" Lawless agrees, "If we, like our sighted counterparts are skilled, focused, and determined, then there will be enumerable possibilities.
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A hacker attempting to modify certificates would be unable to pull off such a fraud, as it would mean changing data stored on enumerable diversified blocks spread out across the cyber sphere.
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His spellbinding images are packed with machinery and infrastructure so complex it seems alive, from knitting machines with enumerable arms to solar furnaces that look like giant compound eyes—more Roald Dahl than Charles Dickens.
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In enumerable ways large and small, many of the 3.4 million inhabitants of Puerto Rico struggled through a 10th day with little or no access to basic necessities - from electricity and clean, running water to communications, food and medicine.
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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.
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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.
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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.
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The class of all recursively enumerable languages is called RE.
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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.
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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.
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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.
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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.
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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.
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It is equivalent to the standard Turing machine and therefore accepts precisely the recursively enumerable languages.
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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.
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The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory.
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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.
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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.
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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".
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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".
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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).
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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.
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The basic idea of Chomsky's work is to divide language into four levels, or the language hierarchy. The four levels are: regular, context-free, context-sensitive, and computably enumerable or unrestricted. Regular is the least complex while computably enumerable is the most complex. While his work grew out of combinatorics on words, it drastically affected other disciplines, especially computer science.
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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.
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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.
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Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.
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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.
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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.
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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.
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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.
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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.
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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.
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Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: :Every computably enumerable set is Diophantine. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0. Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0.
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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.
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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.
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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.
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The question "Does any arbitrary "Diophantine equation" have an integer solution?" is undecidable.That is, it is impossible to answer the question for all cases. Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich- Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can decide whether or not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an example of a computably enumerable but not decidable set, and the set of unsolvable Diophantine equations is not computably enumerable" (p. 71).
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.Recursively enumerable sets and degrees, Soare, Robert I.
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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.
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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.
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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.
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The topography of the land within the village boundary varies from hilly to plains. The forest areas are hilly with plateau at the top. They are cris- crossed with enumerable streams. The areas under the cultivation is generally flat especially where it is irrigated.
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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.
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Wolfram argues that science is far too ad hoc, in part because the models used are too complicated and unnecessarily organized around the limited primitives of traditional mathematics. Wolfram advocates using models whose variations are enumerable and whose consequences are straightforward to compute and analyze.
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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.
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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.
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Yuri Matiyasevich utilized a method involving Fibonacci numbers, which grow exponentially, in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam – hence, MRDP – had shown that this suffices to show that every computably enumerable set is Diophantine.
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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.
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In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable.
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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.
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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.
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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.
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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.
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To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.
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Here, the difference between the two notions of program mentioned in the last paragraph becomes clear; one is easily recognized by some grammar, while the other requires arbitrary computation to recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent to the halting problem.
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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.
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In Ayers' words, speckles are enumerable only if in fact they have been enumerated. A number of philosophers analyzed the merits of this proposition. Chisholm concludes that the problem of the speckled hen emphasizes the fact that there are basic propositions (synthetic propositions which do not refer beyond the content of the immediate experience) that are necessarily imprecise.
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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.
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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.
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He was among the first to claim that Gödel's incompleteness theorem is relevant for theories of everything (TOE) in theoretical physics.Cf. Jaki's "A Late Awakening to Gödel in Physics" Gödel's theorem states that any theory that includes certain basic facts of number theory and is computably enumerable will be either incomplete or inconsistent. Since any 'theory of everything' must be consistent, it also must be incomplete.
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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.
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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).
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Gács, P. and Vitányi, P., "In Memoriam Raymond J. Solomonoff", IEEE Information Theory Society Newsletter, Vol. 61, No. 1, March 2011, p 11. Solomonoff's enumerable measure is universal in a certain powerful sense, but the computation time can be infinite. One way of dealing with this issue is a variant of Leonid Levin's Search Algorithm,Levin, L.A., "Universal Search Problems", in Problemy Peredaci Informacii 9, pp.
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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.
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Flowchart for creating, searching and updating WinFS data instances The primary mode of data retrieval from a WinFS store is querying the WinFS store according to some criteria, which returns an enumerable set of items matching the criteria. The criteria for the query is specified using the OPath query language. The returned data are made available as instances of the type schemas, conforming to the .NET object model.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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In total functional programming languages, such as Charity and Epigram, all functions are total and must terminate. Charity uses a type system and control constructs based on category theory, whereas Epigram uses dependent types. The LOOP language is designed so that it computes only the functions that are primitive recursive. All of these compute proper subsets of the total computable functions, since the full set of total computable functions is not computably enumerable.
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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.
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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.
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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.
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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.
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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.
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Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system.
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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.
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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.
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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.
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For instance, in 1970, it was proven, as a solution to Hilbert's 10th problem, that there is no Turing machine which can solve all Diophantine equations. Reprinted in The Collected Works of Julia Robinson, Solomon Feferman, editor, pp. 269–378, American Mathematical Society 1996. In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions.
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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.
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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.
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The Zorro bus and ISA bus can be connected by means of a "bridgeboard", such as, the Janus Hardware Emulator, which allows emulation of Intel 80286 or 80386 systems. Zorro III is the 32 bit auto-configuring expansion bus of Amiga 3000 and Amiga 4000 systems. From the A3000 design onwards, it was deemed desirable for all enumerable hardware expansions to use Autoconfig. It is OS-legal for non-Autoconfig hardware to be completely ignored and the standard was adopted in AmigaOS 3.1.
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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.
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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.
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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.
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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.
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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.
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Assigning a Gödel number to each Turing machine definition produces a subset S of the natural numbers corresponding to the computable numbers and identifies a surjection from S to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable. The set S of these Gödel numbers, however, is not computably enumerable (and consequently, neither are subsets of S that are defined in terms of it). This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals.
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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, .
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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.
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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.
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Although Jainism is a religion and philosophy predates its most famous exponent, the great Mahaviraswami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period." A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite.
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He also showed that the K-trivials are computable in the halting problem. This class of sets is commonly known as \Delta_2^0 sets in arithmetical hierarchy. Robert M. Solovay was the first to construct a noncomputable K-trivial set, while construction of a computably enumerable such A was attempted by Calude, Coles Cristian Calude, Richard J. Coles, Program-Size Complexity of Initial Segments and Domination Reducibility, (1999), proceeding of: Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa and other unpublished constructions by Kummer of a K-trivial, and Muchnik junior of a low for K set.
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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.
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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.
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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.
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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.
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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.
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George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388).
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This is a method for producing complete theories through the semantic route, with examples including the set of true sentences under the structure (N, +, ×, 0, 1, =), where N is the set of natural numbers, and the set of true sentences under the structure (R, +, ×, 0, 1, =), where R is the set of real numbers. The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms. The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).
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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.
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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).
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The above applies to first order theories, such as Peano arithmetic. However, for a specific model that may be described by a first order theory, some statements may be true but undecidable in the theory used to describe the model. For example, by Gödel's incompleteness theorem, we know that any theory whose proper axioms are true for the natural numbers cannot prove all first order statements true for the natural numbers, even if the list of proper axioms is allowed to be infinite enumerable. It follows that an automated theorem prover will fail to terminate while searching for a proof precisely when the statement being investigated is undecidable in the theory being used, even if it is true in the model of interest.
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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.
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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.
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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.
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LINQ extends the language by the addition of query expressions, which are akin to SQL statements, and can be used to conveniently extract and process data from arrays, enumerable classes, XML documents, relational databases, and third-party data sources. Other uses, which utilize query expressions as a general framework for readably composing arbitrary computations, include the construction of event handlers or monadic parsers. It also defines a set of method names (called standard query operators, or standard sequence operators), along with translation rules used by the compiler to translate query syntax expressions into expressions using fluent-style (called method syntax by Microsoft) with these method names, lambda expressions and anonymous types. Many of the concepts that LINQ introduced were originally tested in Microsoft's Cω research project.
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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.
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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.
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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]).
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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.
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As mentioned above, the first n bits of Gregory Chaitin's constant Ω are random or incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O(1) bits. However, consider the short but never halting algorithm which systematically lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first n bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) onto the first n bits of Ω. In other words, the enumerable first n bits of Ω are highly compressible in the sense that they are limit-computable by a very short algorithm; they are not random with respect to the set of enumerating algorithms.
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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).
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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.
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