ from the ones we have given in one technical but important way which makes them technically more convenient although intuitively less transparent. We now explain this. The following definition (by induction on \alpha) is completely equivalent to that of the function \psi above: :Let C(\alpha,\beta) be the set of ordinals generated starting from 0, 1, \omega, \Omega and all ordinals less than \beta by recursively applying the following functions: ordinal addition, multiplication and exponentiation, and the function \psi\upharpoonright_\alpha. Then \psi(\alpha) is defined as the smallest ordinal \rho such that C(\alpha,\rho) \cap \Omega = \rho.
|
|
Art can exemplify logical paradoxes, as in some paintings by the surrealist René Magritte, which can be read as semiotic jokes about confusion between levels. In La condition humaine (1933), Magritte depicts an easel (on the real canvas), seamlessly supporting a view through a window which is framed by "real" curtains in the painting. Similarly, Escher's Print Gallery (1956) is a print which depicts a distorted city which contains a gallery which recursively contains the picture, and so ad infinitum. Magritte made use of spheres and cuboids to distort reality in a different way, painting them alongside an assortment of houses in his 1931 Mental Arithmetic as if they were children's building blocks, but house-sized.
|
|
The original 1904 Droste cacao tin, designed by Jan Misset (1861–1931) The Droste effect (), known in art as an example of mise en abyme, is the effect of a picture recursively appearing within itself, in a place where a similar picture would realistically be expected to appear, creating a loop which theoretically could go on forever, but realistically only goes on as far as the image's quality allows. The effect is named for a Dutch brand of cocoa, with an image designed by Jan Misset in 1904. It has since been used in the packaging of a variety of products. The effect was anticipated in medieval works of art such as Giotto's Stefaneschi Triptych of 1320.
|
|
Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set in has a point in common with all of the . Since is dense, intersects ; thus, there is a point and such that: : where and denote an open and closed ball, respectively, centered at with radius . Since each is dense, we can continue recursively to find a pair of sequences and such that: :. (This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at .) Since when , we have that is Cauchy, and hence converges to some limit by completeness.
|
|
Deriche has made major contributions to the scientific community, mainly in image processing, computer vision and neuro-imaging. In 1987, Deriche has developed Deriche Edge Detector which is a low-level, recursively implemented, optimal edge detector based on Canny's edge detector criteria for optimal edge detection. In 1998, based on his works in Computational image processing, Early vision, 3D reconstruction, panoramic photography, image-based modeling, and motion analysis, he co-founded with his Inria colleagues (among which O. Faugeras, T. Papadopoulo and L. Robert), Realviz, a start-up specialized in image-based content creation solutions for the film, broadcast, gaming, digital imaging and architecture. The startup was acquired by Autodesk in 2008.
|
|
The classification is done through dividing things into large (or called the head) and small (or called the tail) things around the arithmetic mean or average, and then recursively going on for the division process for the large things or the head until the notion of far more small things than large ones is no longer valid, or with more or less similar things left only.Jiang, Bin (2013). "Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution", The Professional Geographer, 65 (3), 482 – 494. Head/tail breaks is not just for classification, but also for visualization of big data by keeping the head, since the head is self-similar to the whole.
|
|
Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable. One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result: :Corresponding to any given consistent axiomatization of number theory,More precisely, given a \Sigma^0_1-formula representing the set of Gödel numbers of sentences which recursively axiomatize a consistent theory extending Robinson arithmetic. one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization. According to the incompleteness theorems, a powerful-enough consistent axiomatic theory is incomplete, meaning the truth of some of its propositions cannot be established within its formalism.
|
|
Inserting a value into a ternary search can be defined recursively much as lookups are defined. This recursive method is continually called on nodes of the tree given a key which gets progressively shorter by pruning characters off the front of the key. If this method reaches a node that has not been created, it creates the node and assigns it the character value of the first character in the key. Whether a new node is created or not, the method checks to see if the first character in the string is greater than or less than the character value in the node and makes a recursive call on the appropriate node as in the lookup operation.
|
|
PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided by such a function. This includes addition, multiplication, exponentiation, tetration, etc. The Ackermann function is an example of a function that is not primitive recursive, showing that PR is strictly contained in R (Cooper 2004:88). On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input (M, k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing.
|
|
The -calculus is a universal model of computation. This was first observed by Milner in his paper "Functions as Processes", in which he presents two encodings of the lambda-calculus in the -calculus. One encoding simulates the eager (call-by-value) evaluation strategy, the other encoding simulates the normal-order (call-by-name) strategy. In both of these, the crucial insight is the modeling of environment bindings – for instance, " is bound to term M" – as replicating agents that respond to requests for their bindings by sending back a connection to the term M. The features of the -calculus that make these encodings possible are name-passing and replication (or, equivalently, recursively defined agents).
|
|
The system of Presburger arithmetic consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication. Dan Willard (2001) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; these systems are consistent and capable of proving their own consistency (see self-verifying theories).
|
|
In game theory, a strategy refers to the rules that a player uses to choose between the available actionable options. Every player in a non-trivial game has a set of possible strategies to use when choosing what moves to make. A strategy may recursively look ahead and consider what actions can happen in each contingent state of the game—e.g. if the player takes action 1, then that presents the opponent with a certain situation, which might be good or bad, whereas if the player takes action 2 then the opponents will be presented with a different situation, and in each case the choices they make will determine their own future situation.
|
|
Sierpiński triangle Generated using a random algorithm conjunctions of lexicographically ordered arguments. The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... The Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self- similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.
|
|
The bisection method consists roughly of starting from an interval containing all real roots of a polynomial, and divides it recursively into two parts until getting eventually intervals that contain either zero or one root. The starting interval may be of the form , where is an upper bound on the absolute values of the roots, such as those that are given in . For technical reasons (simpler changes of variable, simpler complexity analysis, possibility of taking advantage of the binary analysis of computers), the algorithms are generally presented as starting with the interval . There is no loss of generality, as the changes of variables and move respectively the positive and the negative roots in the interval .
|
|
All loopless outerplanar graphs can be colored using only three colors;. this fact features prominently in the simplified proof of Chvátal's art gallery theorem by . A 3-coloring may be found in linear time by a greedy coloring algorithm that removes any vertex of degree at most two, colors the remaining graph recursively, and then adds back the removed vertex with a color different from the colors of its two neighbors. According to Vizing's theorem, the chromatic index of any graph (the minimum number of colors needed to color its edges so that no two adjacent edges have the same color) is either the maximum degree of any vertex of the graph or one plus the maximum degree.
|
|
Written while Hofstadter was at the University of Oregon, his paper was influential in directing further research. It predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional square lattice, as a function of a magnetic field applied to the system, formed what is now known as a fractal set. That is, the distribution of energy levels for small scale changes in the applied magnetic field recursively repeat patterns seen in the large-scale structure. "Gplot", as Hofstadter called the figure, was described as a recursive structure in his 1976 article in Physical Review B, written before Benoit Mandelbrot's newly coined word "fractal" was introduced in an English text.
|
|
Form a new first-order theory T' from T by adding a new n-ary function symbol f, the logical axioms featuring the symbol f and the new axiom :\forall x_1\dots\forall x_n\phi(f(x_1,\dots,x_n),x_1,\dots,x_n), called the defining axiom of f. Let \psi be any atomic formula of T'. We define formula \psi^\ast of T recursively as follows. If the new symbol f does not occur in \psi, let \psi^\ast be \psi. Otherwise, choose an occurrence of f(t_1,\dots,t_n) in \psi such that f does not occur in the terms t_i, and let \chi be obtained from \psi by replacing that occurrence by a new variable z.
|
|
The more coordination and narratives in the messages being exchanged, the "more meaning contexts recursively affect and are affected by the evolving actions in a conversation" which ideally is critical to point out as a conversation progresses. There are six levels of meaning (listed from lower level to higher level): content, speech act, episodes, relationship, life scripts, and cultural patterns. In the six categories below, we also assign a moral value to the messages we receive when we are conscious of them and or unconscious of them. When consciously aware of them, they can either be, "obligatory, legitimate, undermined, or prohibited" or when unconsciously aware of them, they can be, "caused, probable, random, or blocked".
|
|
The following is a key observation in understanding the modular decomposition: If X is a module of G and Y is a subset of X, then Y is a module of G, if and only if it is a module of G[X]. In (Gallai, 1967), Gallai defined the modular decomposition recursively on a graph with vertex set V, as follows: # As a base case, if G only has one vertex, its modular decomposition is a single tree node. # Gallai showed that if G is connected and so is its complement, then the maximal modules that are proper subsets of V are a partition of V. They are therefore a modular partition. The quotient that they define is prime.
|
|
Working hypercognition is a strong directive-executive function that is responsible for setting and pursuing mental and behavioral goals until they are attained. This function involves processes enabling the person to: (1) set mental and behavioral goals; (2) plan their attainment; (3) evaluate each step's processing demands vis-à-vis the available potentials, knowledge, skills and strategies; (4) monitor planned activities vis-à-vis the goals; and (5) evaluate the outcome attained. These processes operate recursively in such a way that goals and subgoals may be renewed according to the online evaluation of the system's distance from its ultimate objective. These regulatory functions operate under the current structural constraints of the mind that define the current processing potentials.
|
|
The basic backtracking algorithm runs by choosing a literal, assigning a truth value to it, simplifying the formula and then recursively checking if the simplified formula is satisfiable; if this is the case, the original formula is satisfiable; otherwise, the same recursive check is done assuming the opposite truth value. This is known as the splitting rule, as it splits the problem into two simpler sub-problems. The simplification step essentially removes all clauses that become true under the assignment from the formula, and all literals that become false from the remaining clauses. The DPLL algorithm enhances over the backtracking algorithm by the eager use of the following rules at each step: ; Unit propagation : If a clause is a unit clause, i.e.
|
|
The first values in Gould's sequence may be constructed by recursively constructing the first values, and then concatenating the doubles of the first values. For instance, concatenating the first four values 1, 2, 2, 4 with their doubles 2, 4, 4, 8 produces the first eight values. Because of this doubling construction, the first occurrence of each power of two in this sequence is at position . Gould's sequence, the sequence of its exponents, and the Thue–Morse sequence are all self-similar: they have the property that the subsequence of values at even positions in the whole sequence equals the original sequence, a property they also share with some other sequences such as Stern's diatomic sequence... As cited by Gilleland.
|
|
In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics.Solomon Feferman, Turing in the Land of O(z) in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 page 111Concise Routledge encyclopedia of philosophy 2000 page 647 The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228. While Gödel showed that every system of logic suffers from some form of incompleteness, Turing focused on a method so that from a given system of logic a more complete system may be constructed.
|
|
An arbitrary game tree that has been fully colored With a complete game tree, it is possible to "solve" the game – that is to say, find a sequence of moves that either the first or second player can follow that will guarantee the best possible outcome for that player (usually a win or a tie). The algorithm (which is generally called backward induction or retrograde analysis) can be described recursively as follows. #Color the final ply of the game tree so that all wins for player 1 are colored one way (Blue in the diagram), all wins for player 2 are colored another way (Red in the diagram), and all ties are colored a third way (Grey in the diagram). #Look at the next ply up.
|
|
The selected element is removed from all the lists where it appears as a head and appended to the output list. The process of selecting and removing a good head to extend the output list is repeated until all remaining lists are exhausted. If at some point no good head can be selected, because the heads of all remaining lists appear in any one tail of the lists, then the merge is impossible to compute due to inconsistent orderings of dependencies in the inheritance hierarchy and no linearization of the original class exists. A naive divide and conquer approach to computing the linearization of a class may invoke the algorithm recursively to find the linearizations of parent classes for the merge-subroutine.
|
|
Programs need this information because the child process, a few steps after process duplication, usually invokes the execve(2) system call (possibly via the family of exec(3) wrapper functions in glibC) and replace the program that is currently being run by the calling process with a new program, with newly initialized stack, heap, and (initialized and uninitialized) data segments. When it's done, it results in two processes that run two different programs. Depending on the effective user id (euid), and on the effective group id (egid), a process running with user zero privileges (root, the system administrator, owns the identifier 0) can perform everything (e.g., kill all the other processes or recursively wipe out whole filesystems), instead non zero user processes cannot.
|
|
Tagsistant features a simple reasoner which expands the results of a query by including objects tagged with related tags. A relation between two tags can be established inside the `relations/` directory following a three level pattern: :`relations/tag1/rel/tag2/` The `rel` element can be includes or is_equivalent. To include the rock tag in the music tag, the UNIX command `mkdir` can be used: :`mkdir -p relations/music/includes/rock` The reasoner can recursively resolve relations, allowing the creation of complex structures: :`mkdir -p relations/music/includes/rock` :`mkdir -p relations/rock/includes/hard_rock` :`mkdir -p relations/rock/includes/grunge` :`mkdir -p relations/rock/includes/heavy_metal` :`mkdir -p relations/heavy_metal/includes/speed_metal` The web of relations created inside the `relations/` directory constitutes a basic form of ontology.
|
|
Roughly speaking, an ECN specification does two things: it says how to modify a hidden ECN type to produce a new (colored; see below) hidden ECN type, and it says how an ECN type (as well as each of its components if it's a complex type) is to be encoded. The latter can be applied recursively, in the sense that an encoding step for a component of an ECN type may result in a further in-place modification of the remaining part of the ECN type that is being encoded. This process can go on through any number of cycles, until the final ECN type has been completely encoded, that is, all the bits representing the value of the original ASN.1 type have been generated.
|
|
Although this sequence may be generated by a recursive algorithm that constructs the sequence of smaller permutations and then performs all possible insertions of the largest number into the recursively-generated sequence, the actual Steinhaus–Johnson–Trotter algorithm avoids recursion, instead computing the same sequence of permutations by an iterative method. There is an equivalent and conceptually somewhat simpler definition of the Steinhaus-Johnson-Trotter ordering of permutations via the following greedy algorithm: We start with the identity permutation 1\;2\;\ldots\;n. Now we repeatedly transpose the largest possible entry with the entry to its left or right, such that in each step, a new permutation is created that has not been encountered in the list of permutations before.
|
|
Binary space partitioning is a generic process of recursively dividing a scene into two until the partitioning satisfies one or more requirements. It can be seen as a generalisation of other spatial tree structures such as k-d trees and quadtrees, one where hyperplanes that partition the space may have any orientation, rather than being aligned with the coordinate axes as they are in k-d trees or quadtrees. When used in computer graphics to render scenes composed of planar polygons, the partitioning planes are frequently chosen to coincide with the planes defined by polygons in the scene. The specific choice of partitioning plane and criterion for terminating the partitioning process varies depending on the purpose of the BSP tree.
|
|
A vantage-point tree (or VP tree) is a metric tree that segregates data in a metric space by choosing a position in the space (the "vantage point") and partitioning the data points into two parts: those points that are nearer to the vantage point than a threshold, and those points that are not. By recursively applying this procedure to partition the data into smaller and smaller sets, a tree data structure is created where neighbors in the tree are likely to be neighbors in the space. One generalization is called a multi- vantage-point tree, or MVP tree: a data structure for indexing objects from large metric spaces for similarity search queries. It uses more than one point to partition each level.
|
|
In the Plan 9 operating system from Bell Labs (mid-1980s onward), union mounting is a central concept, replacing several older Unix conventions with union directories; for example, several directories containing executables, unioned together at a single directory, replace the variable for command lookup in the shell. Plan 9 union semantics are greatly simplified compared to the implementations for POSIX-style operating systems: the union of two directories is simply the concatenation of their contents, so a directory listing of the union may display duplicate names. Also, no effort is made to recursively merge subdirectories, leading to an extremely simple implementation. Directories are unioned in a controllable order; , where is a union directory, denotes the file called in the first constituent directory that contains such a file.
|
|
An important practical application of smooth numbers is the fast Fourier transform (FFT) algorithms (such as the Cooley–Tukey FFT algorithm), which operates by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.) 5-smooth or regular numbers play a special role in Babylonian mathematics.. They are also important in music theory (see Limit (music)),. and the problem of generating these numbers efficiently has been used as a test problem for functional programming.. Smooth numbers have a number of applications to cryptography.
|
|
Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on n vertices contains a transitive subtournament on 1+\lfloor\log_2 n\rfloor vertices.. The proof is simple: choose any one vertex v to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of v or the set of outgoing neighbors of v, whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed . However, showed that this bound is not tight for some larger values of n.
|
|
A slight variation of TDPL, known as Generalized TDPL or GTDPL, greatly increases the apparent expressiveness of TDPL while retaining the same minimalist approach (though they are actually equivalent). In GTDPL, in place of TDPL's recursive rule form A ← BC/D, we instead use the alternate rule form A ← B[C,D], which is interpreted as follows. When nonterminal A is invoked on some input string, it first recursively invokes B. If B succeeds, then A subsequently invokes C on the remainder of the input left unconsumed by B, and returns the result of C to the original caller. If B fails, on the other hand, then A invokes D on the original input string, and passes the result back to the caller.
|
|
There have been several claims for the longest sentence in the English language, usually with claims that revolve around the longest printed sentence. At least one linguistics textbook concludes that, in theory, "there is no longest English sentence." A sentence can be made arbitrarily long by successive iterations, such as "Someone thinks that someone thinks that someone thinks that...," or by combining shorter clauses in various ways. For example, sentences can be extended by recursively embedding clauses one into another, such as :"The mouse ran away" :"The mouse that the cat hit ran away" :... :"The mouse that the cat that the dog that the man frightened and chased ran away" The ability to embed structures within larger ones is called recursion.
|
|
One of the most notorious pathologies in topology is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to separate the space cleanly. As a counter-example, it motivated the extra condition of tameness, which suppresses the kind of wild behavior the horned sphere exhibits. Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same.
|
|
Conversely, on machines without or operators, can be computed using , albeit inefficiently: : (which depends on returning for the zero input) On platforms with an efficient Hamming weight (population count) operation such as SPARC's `POPC` or Blackfin's `ONES`, there is: : , : : where denotes bitwise exclusive-OR and denotes bitwise negation. The inverse problem (given , produce an such that ) can be computed with a left-shift (). Find first set and related operations can be extended to arbitrarily large bit arrays in a straightforward manner by starting at one end and proceeding until a word that is not all-zero (for , , ) or not all-one (for , , ) is encountered. A tree data structure that recursively uses bitmaps to track which words are nonzero can accelerate this.
|
|
The basic idea is to reduce the transpose of two large matrices into the transpose of small (sub)matrices. We do this by dividing the matrices in half along their larger dimension until we just have to perform the transpose of a matrix that will fit into the cache. Because the cache size is not known to the algorithm, the matrices will continue to be divided recursively even after this point, but these further subdivisions will be in cache. Once the dimensions and are small enough so an input array of size m \times n and an output array of size n \times m fit into the cache, both row-major and column-major traversals result in O(mn) work and O(mn/B) cache misses.
|
|
For instance, the earth's rotation caused the ground receiver to move toward or away from the satellite's orbit, creating a non- symmetric Doppler shift for approach and recession, allowing the receiver to determine whether it was east or west of the satellite's north-south ground track. Calculating the most likely receiver location was not a trivial exercise. The navigation software used the satellite's motion to compute a 'trial' Doppler curve, based on an initial 'trial' location for the receiver. The software would then perform a least squares curve fit for each two-minute section of the Doppler curve, recursively moving the trial position until the trial Doppler curve 'most closely' matched the actual Doppler received from the satellite for all two-minute curve segments.
|
|
A common computer programming tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as dynamic programming or memoization. A recursive function definition has one or more base cases, meaning input(s) for which the function produces a result trivially (without recurring), and one or more recursive cases, meaning input(s) for which the program recurs (calls itself). For example, the factorial function can be defined recursively by the equations and, for all , .
|
|
In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount of time. Informally, it means that if the answer to a problem instance is 'yes', then there is some procedure which takes finite time to determine this, and this procedure never falsely reports 'yes' when the true answer is 'no'. However, when the true answer is 'no', the procedure is not required to halt; it may go into an "infinite loop" for some 'no' cases. Such a procedure is sometimes called a semi-algorithm, to distinguish it from an algorithm, defined as a complete solution to a decision problem.
|
|
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and inconsistent otherwise. Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if is the least such cardinal, then sitting inside the von Neumann universe is a model of ZFC, and a theory is consistent if and only if it has a model. If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication.
|
|
In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not. A set which is not computable is called noncomputable or undecidable. A more general class of sets than the decidable ones consists of the recursively enumerable sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithm may give no answer (but not the wrong answer) for numbers not in the set.
|
|
The distance within an ultrametric is the same as the minimax path weight in the minimum spanning tree of the metric.; From the minimum spanning tree, one can construct a Cartesian tree, the root node of which represents the heaviest edge of the minimum spanning tree. Removing this edge partitions the minimum spanning tree into two subtrees, and Cartesian trees recursively constructed for these two subtrees form the children of the root node of the Cartesian tree. The leaves of the Cartesian tree represent points of the metric space, and the lowest common ancestor of two leaves in the Cartesian tree is the heaviest edge between those two points in the minimum spanning tree, which has weight equal to the distance between the two points.
|
|
Suppose that \phi, \psi, and \rho are quantifier-free formulas and no two of these formulas share any free variable. Consider the formula : (\phi \lor \exists x \psi) \rightarrow \forall z \rho. By recursively applying the rules starting at the innermost subformulas, the following sequence of logically equivalent formulas can be obtained: : (\phi \lor \exists x \psi) \rightarrow \forall z \rho. : ( \exists x (\phi \lor \psi) ) \rightarrow \forall z \rho, : eg( \exists x (\phi \lor \psi) ) \lor \forall z \rho, : (\forall x eg(\phi \lor \psi)) \lor \forall z \rho, : \forall x ( eg(\phi \lor \psi) \lor \forall z \rho), : \forall x ( ( \phi \lor \psi) \rightarrow \forall z \rho ), : \forall x ( \forall z (( \phi \lor \psi) \rightarrow \rho )), : \forall x \forall z ( ( \phi \lor \psi) \rightarrow \rho ).
|
|
This is really just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another. Over the years, languages in general have proved amenable to this kind of analysis. Recently, however, the generally accepted idea that recursion is an essential property of human language has been challenged by Daniel Everett on the basis of his claims about the Pirahã language.
|
|
A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be queried in constant time per query, using lowest common ancestor queries in a Cartesian tree. The root of the Cartesian tree represents the heaviest minimum spanning tree edge, and the children of the root are Cartesian trees recursively constructed from the subtrees of the minimum spanning tree formed by removing the heaviest edge. The leaves of the Cartesian tree represent the vertices of the input graph, and the minimax distance between two vertices equals the weight of the Cartesian tree node that is their lowest common ancestor. Once the minimum spanning tree edges have been sorted, this Cartesian tree can be constructed in linear time.
|
|
By way of constructing a fair split tree, it is possible to construct a WSPD of size O(s^d n) in O(n \lg n) time. The general principle of the split tree of a point set is that each node of the tree represents a set of points and that the bounding box of is split along its longest side in two equal parts which form the two children of and their point set. It is done recursively until there is only one point in the set. Let denote the size of the longest interval of the bounding hyperrectangle of point set and let denote the size of the i-th dimension of the bounding hyperrectangle of point set .
|
|
The convex layers of a point set and their intersection with a halfplane In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their vertices. The outermost one is the convex hull of the points and the rest are formed in the same way recursively. The innermost layer may be degenerate, consisting only of one or two points. The problem of constructing convex layers has also been called onion peeling or onion decomposition.. Although constructing the convex layers by repeatedly finding convex hulls would be slower, it is possible to partition any set of n points into its convex layers in time O(n\log n).
|
|
It was improved in 2013 to by Virginia Vassilevska Williams, giving a time only slightly worse than Le Gall's improvement: The Le Gall algorithm, and the Coppersmith–Winograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two -matrices with fewer than multiplications, and this technique is applied recursively. However, the constant coefficient hidden by the Big O notation is so large that these algorithms are only worthwhile for matrices that are too large to handle on present-day computers. Since any algorithm for multiplying two -matrices has to process all entries, there is an asymptotic lower bound of operations. Raz proved a lower bound of for bounded coefficient arithmetic circuits over the real or complex numbers.
|
|
In order to compute the DFT, we need to evaluate the remainder of x(z) modulo N degree-1 polynomials as described above. Evaluating these remainders one by one is equivalent to the evaluating the usual DFT formula directly, and requires O(N2) operations. However, one can combine these remainders recursively to reduce the cost, using the following trick: if we want to evaluate x(z) modulo two polynomials U(z) and V(z), we can first take the remainder modulo their product U(z) V(z), which reduces the degree of the polynomial x(z) and makes subsequent modulo operations less computationally expensive. The product of all of the monomials (z - \omega_N^k) for k=0..N-1 is simply z^N-1 (whose roots are clearly the N roots of unity).
|
|
In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved.
|
|
The coordinator can be the node that originated the transaction (invoked recursively (transitively) the other participants), but also another node in the same tree can take the coordinator role instead. 2PC messages from the coordinator are propagated "down" the tree, while messages to the coordinator are "collected" by a participant from all the participants below it, before it sends the appropriate message "up" the tree (except an abort message, which is propagated "up" immediately upon receiving it or if the current participant initiates the abort). The Dynamic two-phase commit (Dynamic two-phase commitment, D2PC) protocolYoav Raz (1995): "The Dynamic Two Phase Commitment (D2PC) protocol ",Database Theory — ICDT '95, Lecture Notes in Computer Science, Volume 893/1995, pp. 162-176, Springer, is a variant of Tree 2PC with no predetermined coordinator.
|
|
There are three primary categories of tree construction methods: top-down, bottom- up, and insertion methods. Top-down methods proceed by partitioning the input set into two (or more) subsets, bounding them in the chosen bounding volume, then keep partitioning (and bounding) recursively until each subset consists of only a single primitive (leaf nodes are reached). Top-down methods are easy to implement, fast to construct and by far the most popular, but do not result in the best possible trees in general. Bottom-up methods start with the input set as the leaves of the tree and then group two (or more) of them to form a new (internal) node, proceed in the same manner until everything has been grouped under a single node (the root of the tree).
|
|
For efficiency, we might use a packed bit vector representation for arrays of Boolean values, while using a normal array data structure for integer values. The data structure for arrays of ordered pairs is defined recursively as a pair of arrays of each of the element types. instance ArrayElem Bool where data Array Bool = BoolArray BitVector index (BoolArray ar) i = indexBitVector ar i instance ArrayElem Int where data Array Int = IntArray UIntArr index (IntArray ar) i = indexUIntArr ar i instance (ArrayElem a, ArrayElem b) => ArrayElem (a, b) where data Array (a, b) = PairArray (Array a) (Array b) index (PairArray ar br) = (index ar i, index br i) With these definitions, when a client refers to an `Array (Int, Bool)`, an implementation is automatically selected using the defined instances.
|
|
In any recursive algorithm, there is considerable freedom in the choice of the base cases, the small subproblems that are solved directly in order to terminate the recursion. Choosing the smallest or simplest possible base cases is more elegant and usually leads to simpler programs, because there are fewer cases to consider and they are easier to solve. For example, an FFT algorithm could stop the recursion when the input is a single sample, and the quicksort list-sorting algorithm could stop when the input is the empty list; in both examples there is only one base case to consider, and it requires no processing. On the other hand, efficiency often improves if the recursion is stopped at relatively large base cases, and these are solved non-recursively, resulting in a hybrid algorithm.
|
|
Nimber addition (also known as nim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively by :, where the minimum excludant of a set of ordinals is defined to be the smallest ordinal that is not an element of . For finite ordinals, the nim-sum is easily evaluated on a computer by taking the bitwise exclusive or (XOR, denoted by ) of the corresponding numbers. For example, the nim-sum of 7 and 14 can be found by writing 7 as 111 and 14 as 1110; the ones place adds to 1; the twos place adds to 2, which we replace with 0; the fours place adds to 2, which we replace with 0; the eights place adds to 1.
|
|
This is the recursion-theoretic branch of learning theory. It is based on Gold's model of learning in the limit from 1967 and has developed since then more and more models of learning. The general scenario is the following: Given a class S of computable functions, is there a learner (that is, recursive functional) which outputs for any input of the form (f(0),f(1),...,f(n)) a hypothesis. A learner M learns a function f if almost all hypotheses are the same index e of f with respect to a previously agreed on acceptable numbering of all computable functions; M learns S if M learns every f in S. Basic results are that all recursively enumerable classes of functions are learnable while the class REC of all computable functions is not learnable.
|
|
Linguist Noam Chomsky, among many others, has argued that the lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as the time available to utter one), can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence.
|
|
The original motivation of Erdős in studying this problem was to extend from finite to infinite graphs the theorem that, whenever a graph has an orientation with finite maximum out-degree k, it also has a (2k+1)-coloring. For finite graphs this follows because such graphs always have a vertex of degree at most 2k, which can be colored with one of 2k+1 colors after all the remaining vertices are colored recursively. Infinite graphs with such an orientation do not always have a low-degree vertex (for instance, Bethe lattices have k=1 but arbitrarily large minimum degree), so this argument requires the graph to be finite. But the De Bruijn–Erdős theorem shows that a (2k+1)-coloring exists even for infinite graphs.. See in particular p.
|
|
Other studies have tried to reduce the bias through reducing distance, but overall it still remains. This seemingly paradoxical situation – in which an attempt to reduce bias can sometimes actually increase it – may be related to the insight behind the semi-jocular and recursively worded "Hofstadter's law", which states that: Although research has suggested that it is very difficult to eliminate the bias, some factors may help in closing the gap of the optimistic bias between an individual and their target risk group. First, by placing the comparison group closer to the individual, the optimistic bias can be reduced: studies found that when individuals were asked to make comparisons between themselves and close friends, there was almost no difference in the likelihood of an event occurring. Additionally, actually experiencing an event leads to a decrease in the optimistic bias.
|
|
Structural induction is used to prove that some proposition P(x) holds for all x of some sort of recursively defined structure, such as formulas, lists, or trees. A well-founded partial order is defined on the structures ("subformula" for formulas, "sublist" for lists, and "subtree" for trees). The structural induction proof is a proof that the proposition holds for all the minimal structures and that if it holds for the immediate substructures of a certain structure S, then it must hold for S also. (Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.) A structurally recursive function uses the same idea to define a recursive function: "base cases" handle each minimal structure and a rule for recursion.
|
|
Whenever a formal series :f(X)=\sum_k f_k X^k \in RX has f0 = 0 and f1 being an invertible element of R, there exists a series :g(X)=\sum_k g_k X^k that is the composition inverse of f, meaning that composing f with g gives the series representing the identity function x = 0 + 1x + 0x^2+ 0x^3+\cdots. The coefficients of g may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity X (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.
|
|
To see that this language is not recursively enumerable, imagine that we construct a Turing machine M which is able to give a definite answer for all such Turing machines, but that it may run forever on any Turing machine that does eventually halt. We can then construct another Turing machine M' that simulates the operation of this machine, along with simulating directly the execution of the machine given in the input as well, by interleaving the execution of the two programs. Since the direct simulation will eventually halt if the program it is simulating halts, and since by assumption the simulation of M will eventually halt if the input program would never halt, we know that M' will eventually have one of its parallel versions halt. M' is thus a decider for the halting problem.
|
|
SUBCLU uses a monotonicity criteria: if a cluster is found in a subspace S, then each subspace T \subseteq S also contains a cluster. However, a cluster C \subseteq DB in subspace S is not necessarily a cluster in T \subseteq S, since clusters are required to be maximal, and more objects might be contained in the cluster in T that contains C. However, a density-connected set in a subspace S is also a density-connected set in T \subseteq S. This downward-closure property is utilized by SUBCLU in a way similar to the Apriori algorithm: first, all 1-dimensional subspaces are clustered. All clusters in a higher-dimensional subspace will be subsets of the clusters detected in this first clustering. SUBCLU hence recursively produces k+1-dimensional candidate subspaces by combining k-dimensional subspaces with clusters sharing k-1 attributes.
|
|
The case k = 2 is trivial: a graph requires more than one color if and only if it has an edge, and that edge is itself a K2 minor. The case k = 3 is also easy: the graphs requiring three colors are the non-bipartite graphs, and every non-bipartite graph has an odd cycle, which can be contracted to a 3-cycle, that is, a K3 minor. In the same paper in which he introduced the conjecture, Hadwiger proved its truth for k ≤ 4\. The graphs with no K4 minor are the series- parallel graphs and their subgraphs. Each graph of this type has a vertex with at most two incident edges; one can 3-color any such graph by removing one such vertex, coloring the remaining graph recursively, and then adding back and coloring the removed vertex.
|
|
The Distributed Tree Search Algorithm (also known as Korf–Ferguson algorithm) was created to solve the following problem: "Given a tree with non-uniform branching factor and depth, search it in parallel with an arbitrary number of processors as fast as possible." The top-level part of this algorithm is general and does not use a particular existing type of tree-search, but it can be easily specialized to fit any type of non-distributed tree-search. DTS consists of using multiple processes, each with a node and a set of processors attached, with the goal of searching the sub-tree below the said node. Each process then divides itself into multiple coordinated sub-processes which recursively divide themselves again until an optimal way to search the tree has been found based on the number of processors available to each process.
|
|
Hn(0, b) = :b + 1, when n = 0 :b, when n = 1 :0, when n = 2 :1, when n = 3 and b = 0 For more details, see Powers of zero.For more details, see Zero to the power of zero. :0, when n = 3 and b > 0 :1, when n > 3 and b is even (including 0) :0, when n > 3 and b is odd Hn(1, b) = :1, when n ≥ 3 Hn(a, 0) = :0, when n = 2 :1, when n = 0, or n ≥ 3 :a, when n = 1 Hn(a, 1) = :a, when n ≥ 2 Hn(a, a) = :Hn+1(a, 2), when n ≥ 1 Hn(a, −1) = :0, when n = 0, or n ≥ 4 :a − 1, when n = 1 :−a, when n = 2 : , when n = 3 Hn(2, 2) = : 3, when n = 0 : 4, when n ≥ 1, easily demonstrable recursively.
|
|
The prime-factor algorithm (PFA), also called the Good–Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the discrete Fourier transform (DFT) of a size N = N1N2 as a two-dimensional N1×N2 DFT, but only for the case where N1 and N2 are relatively prime. These smaller transforms of size N1 and N2 can then be evaluated by applying PFA recursively or by using some other FFT algorithm. PFA should not be confused with the mixed-radix generalization of the popular Cooley–Tukey algorithm, which also subdivides a DFT of size N = N1N2 into smaller transforms of size N1 and N2. The latter algorithm can use any factors (not necessarily relatively prime), but it has the disadvantage that it also requires extra multiplications by roots of unity called twiddle factors, in addition to the smaller transforms.
|
|
Alan V. Oppenheim, Ronald W. Schafer, and John R. Buck, Discrete-Time Signal Processing, 2nd edition (Upper Saddle River, NJ: Prentice Hall, 1989) The earliest occurrence in print of the term is thought to be in a 1969 MIT technical report. The same structure can also be found in the Viterbi algorithm, used for finding the most likely sequence of hidden states. Most commonly, the term "butterfly" appears in the context of the Cooley–Tukey FFT algorithm, which recursively breaks down a DFT of composite size n = rm into r smaller transforms of size m where r is the "radix" of the transform. These smaller DFTs are then combined via size-r butterflies, which themselves are DFTs of size r (performed m times on corresponding outputs of the sub- transforms) pre-multiplied by roots of unity (known as twiddle factors).
|
|
In addition to individual adaptive performance, psychologists are also interested in adaptive performance at team level. Team adaptive performance is defined as an emergent phenomenon that compiles over time from the unfolding of a recursive cycle whereby one or more team members use their resources to functionally change current cognitive or behavioral goal-directed action or structures to meet expected or unexpected demands. It is a multilevel phenomenon that emanates as team members and teams recursively display behavioral processes and draw on and update emergent cognitive states to engage in change. Team adaptive performance is considered as the core and proximal temporal antecedents to team adaptation, which could be seen as a change in team performance in response to a salient cue or cue stream that leads to a functional outcome for the entire team.Burke, C. S., Stagl, K. C., Salas, E., Pierce, L., & Kendall, D. (2006).
|
|
Some stories feature what might be called a literary version of the Droste effect, where an image contains a smaller version of itself (also a common feature in many fractals). An early version is found in an ancient Chinese proverb, in which an old monk situated in a temple found on a high mountain recursively tells the same story to a younger monk about an old monk who tells a younger monk a story regarding an old monk sitting in a temple located on a high mountain, and so on. The same concept is at the heart of Michael Ende's classic children's novel The Neverending Story, which prominently features a book of the same title. This is later revealed to be the same book the audience is reading, when it begins to be retold again from the beginning, thus creating an infinite regression that features as a plot element.
|
|
We say R is a reduction procedure if it is \alpha recursively enumerable and every member of R is of the form \langle H,J,K \rangle where H, J, K are all α-finite. A is said to be α-recursive in B if there exist R_0,R_1 reduction procedures such that: : K \subseteq A \leftrightarrow \exists H: \exists J:[\langle H,J,K \rangle \in R_0 \wedge H \subseteq B \wedge J \subseteq \alpha / B ], : K \subseteq \alpha / A \leftrightarrow \exists H: \exists J:[\langle H,J,K \rangle \in R_1 \wedge H \subseteq B \wedge J \subseteq \alpha / B ]. If A is recursive in B this is written \scriptstyle A \le_\alpha B. By this definition A is recursive in \scriptstyle\varnothing (the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being \Sigma_1(L_\alpha[B]).
|
|
To install the latest stable release download the rclone installer archive, from the rclone website, and extract it. To interactively configure xmpl: remote from a backend such as S3, Google Drive, or an sftp server:- >rclone config To test the remote and obtain information about it:- >rclone about xmpl: To recursively copy files from folder remote_files, at the remote, to folder files on the E drive:- >rclone copy -v -P xmpl:remote_files E:\files `-v` enables basic logging and `-P`, progress information. By default rclone checks the file integrity (hash) after copy; can retry each file up to three times if the operation is interrupted; uses up to four parallel transfer threads, and does not apply bandwidth throttling. Running the above command again copies any new or changed files at the remote to the local folder but will not delete files from the local folder which have been removed from the remote.
|
|
Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent recursively enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own consistency. There are some technical caveats as to what requirements the formal statement representing the metamathematical statement "The theory is consistent" needs to satisfy, but the outcome is that if a (sufficiently strong) theory can prove its own consistency then either there is no computable way of identifying whether a statement is even an axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, including false statements such as its own consistency). Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories.
|
|
In recreational mathematics, van Eck's sequence is an integer sequence defined recursively as follows. Let a0 = 0\. Then, for n ≥ 0, if there exists an m < n such that am = an, take the largest such m and set an+1 = n − m; otherwise an+1 = 0. Thus the first occurrence of an integer in the sequence is followed by a 0, and the second and subsequent occurrences are followed by the size of the gap between the two most-recent occurrences. The first few terms of the sequence are (OEIS: A181391): :0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 ... van Eck's sequence (A181391) at the On-Line Encyclopedia of Integer Sequences The sequence was named by Neil Sloane after Jan Ritsema van Eck, who contributed it to the On-Line Encyclopedia of Integer Sequences in 2010.
|
|
Although Kahan's algorithm achieves O(1) error growth for summing n numbers, only slightly worse O(\log n) growth can be achieved by pairwise summation: one recursively divides the set of numbers into two halves, sums each half, and then adds the two sums. This has the advantage of requiring the same number of arithmetic operations as the naive summation (unlike Kahan's algorithm, which requires four times the arithmetic and has a latency of four times a simple summation) and can be calculated in parallel. The base case of the recursion could in principle be the sum of only one (or zero) numbers, but to amortize the overhead of recursion, one would normally use a larger base case. The equivalent of pairwise summation is used in many fast Fourier transform (FFT) algorithms and is responsible for the logarithmic growth of roundoff errors in those FFTs.
|
|
Calculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete.. A number of other papers state that finding a shortest addition chain for a single number is NP-complete, citing this paper, but it does not claim or prove such a result. There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. However, several techniques are known to calculate relatively short chains that are not always optimal.. One very well known technique to calculate relatively short addition chains is the binary method, similar to exponentiation by squaring. In this method, an addition chain for the number n is obtained recursively, from an addition chain for n'=\lfloor n/2\rfloor.
|
|
Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers. Given two large integers, a and b, Toom–Cook splits up a and b into k smaller parts each of length l, and performs operations on the parts. As k grows, one may combine many of the multiplication sub-operations, thus reducing the overall complexity of the algorithm. The multiplication sub-operations can then be computed recursively using Toom–Cook multiplication again, and so on. Although the terms "Toom-3" and "Toom–Cook" are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom–Cook algorithm, where k = 3. Toom-3 reduces 9 multiplications to 5, and runs in Θ(nlog(5)/log(3)) ≈ Θ(n1.46).
|
|
The parent of x in the Cartesian tree is either the left neighbor of x or the right neighbor of x, whichever exists and has a larger value. The left and right neighbors may also be constructed efficiently by parallel algorithms, so this formulation may be used to develop efficient parallel algorithms for Cartesian tree construction.. Another linear-time algorithm for Cartesian tree construction is based on divide-and-conquer. In particular, the algorithm recursively constructs the tree on each half of the input, and then merging the two trees by taking the right spine of the left tree and left spine of the right tree and performing a standard merging operation. The algorithm is also parallelizable since on each level of recursion, each of the two sub-problems can be computed in parallel, and the merging operation can be efficiently parallelized as well.
|
|
Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving a circular definition or self-reference, in which the putative recursive step does not get closer to a base case, but instead leads to an infinite regress. It is not unusual for such books to include a joke entry in their glossary along the lines of: :Recursion, see Recursion. A variation is found on page 269 in the index of some editions of Brian Kernighan and Dennis Ritchie's book The C Programming Language; the index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in "Let's talk Lisp" by Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975 with a copyright date of 1976) and in "Software Tools" by Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976).
|
|
By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size N = N1N2 into many smaller DFTs of sizes N1 and N2, along with O(N) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966). This method (and the general idea of an FFT) was popularized by a publication of Cooley and Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). The best known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size N/2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey).
|
|
In numerical analysis, pairwise summation, also called cascade summation, is a technique to sum a sequence of finite-precision floating-point numbers that substantially reduces the accumulated round-off error compared to naively accumulating the sum in sequence. Although there are other techniques such as Kahan summation that typically have even smaller round-off errors, pairwise summation is nearly as good (differing only by a logarithmic factor) while having much lower computational cost--it can be implemented so as to have nearly the same cost (and exactly the same number of arithmetic operations) as naive summation. In particular, pairwise summation of a sequence of n numbers xn works by recursively breaking the sequence into two halves, summing each half, and adding the two sums: a divide and conquer algorithm. Its worst-case roundoff errors grow asymptotically as at most O(ε log n), where ε is the machine precision (assuming a fixed condition number, as discussed below).
|
|
More efficient than the aforementioned are specialized partial sorting algorithms based on mergesort and quicksort. In the quicksort variant, there is no need to recursively sort partitions which only contain elements that would fall after the 'th place in the final sorted array (starting from the "left" boundary). Thus, if the pivot falls in position or later, we recurse only on the left partition: function partial_quicksort(A, i, j, k) is if i < j then p ← pivot(A, i, j) p ← partition(A, i, j, p) partial_quicksort(A, i, p-1, k) if p < k-1 then partial_quicksort(A, p+1, j, k) The resulting algorithm is called partial quicksort and requires an expected time of only , and is quite efficient in practice, especially if a selection sort is used as a base case when becomes small relative to . However, the worst-case time complexity is still very bad, in the case of a bad pivot selection.
|
|
Estrin's scheme operates recursively, converting a degree-n polynomial in x (for n≥2) to a degree- polynomial in x2 using independent operations (plus one to compute x2). Given an arbitrary polynomial P(x) = C0 \+ C1x + C2x2 \+ C3x3 \+ ⋯ + Cnxn, one can group adjacent terms into sub-expressions of the form (A + Bx) and rewrite it as a polynomial in x2: P(x) = (C0 \+ C1x) + (C2 \+ C3x)x2 \+ (C4 \+ C5x)x4 \+ ⋯ = Q(x2). Each of these sub-expressions, and x2, may be computed in parallel. They may also be evaluated using a native multiply–accumulate instruction on some architectures, an advantage that is shared with Horner's method. This grouping can then be repeated to get a polynomial in x4: P(x) = Q(x2) = ((C0 \+ C1x) + (C2 \+ C3x)x2) + ((C4 \+ C5x) + (C6 \+ C7x)x2)x4 \+ ⋯ = R(x4). Repeating this +1 times, one arrives at Estrin's scheme for parallel evaluation of a polynomial: # Compute Di = C2i \+ C2i+1x for all 0 ≤ i ≤ .
|
|
In the intelligence explosion scenario hypothesized by I. J. Good, recursively self-improving AI systems quickly transition from subhuman general intelligence to superintelligent. Nick Bostrom's 2014 book Superintelligence: Paths, Dangers, Strategies sketches out Good's argument in detail, while citing writing by Yudkowsky on the risk that anthropomorphizing advanced AI systems will cause people to misunderstand the nature of an intelligence explosion. "AI might make an apparently sharp jump in intelligence purely as the result of anthropomorphism, the human tendency to think of 'village idiot' and 'Einstein' as the extreme ends of the intelligence scale, instead of nearly indistinguishable points on the scale of minds-in-general." In their textbook on artificial intelligence, Stuart Russell and Peter Norvig raise the objection that there are known limits to intelligent problem-solving from computational complexity theory; if there are strong limits on how efficiently algorithms can solve various computer science tasks, then intelligence explosion may not be possible.
|
|
The process of making a BSP tree In computer science, binary space partitioning (BSP) is a method for recursively subdividing a space into two convex sets by using hyperplanes as partitions. This process of subdividing gives rise to a representation of objects within the space in the form of a tree data structure known as a BSP tree. Binary space partitioning was developed in the context of 3D computer graphics in 1969, where the structure of a BSP tree allows for spatial information about the objects in a scene that is useful in rendering, such as objects being ordered from front-to-back with respect to a viewer at a given location, to be accessed rapidly. Other applications of BSP include: performing geometrical operations with shapes (constructive solid geometry) in CAD, collision detection in robotics and 3D video games, ray tracing, and other applications that involve the handling of complex spatial scenes.
|
|
In his 1975 article "Outline of a Theory of Truth", Kripke showed that a language can consistently contain its own truth predicate, something deemed impossible by Alfred Tarski, a pioneer in formal theories of truth. The approach involves letting truth be a partially defined property over the set of grammatically well-formed sentences in the language. Kripke showed how to do this recursively by starting from the set of expressions in a language that do not contain the truth predicate, and defining a truth predicate over just that segment: this action adds new sentences to the language, and truth is in turn defined for all of them. Unlike Tarski's approach, however, Kripke's lets "truth" be the union of all of these definition-stages; after a denumerable infinity of steps the language reaches a "fixed point" such that using Kripke's method to expand the truth-predicate does not change the language any further.
|
|
Early versions of Digital Research's CP/M and CP/M-86 kept the DEC name DDT (and DDT-86 and DDT-68K) for their debugger, however, now meaning "Dynamic Debugging Tool". The CP/M DDT was later superseded by the Symbolic Instruction Debugger (SID, ZSID, SID86, and GEMSID) in DR DOS and GEM. In addition to its normal function as a debugger, DDT was also used as a top- level command shell for the Massachusetts Institute of Technology (MIT) Incompatible Timesharing System (ITS) operating system; on some more recent ITS systems, it is replaced with a "PWORD" which implements a restricted subset of DDT's functionality. DDT could run and debug up to eight processes (called "jobs" on ITS) at a time, such as several sessions of TECO, and DDT could be run recursively - that is, some or all of those jobs could themselves be DDTs (which could then run another eight jobs, and so on).
|
|
Geometrical dissection of an L-tromino (rep-4) Both types of tromino can be dissected into n2 smaller trominos of the same type, for any integer n > 1\. That is, they are rep-tiles.. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.. Motivated by the mutilated chessboard problem, Solomon W. Golomb used this tiling as the basis for what has become known as Golomb's tromino theorem: if any square is removed from a 2n × 2n chessboard, the remaining board can be completely covered with L-trominoes. To prove this by mathematical induction, partition the board into a quarter-board of size 2n−1 × 2n−1 that contains the removed square, and a large tromino formed by the other three quarter-boards.
|
|
So the maximal sets form an orbit, that is, every automorphism preserves maximality and any two maximal sets are transformed into each other by some automorphism. Harrington gave a further example of an automorphic property: that of the creative sets, the sets which are many-one equivalent to the halting problem. Besides the lattice of recursively enumerable sets, automorphisms are also studied for the structure of the Turing degrees of all sets as well as for the structure of the Turing degrees of r.e. sets. In both cases, Cooper claims to have constructed nontrivial automorphisms which map some degrees to other degrees; this construction has, however, not been verified and some colleagues believe that the construction contains errors and that the question of whether there is a nontrivial automorphism of the Turing degrees is still one of the main unsolved questions in this area (Slaman and Woodin 1986, Ambos-Spies and Fejer 2006).
|
|
The thesis that AI poses an existential risk, and that this risk needs much more attention than it currently gets, has been endorsed by many public figures; perhaps the most famous are Elon Musk, Bill Gates, and Stephen Hawking. The most notable AI researcher to endorse the thesis is Stuart J. Russell. Endorsers of the thesis sometimes express bafflement at skeptics: Gates states he does not "understand why some people are not concerned", and Hawking criticized widespread indifference in his 2014 editorial: Many of the scholars who are concerned about existential risk believe that the best way forward would be to conduct (possibly massive) research into solving the difficult "control problem" to answer the question: what types of safeguards, algorithms, or architectures can programmers implement to maximize the probability that their recursively- improving AI would continue to behave in a friendly, rather than destructive, manner after it reaches superintelligence? The thesis that AI can pose existential risk also has many strong detractors.
|
|
In practice, the O(N) additions can often be performed by absorbing the additions into the convolution: if the convolution is performed by a pair of FFTs, then the sum of xn is given by the DC (0th) output of the FFT of aq plus x0, and x0 can be added to all the outputs by adding it to the DC term of the convolution prior to the inverse FFT. Still, this algorithm requires intrinsically more operations than FFTs of nearby composite sizes, and typically takes 3-10 times as long in practice. If Rader's algorithm is performed by using FFTs of size N-1 to compute the convolution, rather than by zero padding as mentioned above, the efficiency depends strongly upon N and the number of times that Rader's algorithm must be applied recursively. The worst case would be if N-1 were 2N2 where N2 is prime, with N2-1 = 2N3 where N3 is prime, and so on.
|
|
A tree ear decomposition is a proper ear decomposition in which the first ear is a single edge and for each subsequent ear P_i , there is a single ear P_j , i>j , such that both endpoints of P_i lie on P_j . A nested ear decomposition is a tree ear decomposition such that, within each ear P_j , the set of pairs of endpoints of other ears P_i that lie within P_j form a set of nested intervals. A series-parallel graph is a graph with two designated terminals s and t that can be formed recursively by combining smaller series-parallel graphs in one of two ways: series composition (identifying one terminal from one smaller graph with one terminal from the other smaller graph, and keeping the other two terminals as the terminals of the combined graph) and parallel composition (identifying both pairs of terminals from the two smaller graphs). The following result is due to : :A 2-vertex-connected graph is series-parallel if and only if it has a nested ear decomposition.
|
|
Separators may be combined into a separator hierarchy of a planar graph, a recursive decomposition into smaller graphs. A separator hierarchy may be represented by a binary tree in which the root node represents the given graph itself, and the two children of the root are the roots of recursively constructed separator hierarchies for the induced subgraphs formed from the two subsets A and B of a separator. A separator hierarchy of this type forms the basis for a tree decomposition of the given graph, in which the set of vertices associated with each tree node is the union of the separators on the path from that node to the root of the tree. Since the sizes of the graphs go down by a constant factor at each level of the tree, the upper bounds on the sizes of the separators also go down by a constant factor at each level, so the sizes of the separators on these paths add in a geometric series to O(√n).
|
|
Consider the following pseudocode function to calculate the factorial of n: function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else return factorial(n – 1) times n [recursively invoke factorial with the parameter 1 less than n] end if end function For every integer n such that `n≥0`, the final result of the function `factorial` is invariant; if invoked as `x = factorial(3)`, the result is such that x will always be assigned the value 6. The non-memoized implementation above, given the nature of the recursive algorithm involved, would require n + 1 invocations of `factorial` to arrive at a result, and each of these invocations, in turn, has an associated cost in the time it takes the function to return the value computed. Depending on the machine, this cost might be the sum of: # The cost to set up the functional call stack frame. # The cost to compare n to 0. # The cost to subtract 1 from n.
|
|
See general set theory for more details. Q is fascinating because it is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier, yet like PA is incomplete and incompletable in the sense of Gödel's incompleteness theorems, and essentially undecidable. Robinson (1950) derived the Q axioms (1)–(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA. The only use this proof makes of the PA axiom schema of induction is to prove a statement that is axiom (3) above, and so, all computable functions are representable in Q (Mendelson 1997: Th. 3.33, Rautenberg 2010: 246). The conclusion of Gödel's second incompleteness theorem also holds for Q: no consistent recursively axiomatized extension of Q can prove its own consistency, even if we additionally restrict Gödel numbers of proofs to a definable cut (Bezboruah and Shepherdson 1976; Pudlák 1985; Hájek & Pudlák 1993:387).
|
|
In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system “runs out of fuel” and cannot name a certain ordinal, a much larger ordinal is brought “from above” to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e.
|
|
In the case where all elements are equal, Hoare partition scheme needlessly swaps elements, but the partitioning itself is best case, as noted in the Hoare partition section above. To solve the Lomuto partition scheme problem (sometimes called the Dutch national flag problem), an alternative linear-time partition routine can be used that separates the values into three groups: values less than the pivot, values equal to the pivot, and values greater than the pivot. (Bentley and McIlroy call this a "fat partition" and it was already implemented in the of Version 7 Unix.) The values equal to the pivot are already sorted, so only the less-than and greater-than partitions need to be recursively sorted. In pseudocode, the quicksort algorithm becomes algorithm quicksort(A, lo, hi) is if lo < hi then p := pivot(A, lo, hi) left, right := partition(A, p, lo, hi) // note: multiple return values quicksort(A, lo, left - 1) quicksort(A, right + 1, hi) The `partition` algorithm returns indices to the first ('leftmost') and to the last ('rightmost') item of the middle partition.
|
|
Thus an oracle machine with a noncomputable oracle will be able to compute sets that a Turing machine without an oracle cannot. Informally, a set of natural numbers A is Turing reducible to a set B if there is an oracle machine that correctly tells whether numbers are in A when run with B as the oracle set (in this case, the set A is also said to be (relatively) computable from B and recursive in B). If a set A is Turing reducible to a set B and B is Turing reducible to A then the sets are said to have the same Turing degree (also called degree of unsolvability). The Turing degree of a set gives a precise measure of how uncomputable the set is. The natural examples of sets that are not computable, including many different sets that encode variants of the halting problem, have two properties in common: #They are recursively enumerable, and #Each can be translated into any other via a many-one reduction.
|
|
For example, the path length of point xi in Fig. 2 is greater than the path length of xj in Fig. 3. More formally, let X = { x1, ..., xn } be a set of d-dimensional points and X’ ⊂ X a subset of X. An Isolation Tree (iTree) is defined as a data structure with the following properties: # for each node T in the Tree, T is either an external-node with no child, or an internal-node with one “test” and exactly two daughter nodes (Tl, Tr) # a test at node T consists of an attribute q and a split value p such that the test q < p determines the traversal of a data point to either Tl or Tr. In order to build an iTree, the algorithm recursively divides X’ by randomly selecting an attribute q and a split value p, until either (i) the node has only one instance or (ii) all data at the node have the same values. When the iTree is fully grown, each point in X is isolated at one of the external nodes.
|
|
A tree data structure can be defined recursively as a collection of nodes (starting at a root node), where each node is a data structure consisting of a value, together with a list of references to nodes (the "children"), with the constraints that no reference is duplicated, and none points to the root. Alternatively, a tree can be defined abstractly as a whole (globally) as an ordered tree, with a value assigned to each node. Both these perspectives are useful: while a tree can be analyzed mathematically as a whole, when actually represented as a data structure it is usually represented and worked with separately by node (rather than as a set of nodes and an adjacency list of edges between nodes, as one may represent a digraph, for instance). For example, looking at a tree as a whole, one can talk about "the parent node" of a given node, but in general as a data structure a given node only contains the list of its children, but does not contain a reference to its parent (if any).
|
|
Ohad Rodeh's original proposal at USENIX 2007 noted that B+ trees, which are widely used as on-disk data structures for databases, could not efficiently allow copy-on-write-based snapshots because its leaf nodes were linked together: if a leaf was copy-on-written, its siblings and parents would have to be as well, as would their siblings and parents and so on until the entire tree was copied. He suggested instead a modified B-tree (which has no leaf linkage), with a refcount associated to each tree node but stored in an ad-hoc free map structure and certain relaxations to the tree's balancing algorithms to make them copy-on-write friendly. The result would be a data structure suitable for a high-performance object store that could perform copy-on-write snapshots, while maintaining good concurrency. At Oracle later that year, Chris Mason began work on a snapshot-capable file system that would use this data structure almost exclusively—not just for metadata and file data, but also recursively to track space allocation of the trees themselves.
|
|
The randomized binary search tree, introduced by Martínez and Roura subsequently to the work of Aragon and Seidel on treaps, stores the same nodes with the same random distribution of tree shape, but maintains different information within the nodes of the tree in order to maintain its randomized structure. Rather than storing random priorities on each node, the randomized binary search tree stores a small integer at each node, the number of its descendants (counting itself as one); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation. When a key x is to be inserted into a tree that already has n nodes, the insertion algorithm chooses with probability 1/(n + 1) to place x as the new root of the tree, and otherwise it calls the insertion procedure recursively to insert x within the left or right subtree (depending on whether its key is less than or greater than the root). The numbers of descendants are used by the algorithm to calculate the necessary probabilities for the random choices at each step.
|
|
One structure that can be used for this purpose is the convex layers of the input point set, a family of nested convex polygons consisting of the convex hull of the point set and the recursively-constructed convex layers of the remaining points. Within a single layer, the points inside the query half-plane may be found by performing a binary search for the half-plane boundary line's slope among the sorted sequence of convex polygon edge slopes, leading to the polygon vertex that is inside the query half-plane and farthest from its boundary, and then sequentially searching along the polygon edges to find all other vertices inside the query half-plane. The whole half-plane range reporting problem may be solved by repeating this search procedure starting from the outermost layer and continuing inwards until reaching a layer that is disjoint from the query halfspace. Fractional cascading speeds up the successive binary searches among the sequences of polygon edge slopes in each layer, leading to a data structure for this problem with space O(n) and query time O(log n + h).
|
|
Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. (The Rader–Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite N. Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial zN − 1, here into real-coefficient polynomials of the form zM − 1 and z2M + azM + 1\. Another polynomial viewpoint is exploited by the Winograd FFT algorithm, which factorizes zN − 1 into cyclotomic polynomials—these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. Indeed, Winograd showed that the DFT can be computed with only O(N) irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.
|
|
In Douglas Hofstadter's Gödel, Escher, Bach, there is a narrative between Achilles and the Tortoise (characters borrowed from Lewis Carroll, who in turn borrowed them from Zeno), and within this story they find a book entitled "Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe", which they begin to read, the Tortoise taking the part of the Tortoise, and Achilles taking the part of Achilles. Within this narrative, which itself is somewhat self-referential, the two characters find a book entitled "Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe", which they begin to read, the Tortoise taking the part of Achilles, and Achilles taking the part of the Tortoise. Italo Calvino's experimental book, If on a winter's night a traveler, is about a reader, addressed in the second person, trying to read the very same book, but being interrupted by ten other recursively nested incomplete stories. Robert Altman's satirical noir The Player about Hollywood ends with the antihero being pitched a movie version of his own story, complete with an unlikely happy ending.
|
|
In quicksort, there is a subprocedure called `partition` that can, in linear time, group a list (ranging from indices `left` to `right`) into two parts: those less than a certain element, and those greater than or equal to the element. Here is pseudocode that performs a partition about the element `list[pivotIndex]`: function partition(list, left, right, pivotIndex) is pivotValue := list[pivotIndex] swap list[pivotIndex] and list[right] // Move pivot to end storeIndex := left for i from left to right − 1 do if list[i] < pivotValue then swap list[storeIndex] and list[i] increment storeIndex swap list[right] and list[storeIndex] // Move pivot to its final place return storeIndex This is known as the Lomuto partition scheme, which is simpler but less efficient than Hoare's original partition scheme. In quicksort, we recursively sort both branches, leading to best-case O(n log n) time. However, when doing selection, we already know which partition our desired element lies in, since the pivot is in its final sorted position, with all those preceding it in an unsorted order and all those following it in an unsorted order.
|
|
Whether including K_0 as a valid graph is useful depends on context. On the positive side, K_0 follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures K_0 is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children). On the negative side, including K_0 as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph", or the definition of connected graphs has to be modified not to include K0). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies "graph with at least one vertex" unless context suggests otherwise.
|
|
Klarner was born in Fort Bragg, California, and spent his childhood in Napa, California.University of Calgary: Archives and Special Collections: David A. Klarner He married Kara Lynn Klarner in 1961. Their son Carl Eoin Klarner was born on April 21, 1969.Carl is a Political Scientist, receiving tenure at Indiana State University and currently working at the University of Florida as a research associate. Klarner did his undergraduate work at Humboldt State University (1960–63), got his Ph.D. at the University of Alberta (1963–66), and did post- doctoral work at McMaster University in Hamilton, Ontario (1966–68). He also did post-doctoral work at Eindhoven University of Technology in the Netherlands (1968-1970), at the University of Reading in England working with Richard Rado (1970–71),Arithmetic properties of certain recursively defined sets by D. A. Klarner and R. Rado, Stanford University: Computer Science Department, March 1972 and at Stanford University (1971–73). He served as an assistant professor at Binghamton University (1973–79) and was a visiting professor at Humboldt State University in California (1979–80). He returned to Eindhoven as a professor (1980–81), and to Binghamton (1981–82).
|
|
Placing x at the root of a subtree may be performed either as in the treap by inserting it at a leaf and then rotating it upwards, or by an alternative algorithm described by Martínez and Roura that splits the subtree into two pieces to be used as the left and right children of the new node. The deletion procedure for a randomized binary search tree uses the same information per node as the insertion procedure, but unlike the insertion procedure it only needs on average O(1) random decisions to join the two subtrees descending from the left and right children of the deleted node into a single tree. That is because the subtrees to be joined are on average at depth Θ(log n); joining two trees of size n and m needs Θ(log(n+m)) random choices on average. If the left or right subtree of the node to be deleted is empty, the join operation is trivial; otherwise, the left or right child of the deleted node is selected as the new subtree root with probability proportional to its number of descendants, and the join proceeds recursively.
|
|
Fair-share scheduling is a scheduling algorithm for computer operating systems in which the CPU usage is equally distributed among system users or groups, as opposed to equal distribution among processes. One common method of logically implementing the fair-share scheduling strategy is to recursively apply the round-robin scheduling strategy at each level of abstraction (processes, users, groups, etc.) The time quantum required by round-robin is arbitrary, as any equal division of time will produce the same results. This was first developed by Judy Kay and Piers Lauder through their research at Sydney University in the 1980s. For example, if four users (A,B,C,D) are concurrently executing one process each, the scheduler will logically divide the available CPU cycles such that each user gets 25% of the whole (100% / 4 = 25%). If user B starts a second process, each user will still receive 25% of the total cycles, but each of user B's processes will now be attributed 12.5% of the total CPU cycles each, totalling user B's fair share of 25%. On the other hand, if a new user starts a process on the system, the scheduler will reapportion the available CPU cycles such that each user gets 20% of the whole (100% / 5 = 20%).
|
|
Otherwise, we have an assignment for the top row of the board and recursively compute the number of solutions to the remaining board, adding the numbers of solutions for every admissible assignment of the top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occurs for a board. The number of solutions for this board is either zero or one, depending on whether the vector is a permutation of (0, 1) and (1, 0) pairs or not. For example, in the first two boards shown above the sequences of vectors would be ((2, 2) (2, 2) (2, 2) (2, 2)) ((2, 2) (2, 2) (2, 2) (2, 2)) k = 4 0 1 0 1 0 0 1 1 ((1, 2) (2, 1) (1, 2) (2, 1)) ((1, 2) (1, 2) (2, 1) (2, 1)) k = 3 1 0 1 0 0 0 1 1 ((1, 1) (1, 1) (1, 1) (1, 1)) ((0, 2) (0, 2) (2, 0) (2, 0)) k = 2 0 1 0 1 1 1 0 0 ((0, 1) (1, 0) (0, 1) (1, 0)) ((0, 1) (0, 1) (1, 0) (1, 0)) k = 1 1 0 1 0 1 1 0 0 ((0, 0) (0, 0) (0, 0) (0, 0)) ((0, 0) (0, 0), (0, 0) (0, 0)) The number of solutions is : 1,\, 2,\, 90,\, 297200,\, 116963796250,\, 6736218287430460752, \ldots Links to the MAPLE implementation of the dynamic programming approach may be found among the external links.
|
|