535 Sentences With "computable" | Random Sentence Generator

But what else might you have thought to say that is not computable?

Making real-world information computable is challenging, but it has inspired some creative approaches. Cortical.

I watched the students pass by, with their identical backpacks, similar haircuts, and computable faces.

To do this right, we should use one of those computable general equilibrium models I mentioned above.

This is what it is to be computable at all—an assurance that the machine won't run forever.

This was a hypothetical device that could come up with a solution to any problem that is computable.

When economists try to assess changes in trade policy, they normally use some kind of "computable general equilibrium" (CGE) model.

The paper outlines a method for multi-advisor reinforcement learning that breaks problems down to be simpler and more easily computable.

It would also include retail clients who acquired subordinated obligations computable as Tier 2 of the July 29, 2011 and Oct.

Technologists inhabit the worlds of deterministic logic and computable probabilities, but lawyers are at home amid unpredictability, noncompliance, and even the possibility of catastrophe.

But this strategy doesn't count as computable, because useful computations must finish in a predictable time, whether that's one day, or one year, or one billion years.

You need a "computable general equilibrium" model of world trade – something that shows how production and trade flows depend on tariff rates, calibrated to match the actual data.

The five-qubit machine, which is described in the journal Nature, represents a dramatic step toward general-purpose quantum computing—and, with it, an upending of what we can even consider to be computable.

A year later, he published the groundbreaking paper "On Computable Numbers, With an Application to the Entscheidungsproblem" (or "decidability problem"), a reference in German to a celebrated riddle that the American logician Alonzo Church had also explained.

The field's pioneers had convincingly argued that classical computers—epitomized by the mathematical abstraction known as a Turing machine—should be able to compute everything that is computable in the physical universe, from basic arithmetic to stock trades to black hole collisions.

Not much to add to this one: Web series Computerphile visits intelligent systems researcher Rob Miles at the University of Nottingham for a breezy but also pretty dense interview on the enormous question of how computers play games—or how games become computable.

As well as an image of Turing, the new note will feature a table and mathematical formulae from a 1936 paper by Turing on computable numbers, an image of a pilot computer and technical drawings for the machines used to break the Enigma code.

The image of a computable set under a total computable bijection is computable. A set is recursive if and only if it is at level of the arithmetical hierarchy. A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set. The image of a computable set under a nondecreasing total computable function is computable.

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.

Each primitive recursive function is LOOP-computable and vice versa. In contrast to GOTO programs and WHILE programs, LOOP programs always terminate. Therefore, the set of functions computable by LOOP-programs is a proper subset of computable functions (and thus a subset of the computable by WHILE and GOTO program functions). An example of a total computable function that is not LOOP computable is the Ackermann function.

In computer science, Programming Computable Functions' (PCF) is a typed functional language introduced by Gordon Plotkin in 1977, based on previous unpublished material by Dana Scott."PCF is a programming language for computable functions, based on LCF, Scott’s logic of computable functions" . Programming Computable Functions is used by . It is also referred to as Programming with Computable Functions or Programming language for Computable Functions.

Dana Scott's LCF language"PCF is a programming language for computable functions, based on LCF, Scott’s logic of computable functions" . Programming Computable Functions is used by . It is also referred to as Programming with Computable Functions or Programming language for Computable Functions. was a stage in the evolution of lambda calculus into modern functional languages.

The uniform norm operator is also computable. This implies the computability of Riemann integration. The Riemann integral is a computable operator: In other words, there is an algorithm that will numerically evaluate the integral of any computable function. The differentiation operator over real valued functions is not computable, but over complex functions is computable.

The compression theorem is an important theorem about the complexity of computable functions. The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

If the sequence is uniformly computable relative to D, then the function is limit computable in D.

The arithmetical operations on computable numbers are themselves computable in the sense that whenever real numbers a and b are computable then the following real numbers are also computable: a + b, a - b, ab, and a/b if b is nonzero. These operations are actually uniformly computable; for example, there is a Turing machine which on input (A,B,\epsilon) produces output r, where A is the description of a Turing machine approximating a, B is the description of a Turing machine approximating b, and r is an \epsilon approximation of a+b. The fact that computable real numbers form a field was first proved by Henry Gordon Rice in 1954 (Rice 1954). Computable reals however do not form a computable field, because the definition of a computable field requires effective equality.

In computational complexity theory the compression theorem is an important theorem about the complexity of computable functions. The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

In mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory.

The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that they are countable. But, again, this subset is not computable, even though the computable reals are themselves ordered.

A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable.

Therefore, almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.

Unfortunately, Solomonoff also proved that Solomonoff's induction is uncomputable. In fact, he showed that computability and completeness are mutually exclusive: any complete theory must be uncomputable. The proof of this is derived from a game between the induction and the environment. Essentially, any computable induction can be tricked by a computable environment, by choosing the computable environment that negates the computable induction's prediction.

Not all real numbers are computable. The entire set of computable numbers is countable, so most reals are not computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.

However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. The set of computable numbers has the same cardinality as the natural numbers.

Rogers (1967) has suggested that a key property of recursion theory is that its results and structures should be invariant under computable bijections on the natural numbers (this suggestion draws on the ideas of the Erlangen program in geometry). The idea is that a computable bijection merely renames numbers in a set, rather than indicating any structure in the set, much as a rotation of the Euclidean plane does not change any geometric aspect of lines drawn on it. Since any two infinite computable sets are linked by a computable bijection, this proposal identifies all the infinite computable sets (the finite computable sets are viewed as trivial). According to Rogers, the sets of interest in recursion theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers.

This result is closely related to the fact that (as Louise Hay and Joseph Rosenstein proved) there exist computable linear orders with no computable non-identity self-embedding.

The notion of computability of a function can be relativized to an arbitrary set of natural numbers A. A function f is defined to be computable in A (equivalently A-computable or computable relative to A) when it satisfies the definition of a computable function with modifications allowing access to A as an oracle. As with the concept of a computable function relative computability can be given equivalent definitions in many different models of computation. This is commonly accomplished by supplementing the model of computation with an additional primitive operation which asks whether a given integer is a member of A. We can also talk about f being computable in g by identifying g with its graph.

Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable. This term has since come to be identified with the computable functions.

10) Thus we have arrived at Turing's Thesis: : "Every function which would naturally be regarded as computable is computable ... by one of his machines..." (Kleene (1952) p.376) Although Kleene did not give examples of "computable functions" others have. For example, Davis (1958) gives Turing tables for the Constant, Successor and Identity functions, three of the five operators of the primitive recursive functions: : Computable by Turing machine: :: Addition (also is the Constant function if one operand is 0) :: Increment (Successor function) :: Common subtraction (defined only if x ≥ y). Thus "x − y" is an example of a partially computable function.

In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number x, the well ordering principle provides that there is a minimal element in S which corresponds to x, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection.

To actually develop analysis over computable numbers, some care must be taken. For example, if one uses the classical definition of a sequence, the set of computable numbers is not closed under the basic operation of taking the supremum of a bounded sequence (for example, consider a Specker sequence, see the section above). This difficulty is addressed by considering only sequences which have a computable modulus of convergence. The resulting mathematical theory is called computable analysis.

See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant. Similarly, most subsets of the natural numbers are not computable.

In computability theory, a Turing degree [X] is low if the Turing jump [X′] is 0′. A set is low if it has low degree. Since every set is computable from its jump, any low set is computable in 0′, but the jump of sets computable in 0′ can bound any degree r.e. in 0′ (Schoenfield Jump Inversion).

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.

Every computable real function is continuous (Weihrauch 2000, p. 6). The arithmetic operations on real numbers are computable. There is a subset of the real numbers called the computable numbers, which by the results above is a real closed field. While the equality relation is not decidable, the greater-than predicate on unequal real numbers is decidable.

In computability theory, the theorem, or universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions. It affirms the existence of a computable universal function, which is capable of calculating any other computable function. The universal function is an abstract version of the universal Turing machine, thus the name of the theorem. Roger's equivalence theorem provides a characterization of the Gödel numbering of the computable functions in terms of the smn theorem and the UTM theorem.

Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines.

After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of the book concerns the authors' results. These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self- adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators. The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically-characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space. The authors are motivated in part by the computability of solutions to differential equations.

R is equal to the set of all total computable functions.

A real number is a computable number if there is an algorithm that, given a natural number n, produces a decimal expansion for the number accurate to n decimal places. This notion was introduced by Alan Turing in 1936. The computable numbers include the algebraic numbers along with many transcendental numbers including π and e. Like the algebraic numbers, the computable numbers also form a subfield of the real numbers, and the positive computable numbers are closed under taking nth roots for each positive n.

Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology.

Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditions can be shown to exist. Computable model theory was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independently.

A computable number, also known as recursive number, is a real number such that there exists an algorithm which, given a positive number n as input, produces the first n digits of the computable number's decimal representation. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers. The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits.

Type 1 computability is the naive form of computable analysis in which one restricts the inputs to a machine to be computable numbers instead of arbitrary real numbers. The difference between the two models lies in the fact that a function which is well-behaved over computable numbers (in the sense of being total) is not necessarily well-behaved over arbitrary real numbers. For instance, there are continuous and computable functions over the computable real numbers which are total, but which map some closed intervals to unbounded open intervals. These functions clearly cannot be extended to arbitrary real numbers without making them partial, as doing so would violate the Extreme value theorem.

In computational complexity theory, Blum's speedup theorem, first stated by Manuel Blum in 1967, is a fundamental theorem about the complexity of computable functions. Each computable function has an infinite number of different program representations in a given programming language. In the theory of algorithms one often strives to find a program with the smallest complexity for a given computable function and a given complexity measure (such a program could be called optimal). Blum's speedup theorem shows that for any complexity measure there are computable functions that are not optimal with respect to that measure.

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.

The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence. More generally, a Specker sequence is called a recursive counterexample to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals.

Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) can manipulate them (and, essentially, compare them). A different definition uses Kleene's system of ordinal notations.

There is no algorithm that takes as input any two lambda expressions and outputs `TRUE` or `FALSE` depending on whether or not the two expressions are equivalent. More precisely, no computable function can decide the equivalence. This was historically the first problem for which undecidability could be proven. As usual for such a proof, computable means computable by any model of computation that is Turing complete.

In the field of theoretical computer science the computability and complexity of computational problems are often sought-after. Computability theory describes the degree to which problems are computable, whereas complexity theory describes the asymptotic degree of resource consumption. Computational problems are therefore confined into complexity classes. The arithmetical hierarchy and polynomial hierarchy classify the degree to which problems are respectively computable and computable in polynomial time.

Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet }). The real numbers are uncountable so most real numbers are not computable.

Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical. Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable. The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy.

They provide an example of computable and continuous initial conditions for the wave equation (with however a non-computable gradient) that lead to a continuous but not computable solution at a later time. However, they show that this phenomenon cannot occur for the heat equation or for Laplace's equation. The book also includes a collection of open problems, likely to inspire its readers to more research in this area.

Thus for each i, as t increases the value of \phi(t,i) eventually becomes constant and equals \omega(i). As with the case of computable real numbers, it is not possible to effectively move between the two representations of limit computable reals.

Zuse's Thesis He pointed out that a simple explanation of the universe would be a Turing machine programmed to execute all possible programs computing all possible histories for all types of computable physical laws. He also pointed out that there is an optimally efficient way of computing all computable universes based on Leonid Levin's universal search algorithm (published in 1973). (pdf) In 2000, he expanded this work by combining Ray Solomonoff's theory of inductive inference with the assumption that quickly computable universes are more likely than others. This work on digital physics also led to limit-computable generalizations of algorithmic information or Kolmogorov complexity and the concept of Super Omegas, which are limit-computable numbers that are even more random (in a certain sense) than Gregory Chaitin's number of wisdom Omega.

A single-valued numbering of the set of partial computable functions is called a Friedberg numbering.

Given a Blum complexity measure (\varphi, \Phi) and a total computable function f with two parameters, then there exists a total computable predicate g (a boolean valued computable function) so that for every program i for g, there exists a program j for g so that for almost all x :f(x, \Phi_j(x)) \leq \Phi_i(x) \, f is called the speedup function. The fact that it may be as fast-growing as desired (as long as it is computable) means that the phenomenon of always having a program of smaller complexity remains even if by "smaller" we mean "significantly smaller" (for instance, quadratically smaller, exponentially smaller).

The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers. Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers. The question naturally arises of whether it is possible to dispose of the full set of reals and use computable numbers for all of mathematics. This idea is appealing from a constructivist point of view, and has been pursued by what Bishop and Richman call the Russian school of constructive mathematics.

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.

The book concerns computable analysis, a branch of mathematical analysis founded by Alan Turing and concerned with the computability of constructions in analysis. This area is connected to, but distinct from, constructive analysis, reverse mathematics, and numerical analysis. The early development of the field was summarized in a book by Oliver Aberth, Computable Analysis (1980), and Computability in Analysis and Physics provides an update, incorporating substantial developments in this area by its authors. In contrast to the Russian school of computable analysis led by Andrey Markov Jr., it views computability as a distinguishing property of mathematical objects among others, rather than developing a theory that concerns only computable objects.

His work is almost entirely devoted to Computable Economics, Macroeconomic Theory and the History and Philosophy of Economics. Within Computable Economics, his major focus has been an attempt to mathematize economic theory—both micro and macro theory—using the methods of recursion theory and constructive mathematics.

An alternate classification of these sets by way of iterated computable functionals is provided by hyperarithmetical theory.

In recursion theory, \phi_e denotes the computable function with index (program) e in some standard numbering of computable functions, and \phi^B_e denotes the eth computable function using a set B of natural numbers as an oracle. A set A of natural numbers is Turing reducible to a set B if there is a computable function that, given an oracle for set B, computes the characteristic function χA of the set A. That is, there is an e such that \chi_A = \phi^B_e. This relationship is denoted A ≤T B; the relation ≤T is a preorder. Two sets of natural numbers are Turing equivalent if each is Turing reducible to the other.

A subset S of the natural numbers is called recursive if there exists a total computable function f such that f(x) = 1 if x∈S and f(x) = 0 if x∉S. In other words, the set S is recursive if and only if the indicator function 1S is computable.

In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis.

In this context, real numbers are represented as arbitrary infinite sequences of symbols. These sequences could for instance represent the digits of a real number. Such sequences need not be computable. On the other hand, the programs that act on these sequences do need to be computable in a reasonable sense.

An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0\. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable. Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

In constructive mathematics, Church's thesis (CT) is an axiom stating that all total functions are computable. The axiom takes its name from the Church–Turing thesis, which states that every effectively calculable function is a computable function, but the constructivist version is much stronger, claiming that every function is computable. The axiom CT is incompatible with classical logic in sufficiently strong systems. For example, Heyting arithmetic (HA) with CT as an addition axiom is able to disprove some instances of the law of the excluded middle.

In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem.

When using static linking, the compiler can safely assume that methods and variables computable at compile-time may be inlined.

According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm. Note that an algorithm in this sense is understood to be a sequence of steps a person with unlimited time and an unlimited supply of pen and paper could follow. The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions.

In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem which constructs fixed points of a computable function is known as Rogers's theorem and is due to Hartley Rogers, Jr. . The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions.

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.

The company launched Wolfram Alpha, an answer engine on May 16, 2009. It brings a new approach to knowledge generation and acquisition that involves large amounts of curated computable data in addition to semantic indexing of text. Wolfram Research acquired MathCore Engineering AB on March 30, 2011. On July 21, 2011, Wolfram Research launched the Computable Document Format (CDF).

All known laws of physics have consequences that are computable by a series of approximations on a digital computer. A hypothesis called digital physics states that this is no accident because the universe itself is computable on a universal Turing machine. This would imply that no computer more powerful than a universal Turing machine can be built physically.

360) : "Definition 2.5. An n-ary function f(x1, ..., xn) is partially computable if there exists a Turing machine Z such that :: f(x1, ..., xn) = ΨZ(n)(x1, ..., [xn) : In this case we say that [machine] Z computes f. If, in addition, f(x1, ..., xn) is a total function, then it is called computable" (Davis (1958) p.

O. Bournez and M. L. Campagnolo. A Survey on Continuous Time Computations. In New Computational Paradigms. Changing Conceptions of What is Computable.

A popular model for doing Computable Analysis are Turing machines. The tape configuration and interpretation of mathematical structures are described as follows.

Marian Boykan Pour-El (April 29, 1928 – June 10, 2009) was an American mathematical logician who did pioneering work in computable analysis.

We shall take for granted the extension of these ideas to computably convergent complex sequences, and the natural definitions of computable continuity.

In his original paper, Shannon presented a result which stated that functions computable by the GPAC are those functions which are differentially algebraic.

A Computable Universe: Understanding and Exploring Nature As Computation with a Foreword by Sir Roger Penrose. Singapore: World Scientific Publishing Company. Page 791.

Not every total computable function is provably total in Peano arithmetic, however; an example of such a function is provided by Goodstein's theorem.

The theorem states that a partial computable function u of two variables exists such that, for every computable function f of one variable, an e exists such that f(x) \simeq u(e,x) for all x. This means that, for each x, either f(x) and u(e,x) are both defined and are equal, or are both undefined. The theorem thus shows that, defining φe(x) as u(e, x), the sequence φ1, φ2, … is an enumeration of the partial computable functions. The function u in the statement of the theorem is called a universal function.

Hypercomputation or super-Turing computation refers to models of computation that can provide outputs that are not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that can correctly evaluate every statement in Peano arithmetic. The Church–Turing thesis states that any "computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not computable in the Church–Turing sense.

The only assumption is that the environment follows some unknown but computable probability distribution. It is a formal inductive framework that combines two well-studied principles of inductive inference: Bayesian statistics and Occam’s Razor. Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs (for a universal computer) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of x in optimal fashion.

This implies that Pr[x] is also computable in polynomial time. #Polynomial-time samplable distributions (P-samplable): these are distributions from which it is possible to draw random samples in polynomial time. These two formulations, while similar, are not equivalent. If a distribution is P-computable it is also P-samplable, but the converse is not true if P ≠ P#P.

"Existence of an Equilibrium for a Competitive Economy", Econometrica 22(3), pp. 265-290. More concretely, many problems are amenable to analytical (formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.

The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable. The set of all periods is countable, and all periods are computable , and in particular definable.

Ming Li and Paul Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, N.Y., 2008p 339 ff. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.

This is why these sets are studied in the field of algorithmic randomness, which is a subfield of Computability theory and related to algorithmic information theory in computer science. At the same time, K-trivial sets are close to computable. For instance, they are all superlow, i.e. sets whose Turing jump is computable from the Halting problem, and form a Turing ideal, i.e.

The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

There are some computer packages that work with computable real numbers, representing the real numbers as programs computing approximations. One example is the RealLib package .

Gödel's dialectica interpretation realizes (an extension of) intuitionistic arithmetic with computable functions. The connection with lambda calculus is unclear, even in the case of natural deduction.

For various syntactic properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences. The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth.

The definition depends on a suitable Gödel numbering that assigns natural numbers to computable functions. This numbering must be sufficiently effective that, given an index of a computable function and an input to the function, it is possible to effectively simulate the computation of the function on that input. The T predicate is obtained by formalizing this simulation. The ternary relation T1(e,i,x) takes three natural numbers as arguments.

Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at Stanford and Edinburgh by Robin Milner and collaborators in early 1970s, based on the theoretical foundation of logic of computable functions previously proposed by Dana Scott. Work on the LCF system introduced the general-purpose programming language ML to allow users to write theorem- proving tactics, supporting algebraic data types, parametric polymorphism, abstract data types, and exceptions.

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.

Heinrich Scholz (; December 17, 1884 – December 30, 1956) was a German logician, philosopher, and Protestant theologian. He was a peer of Alan Turing who mentioned Scholz when writing with regard to the reception of "On Computable Numbers, with an Application to the Entscheidungsproblem":Alan Turing: "On Computable Numbers, with an Application to the Entscheidungsproblem." In: Proceedings of the London Mathematical Society, 2nd series, vol. 42 (1937), pp. 230–265.

K-triviality turns out to coincide with some computational lowness notions, saying that a set is close to computable. The following notions capture the same class of sets.

A. M. Turing's paper On Computable Numbers, With an Application to the Entscheidungsproblem was delivered to the London Mathematical Society in November 1936. Again the reader must bear in mind a caution: as used by Turing, the word "computer" is a human being, and the action of a "computer" he calls "computing"; for example, he states "Computing is normally done by writing certain symbols on paper" (p. 135). But he uses the word "computation" in (Davis 1967:118) in the context of his machine-definition, and his definition of "computable" numbers is as follows: :"The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means ... .According to my definition, a number is computable if its decimal can be written down by a machine." in (Davis 1967:116) What is Turing's definition of his "machine?" Turing gives two definitions, the first a summary in §1 Computing machines and another very similar in §9.

The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F(w x) = f(x); here w x represents the concatenation of the two strings w and x. This means that F can be used to simulate any computable function of one variable. Informally, w represents a "script" for the computable function f, and F represents an "interpreter" that parses the script as a prefix of its input and then executes it on the remainder of input. The domain of F is the set of all inputs p on which it is defined.

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.

Contemporary research in recursion theory includes the study of applications such as algorithmic randomness, computable model theory, and reverse mathematics, as well as new results in pure recursion theory.

That is, \Sigma^b_1-definable functions in S^1_2 are precisely the functions computable in polynomial time. The characterization can be generalized to higher levels of the polynomial hierarchy.

This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.

In computational complexity theory, a pseudo-polynomial transformation is a function which maps instances of one strongly NP-complete problem into another and is computable in pseudo-polynomial time.

Valentina Harizanov is a Serbian-American mathematician and professor of mathematics at The George Washington University. Her main research contributions are in computable structure theory (roughly at the intersection of computability theory and model theory), where she introduced the notion of degree spectra of relations on computable structures and obtained the first significant results concerning uncountable, countable, and finite Turing degree spectra. Her recent interests include algorithmic learning theory and spaces of orders on groups.

A real number is called computable if there is an algorithm which, given n, returns the first n digits of the number. This is equivalent to the existence of a program that enumerates the digits of the real number. No halting probability is computable. The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n.

The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable." The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multi-tape) universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine.

This has been termed the strong Church–Turing thesis, or Church–Turing–Deutsch principle, and is a foundation of digital physics. #The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer. For example, a universe in which physics involves random real numbers, as opposed to computable reals, would fall into this category.

His argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that super-recursive algorithms are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.

Brainerd and Landweber, 1974 In fact, for showing that a computable function is primitive recursive, it suffices to show that its computational complexity is bounded above by a primitive recursive function of the input size. It follows that it is difficult to devise a computable function that is not primitive recursive, although some are known (see the section on Limitations below). The set of primitive recursive functions is known as PR in computational complexity theory.

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.

Solomonoff's theory of inductive inference is a mathematical theory of induction introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory. Interestingly, Solomonoff's induction naturally formalizes Occam's razorJJ McCall.

The challenge is to find explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor (and disperser) graphs have found many applications in computer science.

In 1995 Leijonhufvud was appointed Professor of Monetary Theory and Policy at the University of Trento in Italy, where he is also part of the CEEL (Computable and Experimental Economics Laboratory).

In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.

N. Soklakov. Occam's razor as a formal basis for a physical theory from arxiv.org – Foundations of Physics Letters, 2002 – SpringerM Hutter. On the existence and convergence of computable universal priors arxiv.

The Frontiers Collection, Springer, 2005, pp 121-152, . Edward Fredkin and Konrad Zuse pioneered the idea of a computable universe, the former by writing a line in his book on how the world might be like a cellular automaton, and later further developed by Fredkin using a toy model called Salt. It has been claimed that NKS tries to take these ideas as its own, but Wolfram's model of the universe is a rewriting network, and not a cellular automaton, as Wolfram himself has suggested a cellular automaton cannot account for relativistic features such as no absolute time frame. Jürgen Schmidhuber has also charged that his work on Turing machine-computable physics was stolen without attribution, namely his idea on enumerating possible Turing-computable universes.

A gap group is a group in which the computational Diffie-Hellman problem is intractable but the decisional Diffie-Hellman problem can be efficiently solved. Non-degenerate, efficiently computable, bilinear pairings permit such groups. Let e\colon G\times G\rightarrow G_T be a non-degenerate, efficiently computable, bilinear pairing where G, G_T are groups of prime order, r. Let g be a generator of G. Consider an instance of the CDH problem, g,g^x, g^y.

If the same group is used for the first two groups (i.e. G_1 = G_2), the pairing is called symmetric and is a mapping from two elements of one group to an element from a second group. Some researchers classify pairing instantiations into three (or more) basic types: # G_1 = G_2; # G_1 e G_2 but there is an efficiently computable homomorphism \phi : G_2 \to G_1; # G_1 e G_2 and there are no efficiently computable homomorphisms between G_1 and G_2.

However, another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities (i.e., any language guaranteeing that every program will eventually finish to a halt). So any such language is not Turing-complete. For example, a language in which programs are guaranteed to complete and halt cannot compute the computable function produced by Cantor's diagonal argument on all computable functions in that language.

On the other hand, any computable system is incomplete. There will always be descriptions outside that system's search space which will never be acknowledged or considered, even in an infinite amount of time. Computable prediction models hide this fact by ignoring such algorithms. In many of his papers he described how to search for solutions to problems and in the 1970s and early 1980s developed what he felt was the best way to update the machine.

In computability theory a numbering is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some language. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. Common examples of numberings include Gödel numberings in first-order logic and admissible numberings of the set of partial computable functions.

In mathematics, a set of natural numbers is called a K-trivial set if its initial segments viewed as binary strings are easy to describe: the prefix- free Kolmogorov complexity is as low as possible, close to that of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable. The Schnorr–Levin theorem says that random sets have a high initial segment complexity. Thus the K-trivials are far from random.

Later, it was adopted by the ASL-BiSL Foundation, founded in 2002.Dutch Article in Computable magazine about the ASL-BiSL Foundation The body of knowledge on BiSL is currently public domain.

Journal of Complexity, 19(5):644–664, 2003 In particular it was shown in 2007 that (a deterministic variant of) the GPAC is equivalent, in computability terms, to Turing machines, thereby proving the physical Church–Turing thesis for the class of systems modelled by the GPAC.O. Bournez, M. L. Campagnolo, D. S. Graça, and E. Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity, 23:317–335, 2007 This was recently strengthened to polynomial time equivalence.

In computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means it can keep a constant number of pointers into the input, along with a logarithmic number of fixed-size integers.Arora & Barak (2009) p. 88 It is possible that such a machine may not have space to write down its own output, so the only requirement is that any given bit of the output be computable in log-space.

Numerical and string constants and expressions in code can and often do imply type in a particular context. For example, an expression might imply a type of floating-point, while might imply a list of integers—typically an array. Type inference is in general possible, if it is computable in the type system in question. Moreover, even if inference is not computable in general for a given type system, inference is often possible for a large subset of real-world programs.

Similarly, Tarski's indefinability theorem can be interpreted both in terms of definability and in terms of computability. Recursion theory is also linked to second order arithmetic, a formal theory of natural numbers and sets of natural numbers. The fact that certain sets are computable or relatively computable often implies that these sets can be defined in weak subsystems of second order arithmetic. The program of reverse mathematics uses these subsystems to measure the noncomputability inherent in well known mathematical theorems.

Leopold Vietoris on his 110th birthday Like the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of linear algebra, many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary.

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.

The main use for log-space computable functions is in log-space reductions. This is a means of transforming an instance of one problem into an instance of another problem, using only logarithmic space.

The NID is called `the similarity metric.' since the function NID(x,y) has been shown to satisfy the basic requirements for a metric distance measure. However, it is not computable or even semicomputable.

Bourbaki–Witt has other applications. In particular in computer science, it is used in the theory of computable functions. It is also used to define recursive data types, e.g. linked lists, in domain theory.

Since that sort of behaviour could be considered pathological, it is natural to insist that a function should only be considered total if it is total over all real numbers, not just the computable ones.

This strategy may be extended for every group operator where the notion of f^{-1} is well defined and easily computable. Finally, this solution can be extended to two-dimensional arrays with a similar preprocessing.

Velupillai, Kumaraswamy. Computable Economics: The Arne Ryde Memorial Lectures. New York: Oxford University Press, 2000. In 1933, Simon entered the University of Chicago, and, following his early influences, decided to study social science and mathematics.

The principle of the modern computer was proposed by Alan Turing in his seminal 1936 paper, On Computable Numbers. Turing proposed a simple device that he called "Universal Computing machine" and that is now known as a universal Turing machine. He proved that such a machine is capable of computing anything that is computable by executing instructions (program) stored on tape, allowing the machine to be programmable. The fundamental concept of Turing's design is the stored program, where all the instructions for computing are stored in memory.

Turing's Thesis hypothesizes the computability of "all computable functions" by the Turing machine model and its equivalents. To do this in an effective manner, Kleene extended the notion of "computable" by casting the net wider—by allowing into the notion of "functions" both "total functions" and "partial functions". A total function is one that is defined for all natural numbers (positive integers including 0). A partial function is defined for some natural numbers but not all—the specification of "some" has to come "up front".

The main form of computability studied in recursion theory was introduced by Turing (1936). A set of natural numbers is said to be a computable set (also called a decidable, recursive, or Turing computable set) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in the set. A function f from the natural numbers to themselves is a recursive or (Turing) computable function if there is a Turing machine that, on input n, halts and returns output f(n). The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.

For instance, the level \Sigma^0_0=\Pi^0_0=\Delta^0_0 of the arithmetical hierarchy classifies computable, partial functions. Moreover, this hierarchy is strict such that at any other class in the arithmetic hierarchy classifies strictly uncomputable functions.

The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated.Hutter, M., Legg, S., and Vitanyi, P., "Algorithmic Probability", Scholarpedia, 2(8):2572, 2007.

What is effectively calculable is computable. :" ... Both Church and Turing had in mind calculation by an abstract human being using some mechanical aids (such as paper and pencil)"Gandy in (Barwise 1980:123) Robert Soare (1995, see below) had issues with this framing, considering Church's paper (1936) published prior to Turing's "Appendix proof" (1937). Gandy attempts to "analyze mechanical processes and so to provide arguments for the following: :"Thesis M. What can be calculated by a machine is computable." Gandy in (Barwise 1980:124 Gandy "exclude[s] from consideration devices which are essentially analogue machines ... .

In computability theory, a Turing degree [X] is high if it is computable in 0′, and the Turing jump [X′] is 0′′, which is the greatest possible degree in terms of Turing reducibility for the jump of a set which is computable in 0′ (Soare 1987:71). Similarly, a degree is high n if its n'th jump is the (n+1)'st jump of 0. Even more generally, a degree d is generalized high n if its n'th jump is the n'th jump of the join of d with 0′.

Reification is the process by which an abstract idea about a computer program is turned into an explicit data model or other object created in a programming language. A computable/addressable object—a resource—is created in a system as a proxy for a non computable/addressable object. By means of reification, something that was previously implicit, unexpressed, and possibly inexpressible is explicitly formulated and made available to conceptual (logical or computational) manipulation. Informally, reification is often referred to as "making something a first-class citizen" within the scope of a particular system.

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.

In computability theory, several closely related terms are used to describe the computational power of a computational system (such as an abstract machine or programming language): ;Turing completeness : A computational system that can compute every Turing- computable function is called Turing-complete (or Turing-powerful). Alternatively, such a system is one that can simulate a universal Turing machine. ;Turing equivalence : A Turing-complete system is called Turing- equivalent if every function it can compute is also Turing-computable; i.e., it computes precisely the same class of functions as do Turing machines.

This does not, however, preclude very long programs from having very high probability. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity. The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs (for a universal computer) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of x in optimal fashion.

In the early days of the development of K-triviality, attention was paid to separation of K-trivial sets and computable sets. Chaitin in his 1976 paper Gregory J. Chaitin (1976), "Information-Theoretic Characterizations of Recursive Infinite Strings", Theoretical Computer Science Volume 2, Issue 1, June 1976, Pages 45–48 mainly studied sets such that there exists b ∈ℕ with : \forall n C(A\upharpoonright n)\leq C(n)+b where C denotes the plain Kolmogorov complexity. These sets are known as C-trivial sets. Chaitin showed they coincide with the computable sets.

Computable number: A real number whose digits can be computed using an algorithm. Definable number: A real number that can be defined uniquely using a first-order formula with one free variable in the language of set theory.

In classical logic, formulas represent true/false statements. In CoL, formulas represent computational problems. In classical logic, the validity of a formula depends only on its form, not on its meaning. In CoL, validity means being always computable.

Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp.Husemoller (1987) pp.116-117 Deciding whether this condition holds is effectively computable by Tate's algorithm.Husemöller (1987) pp.

In the 1980s however, AGE models faded from popularity due to their inability to provide a precise solution and its high cost of computation. Computable general equilibrium (CGE) models surpassed and replaced AGE models in the mid-1980s, as the CGE model was able to provide relatively quick and large computable models for a whole economy, and was the preferred method of governments and the World Bank. CGE models are heavily used today, and while 'AGE' and 'CGE' is used inter-changeably in the literature, Scarf-type AGE models have not been constructed since the mid-1980s, and the CGE literature at current is not based on Arrow-Debreu and General Equilibrium Theory as discussed in this article. CGE models, and what is today referred to as AGE models, are based on static, simultaneously solved, macro balancing equations (from the standard Keynesian macro model), giving a precise and explicitly computable result.

As each of the equivalent definitions of a Martin-Löf random sequence is based on what is computable by some Turing machine, one can naturally ask what is computable by a Turing oracle machine. For a fixed oracle A, a sequence B which is not only random but in fact, satisfies the equivalent definitions for computability relative to A (e.g., no martingale which is constructive relative to the oracle A succeeds on B) is said to be random relative to A. Two sequences, while themselves random, may contain very similar information, and therefore neither will be random relative to the other. Any time there is a Turing reduction from one sequence to another, the second sequence cannot be random relative to the first, just as computable sequences are themselves nonrandom; in particular, this means that Chaitin's Ω is not random relative to the halting problem.

Determining whether an arbitrary Turing machine is a busy beaver is undecidable. This has implications in computability theory, the halting problem, and complexity theory. The concept was first introduced by Tibor Radó in his 1962 paper, "On Non-Computable Functions".

Integer addition and subtraction are computable in AC0, but multiplication is not (at least, not under the usual binary or base-10 representations of integers). Since it is a circuit class, like P/poly, AC0 also contains every unary language.

Now it remains to determine the BiCG constants and and choose a suitable . In BiCG, with :. Since BiCGSTAB does not explicitly keep track of or , is not immediately computable from this formula. However, it can be related to the scalar :.

In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the five primitive recursive operators makes it possible to define all computable functions.

Computable Document Format supports GUI elements such as sliders, menus, and buttons. Content is updated using embedded computation in response to GUI interaction. Contents can include formatted text, tables, images, sounds, and animations. CDF supports Mathematica typesetting and technical notation.

Thus, large amounts of pathway data are available in a computable form to support visualization, analysis and biological discovery. It is supported by a variety of online databases (e.g. Reactome) and tools. The latest released version is BioPAX Level 3.

Thus, he reasons, it is preferred over other theories-of-everything by Occam's Razor. Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis (CUH), which says that the mathematical structure that is our external physical reality is defined by computable functions. The MUH is related to Tegmark's categorization of four levels of the multiverse. This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of initial conditions (level 1), physical constants (level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4).

While the second recursion theorem is about fixed points of computable functions, the first recursion theorem is related to fixed points determined by enumeration operators, which are a computable analogue of inductive definitions. An enumeration operator is a set of pairs (A,n) where A is a (code for a) finite set of numbers and n is a single natural number. Often, n will be viewed as a code for an ordered pair of natural numbers, particularly when functions are defined via enumeration operators. Enumeration operators are of central importance in the study of enumeration reducibility.

Simpson (1999) discusses many aspects of second-order arithmetic and reverse mathematics. The field of proof theory includes the study of second-order arithmetic and Peano arithmetic, as well as formal theories of the natural numbers weaker than Peano arithmetic. One method of classifying the strength of these weak systems is by characterizing which computable functions the system can prove to be total (see Fairtlough and Wainer (1998)). For example, in primitive recursive arithmetic any computable function that is provably total is actually primitive recursive, while Peano arithmetic proves that functions like the Ackermann function, which are not primitive recursive, are total.

See also which also gives these definitions for "effective" – the first ["producing a decided, decisive, or desired effect"] as the definition for sense "1a" of the word "effective", and the second ["capable of producing a result"] as part of the "Synonym Discussion of EFFECTIVE" there, (in the introductory part, where it summarizes the similarities between the meanings of the words "effective", "effectual", "efficient", and "efficacious"). In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis Systems of Logic Based on Ordinals, supervised by Church, are virtually the same: > We shall use the expression "computable function" to mean a function > calculable by a machine, and let "effectively calculable" refer to the > intuitive idea without particular identification with any one of these > definitions. The thesis can be stated as: Every effectively calculable function is a computable function.

The T predicate can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15; Kleene 1943, p. 52--53). This states there exists a primitive recursive function U such that a function f of one integer argument is computable if and only if there is a number e such that for all n one has :f(n) \simeq U( \mu x\, T(e,n,x)), where μ is the μ operator (\mu x\, \phi(x) is the smallest natural number for which \phi(x) holds) and \simeq holds if both sides are undefined or if both are defined and they are equal. Here U is a universal operation (it is independent of the computable function f) whose purpose is to extract, from the number x (encoding a complete computation history) returned by the operator μ, just the value f(n) that was found at the end of the computation.

Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.

Thus i(e,n) \in A \Leftrightarrow (e,n) \in B holds for all e and n. Because the function i is computable, this shows B \leq_T A. The reductions presented here are not only Turing reductions but many-one reductions, discussed below.

It cannot compute everything that is computable. Intrinsically the model is bounded by the size of its (very-) finite state machine's instructions. The counter machine based RASP can compute any primitive recursive function (e.g. multiplication) but not all mu recursive functions (e.g.

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be.

RunGEM is a freeware component of the commercial modelling package, Gempack. It is mainly used for solving (i.e., simulating with) computable general equilibrium (CGE) models. It does not allow a user to change a model's specification or to create a new model.

Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.

Because PA is an effective first-order theory, the completions of PA can be characterized as the infinite paths through a particular computable subtree of 2<ω. Thus the PA degrees are exactly the degrees that compute an infinite path through this tree.

In general, a real is computable if and only if its Dedekind cut is at level \Delta^0_1 of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.

It is an easy numerical task to follow such a path from q to the fixed point so the method is essentially computable. gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

Computable general equilibrium (CGE) models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors. CGE models are also referred to as AGE (applied general equilibrium) models.

J. Schmidhuber (1997): A Computer Scientist's View of Life, the Universe, and Everything. Lecture Notes in Computer Science, pp. 201–208, Springer: IDSIA – Dalle Molle Institute for Artificial IntelligenceJ. Schmidhuber (2002): Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit.

Computable Document Format (CDF) is an electronic document formatWolfram Alpha Creator plans to delete the PDF The Telegraph (UK) designed to allow easy authoringWolfram makes data interactive PC World of dynamically generated interactive content. CDF is a published public format created by Wolfram Research.

Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by a computable function. For this reason, they are sometimes referred to as bounded Turing (bT) reductions rather than as weak truth-table (wtt) reductions.

Jürgen Schmidhuber (2000) constructed a limit-computable "Super Ω" which in a sense is much more random than the original limit-computable Ω, as one cannot significantly compress the Super Ω by any enumerating non-halting algorithm. For an alternative "Super Ω", the universality probability of a prefix-free Universal Turing Machine (UTM) namely, the probability that it remains universal even when every input of it (as a binary string) is prefixed by a random binary string can be seen as the non-halting probability of a machine with oracle the third iteration of the halting problem (i.e., O^{(3)}using Turing Jump notation).

Any model (structure) that satisfies all axioms of Q except possibly axiom (3) has a unique submodel ("the standard part") isomorphic to the standard natural numbers . (Axiom (3) need not be satisfied; for example the polynomials with non-negative integer coefficients forms a model that satisfies all axioms except (3).) Q, like Peano arithmetic, has nonstandard models of all infinite cardinalities. However, unlike Peano arithmetic, Tennenbaum's theorem does not apply to Q, and it has computable non-standard models. For instance, there is a computable model of Q consisting of integer-coefficient polynomials with positive leading coefficient, plus the zero polynomial, with their usual arithmetic.

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.

Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.

The terminology for recursive functions and sets is not completely standardized. The definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine. The word decidable stems from the German word Entscheidungsproblem which was used in the original papers of Turing and others. In contemporary use, the term "computable function" has various definitions: according to Cutland (1980), it is a partial recursive function (which can be undefined for some inputs), while according to Soare (1987) it is a total recursive (equivalently, general recursive) function.

In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if- then-else statement). A property is non-trivial if it is neither true for every computable function, nor false for every computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property.

Easily computable graph invariants are instrumental for fast recognition of graph isomorphism, or rather non-isomorphism, since for any invariant at all, two graphs with different values cannot (by definition) be isomorphic. Two graphs with the same invariants may or may not be isomorphic, however. A graph invariant I(G) is called complete if the identity of the invariants I(G) and I(H) implies the isomorphism of the graphs G and H. Finding an efficiently-computable such invariant (the problem of graph canonization) would imply an easy solution to the challenging graph isomorphism problem. However, even polynomial-valued invariants such as the chromatic polynomial are not usually complete.

Thus the inclusion of "partial function" extends the notion of function to "less-perfect" functions. Total- and partial-functions may either be calculated by hand or computed by machine. : Examples: :: "Functions": include "common subtraction m − n" and "addition m + n" :: "Partial function": "Common subtraction" m − n is undefined when only natural numbers (positive integers and zero) are allowed as input – e.g. 6 − 7 is undefined :: Total function: "Addition" m + n is defined for all positive integers and zero. We now observe Kleene's definition of "computable" in a formal sense: : Definition: "A partial function φ is computable, if there is a machine M which computes it" (Kleene (1952) p.

Schmidhuber (2000, 2002) uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines and simple inductive Turing machines are two classes of super-recursive algorithms that are the closest to recursive algorithms (Schmidhuber 2000).

To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet Cantor's diagonal argument shows that real numbers have higher cardinality. Furthermore, the diagonal argument seems perfectly constructive.

Although the Church–Turing thesis states that the computable functions include all functions with algorithms, it is possible to consider broader classes of functions that relax the requirements that algorithms must possess. The field of Hypercomputation studies models of computation that go beyond normal Turing computation.

Over the past decade the increased usage of electronic health records has produced vast amounts of clinical data that is now computable. Predictive informatics integrates this data with other datasets (e.g., genotypic, phenotypic) in centralized and standardized data repositories upon which predictive analytics may be conducted.

He studied computable real numbers, in particular provided few different definitions of these numbers and the ways to development of mathematical analysis based only on these numbers and computable functions determined on these numbers. He investigated computable functionals of higher types and proved undecidability of different weak theories such like elementary topological algebra, he considered axiomatic foundations of geometry by the means of solids instead of points, he showed that mereology is equivalent to the Boolean algebra, he approached intuitionistic logic with a help of semantics of intuitionistic propositional calculus built upon the notion of enforced recognition of sentences in the frames of cognitive procedures, what is similar to the Kripke semantics which was created parallelly, and he studied Kotarbiński's reism. He proposed an interpretation of the Leśniewski ontology as the Boolean algebra without zero, and demonstrated the undecidability of the theory of the Boolean algebras with the operation of closure. He investigated intuitionistic logic, just a modal interpretation of the Grzegorczyk semantics for intuitionism, which predetermined the Kripke semantics, leads to the aforementioned S4.Grz.

Peter Bishop Dixon AO FASSA (born 23 July 1946) is an Australian economist known for his work in general equilibrium theory and computable general equilibrium models."Peter Dixon", The Conversation He has published several books and more than two hundred academic papers on economic modelling and economic policy analysis.

A log-space computable function is a function f\colon \Sigma^\ast \rightarrow \Sigma^\ast that requires only O(\log n) memory to be computed (this restriction does not apply to the size of the output). The computation is generally done by means of a log-space transducer.

For example, a prominent circuit class P/poly consists of Boolean functions computable by circuits of polynomial-size. Proving that NP ot\subseteq P/poly would separate P and NP (see below). Complexity classes defined in terms of Boolean circuits include AC0, AC, TC0, NC1, NC, and P/poly.

The definition of a halting probability relies on the existence of a prefix-free universal computable function. Such a function, intuitively, represents a programming language with the property that no valid program can be obtained as a proper extension of another valid program. Suppose that F is a partial function that takes one argument, a finite binary string, and possibly returns a single binary string as output. The function F is called computable if there is a Turing machine that computes it (in the sense that for any finite binary strings x and y, F(x) = y if and only if the Turing machine halts with y on its tape when given the input x).

Within this theory, it is possible to prove interesting statements such as "The complement of the Mandelbrot set is only partially decidable." These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers. They may be further subdivided into differential and algebraic models (digital computers, in this context, should be thought of as topological, at least insofar as their operation on computable reals is concerned). Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (For example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages.) or vice versa.

In the late 1970s Pour-El began working on computable analysis. Her "most famous and surprising result", co-authored with Minnesota colleague J. Ian Richards, was that for certain computable initial conditions, determining the behavior of the wave equation is an undecidable problem. Their result was later taken up by Roger Penrose in his book The Emperor's New Mind; Penrose used this result as a test case for the Church–Turing thesis, but concluded that the non-smoothness of the initial conditions makes it implausible that a computational device could use this phenomenon to exceed the limits of conventional computing. Freeman Dyson used the same result to argue for the evolutionary superiority of analog to digital forms of life.

In computer science and engineering, a system acts as a computable function. An example of a specific function could be y = f(x) where y is the output of the system and x is the input; however, most systems' inputs are not one-dimensional. When the inputs are multi-dimensional, we could say that the system takes the form y = f(x_1, x_2, ...) ; however, we can generalize this equation to a general form Y = C(X) where Y is the result of the system's execution, C belongs to the set of computable functions, and X is an input vector. While testing the system, various test vectors must be used to examine the system's behavior with differing inputs.

If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula., chapter V. A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

This article follows the second of these conventions. Soare (1996) gives additional comments about the terminology. Not every set of natural numbers is computable. The halting problem, which is the set of (descriptions of) Turing machines that halt on input 0, is a well-known example of a noncomputable set.

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.

James Stanford outlines this issue for the specific Computable General Equilibrium ("CGE") models that were introduced as evidence into the public policy debate, by advocates for NAFTA.Rick Crawford. 1996. in "Invisible Crises: What Conglomerate Control of Media Means for America and the World". Ed. Herbert Schiller, Hamid Mowlana, George Gerbner. Westview. 1996.

Turing's thesis that every function which would naturally be regarded as computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX." Indeed immediately before this statement, Kleene states the Theorem XXX: ::"Theorem XXX (= Theorems XXVIII + XXIX). The following classes of partial functions are coextensive, i.e.

Mihara Reprinted in shows that such a rule violates algorithmic computability.Mihara's definition of a computable aggregation rule is based on computability of a simple game (see Rice's theorem). These results can be seen to establish the robustness of Arrow's theorem.See Chapter 6 of for a concise discussion of social choice for infinite societies.

In 1996, Anand Rao created a logic-based agent programming language based on the BDI architecture and named it AgentSpeak(L).Anand S. Rao, 1996. AgentSpeak(L): BDI Agents Speak Out in a Logical Computable Language. Proceedings of Seventh European Workshop on Modelling Autonomous Agents in a Multi-Agent World (MAAMAW-96).

Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability.Alonzo Church, "On the Concept of Random Sequence," Bull. Amer. Math. Soc., 46 (1940), 254–260Companion encyclopedia of the history and philosophy Volume 2, by Ivor Grattan-Guinness 0801873975 page 1412J.

Knopf, 2006 The universe/nature as computational mechanism is addressed by,Zenil, H. A Computable Universe: Understanding and Exploring Nature as Computation. World Scientific Publishing Company, 2012 exploring nature with help the ideas of computability, and Dodig-Crnkovic, G. and Giovagnoli, R. COMPUTING NATURE. Springer, 2013 studying natural processes as computations (information processing).

Principles of Mathematical Logic. AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950 Turing's solution involved proposing a hypothetical programable computing machine. In spring 1936, Newman was presented by Turing with a draft of "On Computable Numbers with an Application to the Entscheidungsproblem". He realised the paper's importance and helped ensure swift publication.

An effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in both effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

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.

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.

In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

See the footnote at the end of Soare: 1996. showing that a general solution to the ' is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.

Yuri Ivanovich Manin (; born February 16, 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.

Computability in Analysis and Physics is a monograph on computable analysis by Marian Pour-El and J. Ian Richards. It was published by Springer-Verlag in their Perspectives in Mathematical Logic series in 1989, and reprinted by the Association for Symbolic Logic and Cambridge University Press in their Perspectives in Logic series in 2016.

He mentions that it can easily be shown using the elements of recursive functions that every number calculable on the arithmetic machine is computable. A proof of which was given by Lambek on an equivalent two instruction machine : X+ (increment X) and X− else T (decrement X if it not empty, else jump to T).

Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none, and an effective decision method is called general if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation of 1951 at Syracuse University.

But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental. The non-computable numbers are a strict subset of the transcendental numbers. All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion.

All the systems mentioned so far, with the exception of the untyped lambda calculus, are strongly normalizing: all computations terminate. Therefore, they cannot describe all Turing-computable functions.since the halting problem for the latter class was proven to be undecidable As another consequence they are consistent as a logic, i.e. there are uninhabited types.

The speed prior complexity of a program is its size in bits plus the logarithm of the maximum time we are willing to run it to get a prediction. When compared to traditional measures, use of the Speed Prior has the disadvantage of leading to less optimal predictions, and the advantage of providing computable predictions.

The first problem becomes the submodule membership problem. The second one is also called the syzygy problem. A ring such that there are algorithms for the arithmetic operations (addition, subtraction, multiplication) and for the above problems may be called a computable ring, or effective ring. One may also say that linear algebra on the ring is effective.

Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should not be confused with the symbolic computation provided by many computer algebra systems, which represent numbers by expressions such as , and can thus represent any computable number with infinite precision.

Gradient descent can be extended to handle constraints by including a projection onto the set of constraints. This method is only feasible when the projection is efficiently computable on a computer. Under suitable assumptions, this method converges. This method is a specific case of the forward-backward algorithm for monotone inclusions (which includes convex programming and variational inequalities).

Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.

Stanisław Mieczysław Mazur (1 January 1905, Lwów – 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences. Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. He was also interested in summability theory, infinite games and computable functions.

Computable functions are represented as programs on a Type 2 Turing Machine. A program is considered total (in the sense of a total functions as opposed to partial functions) if it takes finite time to write any number of symbols on the output tape regardless of the input. The programs run forever, generating increasingly more digits of the output.

It maximizes the expected total rewards received from the environment. Intuitively, it simultaneously considers every computable hypothesis (or environment). In each time step, it looks at every possible program and evaluates how many rewards that program generates depending on the next action taken. The promised rewards are then weighted by the subjective belief that this program constitutes the true environment.

If the problem is to count the number of solutions, which is denoted by #CSP(Γ), then a similar result by Creignou and Hermann holds. Let Γ be a finite constraint language over the Boolean domain. The problem #CSP(Γ) is computable in polynomial time if Γ has a Mal'tsev operation as a polymorphism. Otherwise, the problem #CSP(Γ) is #P-complete.

And confusingly, since Turing was unable to correct his original paper, some text within the body harks to Turing's flawed first effort. Bernays' corrections may be found in Undecidable, pp. 152–154; the original is to be found as: :"On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction," Proceedings of the London Mathematical Society (2), 43 (1936–37), 544-546.

Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method. Several computational models allow for the computation of (Church-Turing) non-computable functions. These are known as hypercomputers. Mark Burgin argues that super-recursive algorithms such as inductive Turing machines disprove the Church–Turing thesis.

For a literal translation of agent-oriented concepts into a scheme unobfuscated as is JADE, behind Java and Object Orientedness, Agent Speak Anand S. Rao, 1996. AgentSpeak(L): BDI Agents Speak Out in a Logical Computable Language. Proceedings of Seventh European Workshop on Modelling Autonomous Agents in a Multi-Agent World (MAAMAW-96). (Jason) provides a "natural" language for agents. started.

The loop representation has found application in mathematics. If topological quantum field theories are formulated in terms of loops, the resulting quantities should be what are known as knot invariants. Topological field theories only involve a finite number of degrees of freedom and so are exactly solvable. As a result, they provide concrete computable expressions that are invariants of knots.

The language consisting of all Turing machine descriptions paired with all possible input streams on which those Turing machines will eventually halt, is not recursive. The halting problem is therefore called non-computable or undecidable. An extension of the halting problem is called Rice's theorem, which states that it is undecidable (in general) whether a given language possesses any specific nontrivial property.

Parametric polymorphism was first introduced to programming languages in ML in 1975.Milner, R., Morris, L., Newey, M. "A Logic for Computable Functions with reflexive and polymorphic types", Proc. Conference on Proving and Improving Programs, Arc-et-Senans (1975) Today it exists in Standard ML, OCaml, F#, Ada, Haskell, Mercury, Visual Prolog, Scala, Julia, Python, TypeScript, C++ and others. Java, C#, Visual Basic .

Choi et al. propose a p-value derived from the likelihood ratio test based on the conditional distribution of the odds ratio given the marginal success rate. This p-value is inferentially consistent with classical tests of normally distributed data as well as with likelihood ratios and support intervals based on this conditional likelihood function. It is also readily computable.

Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols, it does not employ any commuting or commutative subgroups of a given platform group and can use any nonabelian group with efficiently computable normal forms.

The Annotated Turing: A Guided Tour Through Alan Turing’s Historic Paper on Computability and the Turing Machine is a book by Charles Petzold, published in 2008 by John Wiley & Sons, Inc. Petzold annotates Alan Turing's paper "On Computable Numbers, with an Application to the Entscheidungsproblem". The book takes readers sentence by sentence through Turing's paper, providing explanations, further examples, corrections, and biographical information.

Computational complexity theory deals with the relative computational difficulty of computable functions. By definition, it does not cover problems whose solution is unknown or has not been characterised formally. Since many AI problems have no formalisation yet, conventional complexity theory does not allow the definition of AI-completeness. To address this problem, a complexity theory for AI has been proposed.

In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.

MINLOG is a proof assistant developed at the University of Munich by the team of Helmut Schwichtenberg. MINLOG is based on first ordernatural deduction calculus. It is intended to reason about computable functionals, using minimal rather than classical or intuitionistic logic. The primary motivation behind MINLOG is to exploit the proofs-as-programs paradigm for program development and program verification.

The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function. The Church encoding is not intended as a practical implementation of primitive data types. Its use is to show that other primitive data types are not required to represent any calculation.

In the event that one is unhappy with using Turing Machines (on the grounds that they are low level and somewhat arbitrary), there is a realisability topos called the Kleene-Vesley topos in which one can reduce computable analysis to constructive analysis. This constructive analysis includes everything that is valid in the Brouwer school, and not just the Bishop school.

316-317 These grammars were introduced by Dahlhaus and Warmuth.. Here: p.197-198 They were later shown to be equivalent to the acyclic context-sensitive grammars. Membership in any growing context- sensitive language is polynomial time computable; Here: p.85-86 however, the uniform problem of deciding whether a given string belongs to the language generated by a given growingG.

The difference between this sketch and the actual proof is that in the actual proof, the computable function halts does not directly take a subroutine as an argument; instead it takes the source code of a program. The actual proof requires additional work to handle this issue. Moreover, the actual proof avoids the direct use of recursion shown in the definition of g.

In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computable amount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, called the extremal norm.N. Barabanov.

Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the (often very long) details which would be involved in a rigorous, formal proof.Horsten in . To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive). Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis: In order to make the above example completely rigorous, one would have to carefully construct a Turing machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory.

The set 2<ω consists of all finite sequences of 0s and 1s, while the set 2ω consists of all infinite sequences of 0s and 1s (that is, functions from } to the set }). A tree on 2ω is a subset of 2ω that is closed under taking initial segments. An element f of 2ω is a path through a tree T on 2ω if every finite initial segment of f is in T. A (lightface) Π01 class is a subset C of 2ω for which there is a computable tree T such that C consists of exactly the paths through T. A boldface Π01 class is a subset D of 2ω for which there is an oracle f in 2ω and a subtree tree T of 2< ω from computable from f such that D is the set of paths through T.

If the length of the program K is L(K) bits then its prior probability is, :P(K) = 2^{-L(K)} The length of the shortest program that represents the string of bits is called the Kolmogorov complexity. Kolmogorov complexity is not computable. This is related to the halting problem. When searching for the shortest program some programs may go into an infinite loop.

Turing informs him that he wrote letters to Christopher's mother and even managed to acquire his photograph. He still has the photograph in his wallet, and he shows it to Dr. Greenbaum. Turing later goes on to publish his paper "On Computable Numbers, with an Application to the Entscheidungsproblem". A computer in those days did not mean a machine, it meant a person who calculates or computes.

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems. Gödel published his incompleteness theorems in (called in English "On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g.

As shown by Alan Turing and Alonzo Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis).Church 1934:90 footnote in Davis 1952Turing 1936–7 in Davis 1952:149Barendregt, H.P., The Lambda Calculus Syntax and Semantics. North- Holland Publishing Company. 1981 Lambda-calculus is thus effectively a programming language, from which other languages can be built.

Revised version in Information and Control, 68 (1986), 86–104. and Vardi independently showed the descriptive complexity result that the polynomial-time computable properties of linearly ordered structures are definable in FO(LFP), i.e. in first-order logic with a least fixed point operator. However, FO(LFP) is too weak to express all polynomial-time properties of unordered structures (for instance that a structure has even size).

Complex biological systems may be represented and analyzed as computable networks. For example, ecosystems can be modeled as networks of interacting species or a protein can be modeled as a network of amino acids. Breaking a protein down farther, amino acids can be represented as a network of connected atoms, such as carbon, nitrogen, and oxygen. Nodes and edges are the basic components of a network.

For FourQ it turns that one can guarantee an efficiently computable solution with a_i < 2^{64}. Moreover, as the characteristic of the field is a Mersenne prime, modulations can be carried efficiently. Both properties (four dimensional decomposition and Mersenne prime characteristic), alongside usage of fast multiplication formulae (extended twisted Edwards coordinates), make FourQ the currently fastest elliptic curve for the 128 bit security level.

CoL defines a computational problem as a game played by a machine against its environment. Such problem is computable if there is a machine that wins the game against every possible behavior of the environment. Such game-playing machine generalizes the Church-Turing thesis to the interactive level. The classical concept of truth turns out to be a special, zero-interactivity-degree case of computability.

Some, however, might raise issues with this assessment. At the time (mid-1940s to mid-1950s) a relatively small cadre of researchers were intimately involved with the architecture of the new "digital computers". Hao Wang (1954), a young researcher at this time, made the following observation: :Turing's theory of computable functions antedated but has not much influenced the extensive actual construction of digital computers.

Massimo Egidi (December 1, 1942) is an Italian economist. He is Professor of Economics at Libera Università Internazionale degli Studi Sociali Guido Carli in Rome and former rector of the university. With Axel Leijonhuvfud is co- director of CELL, the Laboratory of Computable and Experimental Economics (University of Trento). His main research interests are related to the study of boundedly rational behaviors in organizations and institutions.

In 1930 Schnirelmann used these ideas in conjunction with the Brun sieve to prove Schnirelmann's theorem, that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant:Nathanson (1996) p.208 Schnirelmann obtained C < 800000.Gelfond & Linnik (1966) p.136 Schnirelmann's constant is the lowest number C with this property.

The speed prior is a complexity measure similar to Kolmogorov complexity, except that it is based on computation speed as well as program length.Schmidhuber, J. (2002) The Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions. In J. Kivinen and R. H. Sloan, editors, Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002). Lecture Notes in Artificial Intelligence, pages 216--228. Springer.

It is impossible to calculate outcomes with any certainty in a decision situation that involves qualitative variables. This is based on the theory of computability which states that a problem is computable if an algorithm exists, the algorithm is efficient/tractable and if there is a well defined solution state. Because of their very nature, qualitative problems lack well-defined state and hence the law.

Though Solomonoff's inductive inference is not computable, several AIXI-derived algorithms approximate it in order to make it run on a modern computer. The more computing power they are given, the closer their predictions are to the predictions of inductive inference (their mathematical limit is Solomonoff's inductive inference).J. Veness, K.S. Ng, M. Hutter, W. Uther, D. Silver. "A Monte Carlo AIXI Approximation" – Arxiv preprint, 2009 arxiv.orgJ.

For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation.

A completion of Peano arithmetic is a set of formulas in the language of Peano arithmetic, such that the set is consistent in first-order logic and such that, for each formula, either that formula or its negation is included in the set. Once a Gödel numbering of the formulas in the language of PA has been fixed, it is possible to identify completions of PA with sets of natural numbers, and thus to speak about the computability of these completions. A Turing degree is defined to be a PA degree if there is a set of natural numbers in the degree that computes a completion of Peano Arithmetic. (This is equivalent to the proposition that every set in the degree computes a completion of PA.) Because there are no computable completions of PA, the degree 0 consisting of the computable sets of natural numbers is not a PA degree.

His PhD thesis, titled "Systems of Logic Based on Ordinals", contains the following definition of "a computable function": When Turing returned to the UK he ultimately became jointly responsible for breaking the German secret codes created by encryption machines called "The Enigma"; he also became involved in the design of the ACE (Automatic Computing Engine), "[Turing's] ACE proposal was effectively self- contained, and its roots lay not in the EDVAC [the USA's initiative], but in his own universal machine" (Hodges p. 318). Arguments still continue concerning the origin and nature of what has been named by Kleene (1952) Turing's Thesis. But what Turing did prove with his computational-machine model appears in his paper "On Computable Numbers, with an Application to the Entscheidungsproblem" (1937): Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable".

A distributional problem (L, D) is in the complexity class AvgP if there is an efficient average-case algorithm for L, as defined above. The class AvgP is occasionally called distP in the literature. A distributional problem (L, D) is in the complexity class distNP if L is in NP and D is P-computable. When L is in NP and D is P-samplable, (L, D) belongs to sampNP.

While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero), this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers , , and e are normal, but a proof remains elusive.

Bemelmans supervised more than thirty doctoral students, and was involved in many other promotions as a committee. Among his doctoral students were Eero Eloranta (1981),Eero Eloranta (b. 1950), Professor in Charge of the Renewal and vice dean of the Atlto University. Jacques Theeuwes (1985), Jan Dietz (1987), Maarten Looijen (1988),Scheidend ict- hoogleraar Maarten Looijen: 'Het traditionele en veelal lokale beheer behoeft aanpassing by Cok de Zwart, in computable.

The principle of the modern computer was first described by computer scientist Alan Turing, who set out the idea in his seminal 1936 paper,. Online versions: Proceedings of the London Mathematical Society Another version online. On Computable Numbers. Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines.

Many fixed-point theorems yield algorithms for locating the least fixed point. Least fixed points often have desirable properties that arbitrary fixed points do not. In mathematical logic and computer science, the least fixed point is related to making recursive definitions (see domain theory and/or denotational semantics for details). Immerman N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1–3) (1986) 86–104.

The Todd–Coxeter algorithm can be applied to infinite groups and is known to terminate in a finite number of steps, provided that the index of H in G is finite. On the other hand, for a general pair consisting of a group presentation and a subgroup, its running time is not bounded by any computable function of the index of the subgroup and the size of the input data.

The group has also hosted several conferences and workshops, with participants and speakers from all around the world. In recent years, De has contributed significantly to the understanding of quantum information and communication, in particular the formulation of a computable entanglement measure and a novel density- matrix recursion method. Her work also involves understanding the theory of quantum channels, the security of quantum cryptography and quantification of quantum correlations.

And suppose that that partial recursive function converges (to something, not necessarily zero) whenever μyR(y, x1, ..., xk) is defined and y is μyR(y, x1, ..., xk) or smaller. Then the function μyR(y, x1, ..., xk) is also a partial recursive function. The μ-operator is used in the characterization of the computable functions as the μ recursive functions. In constructive mathematics, the unbounded search operator is related to Markov's principle.

Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. Correspondence systems were introduced by Post in 1946 to give simple examples of undecidability. He showed that the Post Correspondence Problem (PCP) of satisfying their constraints is, in general, undecidable. With 2 string pairs, PCP was shown to be decidable in 1981.

Carl Jockusch and Robert Soare (1972) proved that the PA degrees are exactly the degrees of DNR2 functions. By definition, a degree is PA if and only if it computes a path through the tree of completions of Peano arithmetic. A stronger property holds: a degree a is a PA degree if and only if a computes a path through every infinite computable subtree of 2<ω (Simpson 1977).

The halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church–Turing thesis, there is no effective procedure (with an algorithm) which can perform these computations.

Several variants of this model are also equivalent to DFAs. In particular, the nondeterministic case (in which the transition from one state can be to multiple states given the same input) is reducible to a DFA. Other variants of this model allow more computational complexity. With a single infinite stack the model can parse (at least) any language that is computable by a Turing machine in linear time.

An interesting question is how large the Nakamura number can be. It has been shown that for a (finite or) algorithmically computable simple game that has no veto player (an individual that belongs to every winning coalition) to have a Nakamura number greater than three, the game has to be non-strong. This means that there is a losing (i.e., not winning) coalition whose complement is also losing.

The motivation behind using the circular convolution approach is that it is based on the DFT. The premise behind circular convolution is to take the DFTs of the input signals, multiply them together, and then take the inverse DFT. Care must be taken such that a large enough DFT is used such that aliasing does not occur. The DFT is numerically computable when dealing with signals of finite-extent.

Contemporary biosophers include Jong Bhak, who defines Biosophy as a "new way of performing philosophy generated from scientific and biological awareness". Bhak developed his theory of Biosophy while studying at Cambridge University in 1995 and afterwards. The main difference of Bhak's biosophy from other philosophy is that his biosophy is a computable philosophy. It borrows Bertrand Russell logicism and extends it to a computational set of ideas and knowledge.

If halts(g) returns false, then g will halt, because it will not call loop_forever; this is also a contradiction. Overall, halts(g) can not return a truth value that is consistent with whether g halts. Therefore, the initial assumption that halts is a total computable function must be false. The method used in the proof is called diagonalization - g does the opposite of what halts says g should do.

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.

There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of Peano arithmetic). Certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formal system might not be sufficiently powerful to show that o is, indeed, an ordinal notation: the system does not show transfinite induction for such large ordinals. For example, the usual first-order Peano axioms do not prove transfinite induction for (or beyond) ε0: while the ordinal ε0 can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε0 proves the consistency of Peano's axioms (a theorem by Gentzen), so by Gödel's second incompleteness theorem, Peano's axioms cannot formalize that reasoning. (This is at the basis of the Kirby–Paris theorem on Goodstein sequences.) We say that ε0 measures the proof-theoretic strength of Peano's axioms.

In his first book on consciousness, The Emperor's New Mind (1989), he argued that while a formal system cannot prove its own consistency, Gödel's unprovable results are provable by human mathematicians. Penrose took this to mean that human mathematicians are not formal proof systems and not running a computable algorithm. According to Bringsjord and Xiao, this line of reasoning is based on fallacious equivocation on the meaning of computation.Bringsjord, S. and Xiao, H. 2000.

A Refutation of Penrose's Gödelian Case Against Artificial Intelligence. Journal of Experimental and Theoretical Artificial Intelligence In the same book, Penrose wrote, "One might speculate, however, that somewhere deep in the brain, cells are to be found of single quantum sensitivity. If this proves to be the case, then quantum mechanics will be significantly involved in brain activity." Penrose determined wave function collapse was the only possible physical basis for a non-computable process.

Like Solomonoff induction, AIXI is incomputable. However, there are computable approximations of it. One such approximation is AIXItl, which performs at least as well as the provably best time t and space l limited agent. Another approximation to AIXI with a restricted environment class is MC-AIXI (FAC-CTW) (which stands for Monte Carlo AIXI FAC-Context-Tree Weighting), which has had some success playing simple games such as partially observable Pac-Man.

All programs in the language must terminate, and this language can only express primitive recursive functions.Stanford Encyclopedia of Philosophy: Computability and Complexity FlooP is identical to BlooP except that it supports unbounded loops; it is a Turing-complete language and can express all computable functions. For example, it can express the Ackermann function, which (not being primitive recursive) cannot be written in BlooP. Borrowing from standard terminology in mathematical logic,Hofstadter (1979), p. 424.

Schnirelmann sought to prove Goldbach's conjecture. In 1930, using the Brun sieve, he proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant.Schnirelmann, L.G. (1930). "On the additive properties of numbers", first published in Proceedings of the Don Polytechnic Institute in Novocherkassk , vol XIV (1930), pp. 3–27, and reprinted in Uspekhi Matematicheskikh Nauk , 1939, no.

The BHK interpretation will depend on the view taken about what constitutes a function that converts one proof to another, or that converts an element of a domain to a proof. Different versions of constructivism will diverge on this point. Kleene's realizability theory identifies the functions with the computable functions. It deals with Heyting arithmetic, where the domain of quantification is the natural numbers and the primitive propositions are of the form x=y.

For example, they showed that: ::All computable functions on the real numbers are the unique solutions to a single finite system of algebraic formulae. The second generalisation, created with Viggo Stoltenberg-Hansen, focuses on implementing data types using approximations contained in the ordered structures of domain theory. The general theories have been applied as formal methods in microprocessor verifications, data types, and tools for volume graphics and modelling excitable media including the heart.

Further work has revealed oblivious transfer to be a fundamental and important problem in cryptography. It is considered one of the critical problems in the field, because of the importance of the applications that can be built based on it. In particular, it is complete for secure multiparty computation: that is, given an implementation of oblivious transfer it is possible to securely evaluate any polynomial time computable function without any additional primitive.

CDF is an electronic document formatWolfram Alpha Creator plans to delete the PDF The Telegraph (UK) designed to allow easy authoringWolfram makes data interactive PC World of dynamically generated interactive content. In June 2014, Wolfram Research officially introduced the Wolfram Language as a new general multi-paradigm programming language. It is the primary programming language used in Mathematica.Slate's article Stephen Wolfram's New Programming Language: He Can Make The World Computable, March 6, 2014.

An open source math-aware question answering system based on Ask Platypus and Wikidata was published in 2018. The system takes an English or Hindi natural language question as input and returns a mathematical formula retrieved from Wikidata as succinct answer. The resulting formula is translated into a computable form, allowing the user to insert values for the variables. Names and values of variables and common constants are retrieved from Wikidata if available.

Lazy linear hybrid automata model the discrete time behavior of control systems containing finite-precision sensors and actuators interacting with their environment under bounded inertial delays. The model permits only linear flow constraints but the invariants and guards can be any computable function. This computational model was proposed by Manindra Agrawal and P. S. Thiagarajan. This model is more realistic and also computationally amenable than the currently popular modeling paradigm of linear hybrid automaton.

The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called analytical. Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical.

The proof is adapted from Barendregt in The Lambda Calculus. Let A and B be closed under beta- convertibility and let a and b be lambda term representations of elements from A and B respectively. Suppose for a contradiction that f is a lambda term representing a computable function such that fx = 0 if x \in A and fx = 1 if x \in B (where equality is β-equality). Then define G \equiv \lambda x.

A-reduction and P-reduction are similar reduction schemes that can be used to show membership in APX and PTAS respectively. Both introduce a new function c, defined on numbers greater than 1, which must be computable. In an A-reduction, we have that :R_B(x', y') \le r \rightarrow R_A(x, y) \le c(r). In a P-reduction, we have that :R_B(x', y') \le c(r) \rightarrow R_A(x, y) \le r.

Milner is generally regarded as having made three major contributions to computer science. He developed Logic for Computable Functions (LCF), one of the first tools for automated theorem proving. The language he developed for LCF, ML, was the first language with polymorphic type inference and type-safe exception handling. In a very different area, Milner also developed a theoretical framework for analyzing concurrent systems, the calculus of communicating systems (CCS), and its successor, the -calculus.

From 1947, he worked at the National Physical Laboratory (NPL) where Alan Turing was designing the Automatic Computing Engine (ACE) computer. It is said that Davies spotted mistakes in Turing's seminal 1936 paper On Computable Numbers, much to Turing's annoyance. These were perhaps some of the first "programming" bugs in existence, even if they were for a theoretical computer, the universal Turing machine. The ACE project was overambitious and floundered, leading to Turing's departure.

Among other contributions, he introduced alternating Turing machines in computational complexity (with Dexter Kozen and Larry Stockmeyer), conjunctive queries in databases (with Philip M. Merlin), computable queries (with David Harel), and multiparty communication complexity (with Merrick L. Furst and Richard J. Lipton). He was a founder of the annual IEEE Symposium on Logic in Computer Science and served as conference chair of the first three conferences, in 1986–8. He was an IEEE Fellow.

The difficulties here were met by radically different proof techniques, taking much more care about proofs by contradiction. The logic involved is closer to proof theory than to that of computability theory and computable functions. It is rather loosely conjectured that the difficulties may lie in the realm of computational complexity theory. Ineffective results are still being proved in the shape A or B, where we have no way of telling which.

In some cases, one can take the mathematical model and using analytical methods, develop closed form solutions such as the Black–Scholes model and the Black model. The resulting solutions are readily computable, as are their "Greeks". Although the Roll–Geske–Whaley model applies to an American call with one dividend, for other cases of American options, closed form solutions are not available; approximations here include Barone-Adesi and Whaley, Bjerksund and Stensland and others.

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.

Biocuration assembles information from patient records, research outputs and medical literature to create a quality-controlled, computable format. She is Director of Translational Data Science at the Linus Pauling Institute. Haendel believes that a globally consistent set of criteria, more comprehensive data collection, sharing and analysis will help to diagnose rare diseases. Rare diseases are thought to impact 10% of the global population, meaning that there are considerable numbers of patients who are underserved by their healthcare systems.

The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set of numerically computable equations that can be applied to analyze a great variety of many-body and/or quantum-optical problems. For example, it is widely applied in semiconductor quantum opticsKira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

One can formally define functions that are not computable. A well-known example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines.

We can view the circuit in PIGEON as a polynomial-time computable hash function. Hence, PPP is the complexity class which captures the hardness of either inverting or finding a collision in hash functions. More generally, the relationship of subclasses of FNP to polynomial-time complexity classes can be used to determine the existence of certain cryptographic primitives, and vice versa. For example, it is known that if FNP = FP, then one-way functions do not exist.

Nodes with high betweenness essentially serve as bridges between different portions of the network (i.e. interactions must pass through this node to reach other portions of the network). In social networks, nodes with high degree or high betweenness may play important roles in the overall composition of a network. As early as the 1980s, researchers started viewing DNA or genomes as the dynamic storage of a language system with precise computable finite states represented as a finite state machine.

XP is the class of parameterized problems that can be solved in time n^{f(k)} for some computable function . These problems are called slicewise polynomial, in the sense that each "slice" of fixed k has a polynomial algorithm, although possibly with a different exponent for each k. Compare this with FPT, which merely allows a different constant prefactor for each value of k. XP contains FPT, and it is known that this containment is strict by diagonalization.

Such a computing structure could define the first > example of an organic computer capable of solving heuristic problems that > would be deemed non-computable by a general Turing-machine. Future works > will elucidate in detail the characteristics of this multi-brain system, its > computational capabilities, and how it compares to other non-Turing > computational architectures Miguel Nicolelis of Duke University, one of the investigators who did the experiment with rats, has done previous work using a brain–computer interface.

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing- computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power.

He advanced the frequency theory of randomness in terms of what he called the collective, i.e. a random sequence. Von Mises regarded the randomness of a collective as an empirical law, established by experience. He related the "disorder" or randomness of a collective to the lack of success of attempted gambling systems. This approach led him to suggest a definition of randomness that was later refined and made mathematically rigorous by Alonzo Church by using computable functions in 1940.

At graduation, she ranked among the top ten students in the School of Science and was a member of Phi Beta Kappa and Phi Kappa Phi. Upon enrolling in doctoral studies at the University of Minnesota Institute of Technology, Zimmerman was named a Corporate Associate Fellow. In 1990, she earned a doctorate in computer science, specializing in computational and recursion theory. Zimmerman completed her dissertation titled Classes of Grzegorczyk-Computable Real Numbers under her doctoral advisor Marian Pour-El.

In its full generality, partitioning cryptanalysis works by dividing the sets of possible plaintexts and ciphertexts into efficiently-computable partitions such that the distribution of ciphertexts is significantly non-uniform when the plaintexts are chosen uniformly from a given block of the partition. Partitioning cryptanalysis has been shown to be more effective than linear cryptanalysis against variants of DES and CRYPTON. A specific partitioning attack called mod n cryptanalysis uses the congruence classes modulo some integer for partitions.

Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.

The field of mathematical logic dealing with computability and its generalizations has been called "recursion theory" since its early days. Robert I. Soare, a prominent researcher in the field, has proposed (Soare 1996) that the field should be called "computability theory" instead. He argues that Turing's terminology using the word "computable" is more natural and more widely understood than the terminology using the word "recursive" introduced by Kleene. Many contemporary researchers have begun to use this alternate terminology.

Wiley Publishing (2009). and says that perhaps Taleb is correct to urge that banks be treated as utilities forbidden to take potentially lethal risks, while hedge funds and other unregulated entities should be able to do what they want. Taleb's writings discuss the error of comparing real-world randomness with the "structured randomness" in quantum physics where probabilities are remarkably computable and games of chance like casinos where probabilities are artificially built. Taleb calls this the "ludic fallacy".

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.

There is active research to make computers out of many promising new types of technology, such as optical computers, DNA computers, neural computers, and quantum computers. Most computers are universal, and are able to calculate any computable function, and are limited only by their memory capacity and operating speed. However different designs of computers can give very different performance for particular problems; for example quantum computers can potentially break some modern encryption algorithms (by quantum factoring) very quickly.

There exist, however, typed lambda calculi that are not strongly normalizing. For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to Girard's paradox. This system is also the simplest pure type system, a formalism which generalizes the Lambda cube. Systems with explicit recursion combinators, such as Plotkin's "Programming language for Computable Functions" (PCF), are not normalizing, but they are not intended to be interpreted as a logic.

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).

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.

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.

In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in number theory. If K is a compact group and m is the associated Haar measure, the Peter–Weyl theorem provides a decomposition of L^2(K,dm) as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of K.

As Nimrod Megiddo showed,. the minimum enclosing circle can be found in linear time, and the same linear time bound also applies to the smallest enclosing sphere in Euclidean spaces of any constant dimension. His article also gives a brief overview of earlier O(n^3) and O(n \log n) algorithms ; in doing so, Megiddo demonstrated that Shamos and Hoey's conjecture – that a solution to the smallest-circle problem was computable in \Omega(n \log n) at best – was incorrect. Emo Welzl.

Permutation City asks whether there is a difference between a computer simulation of a person and a "real" person. It focuses on a model of consciousness and reality, the Dust Theory, similar to the Ultimate Ensemble Mathematical Universe hypothesis proposed by Max Tegmark. It uses the assumption that human consciousness is Turing-computable: that consciousness can be produced by a computer program. The book deals with consequences of human consciousness being amenable to mathematical manipulation, as well as some consequences of simulated realities.

It is interesting about interval chromatic number that it is easily computable. Indeed, by a simple greedy algorithm one can efficiently find an optimal partition of the vertex set of H into X<(H) independent intervals. This is in sharp contrast with the fact that even the approximation of the usual chromatic number of graph is an NP hard task. Let K(H) is the chromatic number of any ordered graph H. Then for any ordered graph H, X<(H) ≥ K(H).

Church's proof first reduces the problem to determining whether a given lambda expression has a normal form. A normal form is an equivalent expression that cannot be reduced any further under the rules imposed by the form. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a Gödel numbering for lambda expressions, he constructs a lambda expression `e` that closely follows the proof of Gödel's first incompleteness theorem.

Some scholars conjecture that a quantum mechanical system which somehow uses an infinite superposition of states could compute a non-computable function.There have been some claims to this effect; see or and the ensuing literature. For a retort see . This is not possible using the standard qubit- model quantum computer, because it is proven that a regular quantum computer is PSPACE-reducible (a quantum computer running in polynomial time can be simulated by a classical computer running in polynomial space).

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.

Alf Rattigan, the first chairman of the Industries Assistance Commission, also enthusiastically supported the development of computable general equilibrium modelling (CGE) in Australia to analyze the effects of tariffs. CGE has been applied to the analysis of proposals for regional trading agreements and for multilateral reductions in tariff levels. The results of these studies have supported those who have advocated trade policy reforms as they invariably show gains from trade liberalization, though the gains are often small relative to existing GDP.

A PRF is an efficient (i.e. computable in polynomial time), deterministic function that maps two distinct sets (domain and range) and looks like a truly random function. Essentially, a truly random function would just be composed of a lookup table filled with uniformly distributed random entries. However, in practice, a PRF is given an input string in the domain and a hidden random seed and runs multiple times with the same input string and seed, always returning the same value.

When studying the complexity class NP and harder classes such as the polynomial hierarchy, polynomial-time reductions are used. When studying classes within P such as NC and NL, log- space reductions are used. Reductions are also used in computability theory to show whether problems are or are not solvable by machines at all; in this case, reductions are restricted only to computable functions. In case of optimization (maximization or minimization) problems, we often think in terms of approximation-preserving reduction.

It turns out that restricting expression to the set of computable functions is not sufficient either if the programming language allows writing non-terminating computations (which is the case if the programming language is Turing complete). Expression must be restricted to the so-called continuous functions (corresponding to continuity in the Scott topology, not continuity in the real analytical sense). Even then, the set of continuous function contains the parallel-or function, which cannot be correctly defined in all programming languages.

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.

A numbering is an enumeration of functions; it has two parameters, e and x and outputs the value of the e-th function in the numbering on the input x. Numberings can be partial-recursive although some of its members are total recursive, that is, computable functions. Admissible numberings are those into which all others can be translated. A Friedberg numbering (named after its discoverer) is a one-one numbering of all partial-recursive functions; it is necessarily not an admissible numbering.

More recent theoretical results are concerned with determining the secrecy capacity and optimal power allocation in broadcast fading channels. There are caveats, as many capacities are not computable unless the assumption is made that Alice knows the channel to Eve. If that were known, Alice could simply place a null in Eve's direction. Secrecy capacity for MIMO and multiple colluding eavesdroppers is more recent and ongoing work, and such results still make the non-useful assumption about eavesdropper channel state information knowledge.

Together with J.A.C. Brown he began the Cambridge Growth Project, which developed the Cambridge Multisectoral Dynamic Model of the British economy (MDM) . In building the Cambridge Growth Project, they used Social Accounting Matrices (SAM), which also formed computable equilibrium model which then developed at the World Bank. He was succeeded as leader of the Cambridge Growth Project by Terry Barker. In 1970, Stone was appointed as the Chairman of the Faculty Board of Economics and Politics for the next two years.

Because computer programs are used to build and work with these complex models, approximations need to be formulated into easily computable forms. Some of these numerical analysis techniques (such as finite differences) require the area of interest to be subdivided into a grid — in this case, over the shape of the Earth. Geodesic grids can be used in video game development to model fictional worlds instead of the Earth. They are a natural analog of the hex map to a spherical surface.

Solomonoff founded the theory of universal inductive inference, which is based on solid philosophical foundations and has its root in Kolmogorov complexity and algorithmic information theory. The theory uses algorithmic probability in a Bayesian framework. The universal prior is taken over the class of all computable measures; no hypothesis will have a zero probability. This enables Bayes' rule (of causation) to be used to predict the most likely next event in a series of events, and how likely it will be.

Except for its two I/O commands, Brainfuck is a minor variation of the formal programming language P′′ created by Corrado Böhm in 1964, which in turn is explicitly based on the Turing machine. In fact, using six symbols equivalent to the respective Brainfuck commands `+`, `-`, `<`, `>`, `[`, `]`, Böhm provided an explicit program for each of the basic functions that together serve to compute any computable function. So the first "Brainfuck" programs appear in Böhm's 1964 paper - and they were programs sufficient to prove Turing completeness.

Recursion theory, also called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work of Rózsa Péter, Alonzo Church and Alan Turing in the 1930s, which was greatly extended by Kleene and Post in the 1940s. Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers.

The busy beaver function quantifies the maximum score attainable by a Busy Beaver on a given measure. This is a noncomputable function. Also, a busy beaver function can be shown to grow faster asymptotically than any computable function. The busy beaver function, Σ: N → N, is defined such that Σ(n) is the maximum attainable score (the maximum number of 1s finally on the tape) among all halting 2-symbol n-state Turing machines of the above-described type, when started on a blank tape.

Soludo graduated with a First Class Honours degree in 1984, an MSc Economics in 1987, and a PhD in 1989, winning prizes for the best student at all three levels.Soludo Appointed CBN Governor, Asia Africa Intelligence Wire, 30 April 2004."N25bn capital base: Senate drills Soludo" BNW News: Biafra Nigeria World News. Chukwuma has been trained and involved in research, teaching, and auditing in such disciplines as multi- country macro-econometric modelling, techniques of computable general equilibrium modelling, survey methodology, and panel data econometrics, among others.

The principle was stated by Deutsch in 1985 with respect to finitary machines and processes. He observed that classical physics, which makes use of the concept of real numbers, cannot be simulated by a Turing machine, which can only represent computable reals. Deutsch proposed that quantum computers may actually obey the CTD principle, assuming that the laws of quantum physics can completely describe every physical process. An earlier version of this thesis for classical computers was stated by Alan Turing's friend and student Robin Gandy in 1980.

Therefore, there is a well-defined function such that all reverses of -state cellular automata with the von Neumann neighborhood use a neighborhood with radius at most : simply let be the maximum, among all of the finitely many reversible -state cellular automata, of the neighborhood size needed to represent the time-reversed dynamics of the automaton. However, because of Kari's undecidability result, there is no algorithm for computing and the values of this function must grow very quickly, more quickly than any computable function..

Some long strings can be described exactly using fewer symbols than those required by their full representation, as is often achieved using data compression. The complexity of a given string is then defined as the minimal length that a description requires in order to (unambiguously) refer to the full representation of that string. The Kolmogorov complexity is defined using formal languages, or Turing machines which avoids ambiguities about which string results from a given description. It can be proven that the Kolmogorov complexity is not computable.

An Armington elasticity is an economic parameter commonly used in models of consumer theory and international trade. It represents the elasticity of substitution between products of different countries, and is based on the assumption made by Paul Armington in 1969 that products traded internationally are differentiated by country of origin. The Armington assumption has become a standard assumption of international computable general equilibrium models. These models generate smaller and more realistic responses of trade to price changes than implied by models of homogeneous products.

Additionally, simulation-based methods are often used to validate inference results, providing test data where the correct answer is known ahead of time. Because computing likelihoods for genealogical data under complex simulation models has proven difficult, an alternative statistical approach called Approximate Bayesian Computation (ABC) is becoming popular in fitting these simulation models to patterns of genetic variation, following successful application of this approach to bacterial diseases. This is because ABC makes use of easily computable summary statistics to approximate likelihoods, rather than the likelihoods themselves.

Chitti is described by Vaseegaran as an advanced "andro-humanoid" robot. Over his metallic body, he sports synthetic inorganic skin molded after Vaseegaran himself. He is designed with a speed capacity of 1 Terahertz (THz) and a memory capacity of 1 Zettabyte. Initially, Chitti has been programmed with almost all the existing world knowledge managed to put in computable terms in his CPU, thus he is knowledgeable and proficient in all forms of academia, martial arts, communication, creative outlets, athletic skills, and scientific ingenuity.

The complexity class \Sigma_2 describes problems of the form :\exists x\forall y\;\psi(x,y) where \psi is any polynomial-time computable predicate. The existential power of the first quantifier in this predicate can be used to guess a correct circuit for SAT, and the universal power of the second quantifier can be used to verify that the circuit is correct. Once this circuit is guessed and verified, the algorithm in class \Sigma_2 can use it as a subroutine for solving other problems.

According to the Church–Turing thesis, Turing machines and the lambda calculus are capable of computing anything that is computable. John von Neumann acknowledged that the central concept of the modern computer was due to Turing's paper."von Neumann ... firmly emphasised to me, and to others I am sure, that the fundamental conception is owing to Turing—insofar as not anticipated by Babbage, Lovelace and others." Letter by Stanley Frankel to Brian Randell, 1972, quoted in Jack Copeland (2004) The Essential Turing, p. 22.

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.

Standard ML (SML) is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers. SML is a modern dialect of ML, the programming language used in the Logic for Computable Functions (LCF) theorem-proving project. It is distinctive among widely used languages in that it has a formal specification, given as typing rules and operational semantics in The Definition of Standard ML.

A language is a subset of the collection of all words on a fixed alphabet. For example, the collection of all binary strings that contain exactly 3 ones is a language over the binary alphabet. A key property of a formal language is the level of difficulty required to decide whether a given word is in the language. Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine.

A classic example using the second recursion theorem is the function Q(x,y)=x. The corresponding index p in this case yields a computable function that outputs its own index when applied to any value . When expressed as computer programs, such indices are known as quines. The following example in Lisp illustrates how the p in the corollary can be effectively produced from the function Q. The function `s11` in the code is the function of that name produced by the S-m-n theorem.

A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle. Even a Turing oracle with random data is not computable (with probability 1), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot.

In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem.See Boolos and Jeffrey (2002, sec. 15) and Mendelson (1997, Prop.

In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions fα: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.

Radó's 1962 paper proved that if f: ℕ → ℕ is any computable function, then Σ(n) > f(n) for all sufficiently large n, and hence that Σ is not a computable function. Moreover, this implies that it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy beaver. (Such an algorithm cannot exist, because its existence would allow Σ to be computed, which is a proven impossibility. In particular, such an algorithm could be used to construct another algorithm that would compute Σ as follows: for any given n, each of the finitely many n-state 2-symbol Turing machines would be tested until an n-state busy beaver is found; this busy beaver machine would then be simulated to determine its score, which is by definition Σ(n).) Even though Σ(n) is an uncomputable function, there are some small n for which it is possible to obtain its values and prove that they are correct. It is not hard to show that Σ(0) = 0, Σ(1) = 1, Σ(2) = 4, and with progressively more difficulty it can be shown that Σ(3) = 6 and Σ(4) = 13 .

The strong reducibilities include: ;One-one reducibility: A is one-one reducible (or 1-reducible) to B if there is a total computable injective function f such that each n is in A if and only if f(n) is in B. ;Many-one reducibility: This is essentially one-one reducibility without the constraint that f be injective. A is many-one reducible (or m-reducible) to B if there is a total computable function f such that each n is in A if and only if f(n) is in B. ;Truth-table reducibility: A is truth-table reducible to B if A is Turing reducible to B via an oracle Turing machine that computes a total function regardless of the oracle it is given. Because of compactness of Cantor space, this is equivalent to saying that the reduction presents a single list of questions (depending only on the input) to the oracle simultaneously, and then having seen their answers is able to produce an output without asking additional questions regardless of the oracle's answer to the initial queries. Many variants of truth-table reducibility have also been studied.

The ease of designing reversible block cellular automata, and of testing block cellular automata for reversibility, is in strong contrast to cellular automata with other non-block neighborhood structures, for which it is undecidable whether the automaton is reversible and for which the reverse dynamics may require much larger neighborhoods than the forward dynamics. Any reversible cellular automaton may be simulated by a reversible block cellular automaton with a larger number of states; however, because of the undecidability of reversibility for non-block cellular automata, there is no computable bound on the radius of the regions in the non-block automaton that correspond to blocks in the simulation, and the translation from a non-block rule to a block rule is also not computable.; Block cellular automata are also a convenient formalism in which to design rules that, in addition to reversibility, implement conservation laws such as the conservation of particle number, conservation of momentum, etc.. For instance, if the rule within each block preserves the number of live cells in the block, then the global evolution of the automaton will also preserve the same number. This property is useful in the applications of cellular automata to physical simulation.

An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the Entscheidungsproblem is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the halting problem, a result with far- ranging implications in both recursion theory and computer science. There are many known examples of undecidable problems from ordinary mathematics.

The HL7 Services-Aware Interoperability Framework Canonical Definition (SAIF-CD) SAIF- CD provides consistency between all artifacts, and enables a standardized approach to enterprise architecture (EA) development and implementation, and a way to measure the consistency. SAIF is a way of thinking about producing specifications that explicitly describe the governance, conformance, compliance, and behavioral semantics that are needed to achieve computable semantic working interoperability. The intended information transmission technology might use a messaging, document exchange, or services approach. SAIF is the framework that is required to rationalize interoperability of standards.

Kalmar defined what are known as elementary functions, number-theoretic functions (i.e. those based on the natural numbers) built up from the notions of composition and variables, the constants 0 and 1, repeated addition + of the constants, proper subtraction ∸, bounded summation and bounded product (Kleene 1952:526). Elimination of the bounded product from this list yields the subelementary or lower elementary functions. By use of the abstract computational model called a register machine Schwichtenberg provides a demonstration that "all elementary functions are computable and totally defined" (Schwichtenberg 58).

Dissatisfied with its randomness, he proposed a new form of wave function collapse that occurs in isolation and called it objective reduction. He suggested each quantum superposition has its own piece of spacetime curvature and that when these become separated by more than one Planck length they become unstable and collapse. Penrose suggested that objective reduction represents neither randomness nor algorithmic processing but instead a non-computable influence in spacetime geometry from which mathematical understanding and, by later extension, consciousness derives. Hameroff provided a hypothesis that microtubules would be suitable hosts for quantum behavior.

Enterprise cognitive systems (ECS) are part of a broader shift in computing, from a programmatic to a probabilistic approach, called cognitive computing. An Enterprise Cognitive System makes a new class of complex decision support problems computable, where the business context is ambiguous, multi-faceted, and fast-evolving, and what to do in such a situation is usually assessed today by the business user. An ECS is designed to synthesize a business context and link it to the desired outcome. It recommends evidence-based actions to help the end-user achieve the desired outcome.

A chart of accounts (COA) is a list of the categories used by an organization to classify and distinguish financial assets, liabilities, and transactions. It is used to organize the entity’s finances and segregate expenditures, revenue, assets and liabilities in order to give interested parties a better understanding of the entity’s financial health. Accounts are typically defined by an identifier (account number) and a caption or header and are coded by account type. In computerized accounting systems with computable quantity accounting, the accounts can have a quantity measure definition.

However, this remained theoretical only - the lesser state of engineering in the lifetime of these two mathematicians proved insufficient to construct the Analytical Engine. The first modern theory of software was proposed by Alan Turing in his 1935 essay Computable numbers with an application to the Entscheidungsproblem (decision problem). This eventually led to the creation of the twin academic fields of computer science and software engineering, which both study software and its creation. Computer science is more theoretical (Turing's essay is an example of computer science), whereas software engineering is focused on more practical concerns.

The fields of constructive analysis and computable analysis were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics. A separate branch of computability theory, computational complexity theory, was also characterized in logical terms as a result of investigations into descriptive complexity. Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title Contributions to the theory of models.

In the late 1960s and early 1970s researchers expanded the counter machine model into the register machine, a close cousin to the modern notion of the computer. Other models include combinatory logic and Markov algorithms. Gurevich adds the pointer machine model of Kolmogorov and Uspensky (1953, 1958): "... they just wanted to ... convince themselves that there is no way to extend the notion of computable function."Gurevich 1988:2 All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be Turing complete.

Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP. It is known that knot Floer homology and Khovanov homology detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot.

A function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f `x` =β `y`, where `x` and `y` are the Church numerals corresponding to x and y, respectively and =β meaning equivalence with β-reduction. This is one of the many ways to define computability; see the Church–Turing thesis for a discussion of other approaches and their equivalence.

PPP contains PPAD as a subclass (strict containment is an open problem). This is because End-of-the-Line, which defines PPAD, admits a straightforward polynomial-time reduction to PIGEON. In End-of-the- Line, the input is a start vertex s in a directed graph G where each vertex has at most one successor and at most one predecessor, represented by a polynomial-time computable successor function f. Define a circuit C whose input is a vertex x and whose output is its successor if there is one, or x if it does not.

An outline (algorithm) for what would have been the first piece of software was written by Ada Lovelace in the 19th century, for the planned Analytical Engine. She created proofs to show how the engine would calculate Bernoulli Numbers. Because of the proofs and the algorithm, she is considered the first computer programmer. The first theory about software—prior to the creation of computers as we know them today—was proposed by Alan Turing in his 1935 essay On Computable Numbers, with an Application to the Entscheidungsproblem (decision problem).

The HL7 Services-Aware Enterprise Architecture Framework (SAIF) provides consistency between all HL7 artifacts, and enables a standardized approach to Enterprise Architecture (EA) development and implementation, and a way to measure the consistency. SAIF is a way of thinking about producing specifications that explicitly describe the governance, conformance, compliance, and behavioral semantics that are needed to achieve computable semantic working interoperability. The intended information transmission technology might use a messaging, document exchange, or services approach. SAIF is the framework that is required to rationalize interoperability of other standards.

Burgin (2005: 13) uses the term recursive algorithms for algorithms that can be implemented on Turing machines, and uses the word algorithm in a more general sense. Then a super-recursive class of algorithms is "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005: 107). Super-recursive algorithms are closely related to hypercomputation in a way similar to the relationship between ordinary computation and ordinary algorithms. Computation is a process, while an algorithm is a finite constructive description of such a process.

There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set.

A black swan event, as analyzed by Nassim Nicholas Taleb, is an important and inherently unpredictable event that, once occurred, is rationalized with the benefit of hindsight. Another position of the black swan theory is that appropriate preparation for these events is frequently hindered by the pretense of knowledge of all the risks; in other words, Knightian uncertainty is presumed to not exist in day-to-day affairs, often with disastrous consequences. Taleb asserts that Knightian risk does not exist in the real world, and instead finds gradations of computable risk.

Turing described such a construction in complete detail in his 1936 paper: :"It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with a tape on the beginning of which is written the S.D ["standard description" of an action table] of some computing machine M, then U will compute the same sequence as M."Boldface replacing script. Turing 1936 in Davis 1965:127–128. An example of Turing's notion of S.D is given at the end of this article.

He has held tenured and visiting appointments at the European University Institute, Fiesole, Italy, UCLA, the People's University in Beijing and several other European Universities and Research Institutions. He is the founder of the Algorithmic Social Sciences Research Unit at the University of Trento. A Festschrift in Vela Velupillai's honour, Computable, Constructive and Behavioural Economic Dynamics, edited by Stefano Zambelli, was published by Routledge. A Special Issue of the journal New Mathematics and Natural Computation, edited by Shu-Heng, in honour of Vela Velupillai, was published in March, 2012.

In the late 1970s, Kotlikoff, together with Berkeley economist, Alan J. Auerbach, developed the first large-scale computable general equilibrium life- cycle model that can track the behavior, over time, of economies comprising large numbers of overlapping generations.Dynamic Fiscal Policy (with Alan Auerbach), Cambridge University Press, 1987. The model and its offspring have been used extensively to study future fiscal and demographic transitions in the U.S. and abroad. Demographically realistic overlapping generations models, in which agents can live for up to, say, 100 years, are very complicated mathematical structures.

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.

In the field of informatics, an archetype is a formal re-usable model of a domain concept. Traditionally, the term archetype is used in psychology to mean an idealized model of a person, personality or behaviour (see Archetype). The usage of the term in informatics is derived from this traditional meaning, but applied to domain modelling instead. An archetype is defined by the OpenEHR Foundation (for health informatics) as follows: :An archetype is a computable expression of a domain content model in the form of structured constraint statements, based on some reference model.

In 2014, it was proposed that existing technology and standard probabilistic methods of generating single photon states could be used as input into a suitable quantum computable linear optical network and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted the sampling problem had similar complexity for inputs other than Fock state photons and identified a transition in computational complexity from classically simulatable to just as hard as the Boson Sampling Problem, dependent on the size of coherent amplitude inputs.

Recursion theory in mathematical logic has traditionally focused on relative computability, a generalization of Turing computability defined using oracle Turing machines, introduced by Turing (1939). An oracle Turing machine is a hypothetical device which, in addition to performing the actions of a regular Turing machine, is able to ask questions of an oracle, which is a particular set of natural numbers. The oracle machine may only ask questions of the form "Is n in the oracle set?". Each question will be immediately answered correctly, even if the oracle set is not computable.

Alternatively, a Turing-equivalent system is one that can simulate, and be simulated by, a universal Turing machine. (All known Turing-complete systems are Turing-equivalent, which adds support to the Church–Turing thesis.) ;(Computational) universality : A system is called universal with respect to a class of systems if it can compute every function computable by systems in that class (or can simulate each of those systems). Typically, the term universality is tacitly used with respect to a Turing-complete class of systems. The term "weakly universal" is sometimes used to distinguish a system (e.g.

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.

Digital physics suggests that there exists, at least in principle, a program for a universal computer that computes the evolution of the universe. The computer could be, for example, a huge cellular automaton (Zuse 1967Zuse, Konrad, 1967, Elektronische Datenverarbeitung vol 8., pages 336–344), or a universal Turing machine, as suggested by Schmidhuber (1997), who pointed out that there exists a short program that can compute all possible computable universes in an asymptotically optimal way. Loop quantum gravity could lend support to digital physics, in that it assumes space-time is quantized.

John A. Wheeler, 1990, "Information, physics, quantum: The search for links" in W. Zurek (ed.) Complexity, Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley. Moreover, computers can manipulate and solve formulas describing real numbers using symbolic computation, thus avoiding the need to approximate real numbers by using an infinite number of digits. A number—in particular a real number, one with an infinite number of digits—was defined by Alan Turing to be computable if a Turing machine will continue to spit out digits endlessly.

A similar Web Services adapter was developed for the DoD to connect to their CHCS-I legacy patient record systems. In this way, both systems hosted a peer Web Services client that is accessible to the other with proper authentication, allowing bi-directional, query-based data exchanges between the disparate systems. Direct cross-Domain write capability and fully computable data storage and transfers are not supported at this time. With the addition of the DSI components to the FHIE system, the entire project was renamed the Federal Bi-Directional Healthcare Information Exchange, or BHIE.

He became Chairman of the Department of Mathematics at Ohio State University in 1948. In the 1920s, he proved that surfaces have an essentially unique triangulation. In 1933, Radó published "On the Problem of Plateau" in which he gave a solution to Plateau's problem, and in 1935, "Subharmonic Functions". His work focused on computer science in the last decade of his life and in May 1962 he published one of his most famous results in the Bell System Technical Journal: the busy beaver function and its non-computability ("On Non-Computable Functions").

The proof that the halting problem is not solvable is a proof by contradiction. To illustrate the concept of the proof, suppose that there exists a total computable function halts(f) that returns true if the subroutine f halts (when run with no inputs) and returns false otherwise. Now consider the following subroutine: def g(): if halts(g): loop_forever() halts(g) must either return true or false, because halts was assumed to be total. If halts(g) returns true, then g will call loop_forever and never halt, which is a contradiction.

From 1984 to 1989, he attended Harvard University, where he undertook postgraduate studies in economics. He was awarded a Master of Arts (AM) degree in June 1986 and a Doctor of Philosophy (PhD) degree in June 1989. His dissertation was titled "Agricultural Change and Rural Depopulation: Ireland 1845–76". Using computable general equilibrium techniques, and detailed statistics on the Irish economy collected by the UK administration, he challenged the hypothesis that Ireland's Great Famine was merely an inevitable acceleration of existing trends. His work showed that rural depopulation was not linked to relative price changes for agricultural goods,O'Rourke, K. (1991).

In the case of computing, concrete, precise information is in general not computable within finite time and memory (see Rice's theorem and the halting problem). Abstraction is used to allow for generalized answers to questions (for example, answering "maybe" to a yes/no question, meaning "yes or no", when we (an algorithm of abstract interpretation) cannot compute the precise answer with certainty); this simplifies the problems, making them amenable to automatic solutions. One crucial requirement is to add enough vagueness so as to make problems manageable while still retaining enough precision for answering the important questions (such as "might the program crash?").

The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak Mordell–Weil theorem that its subgroup B(K)/f(A(K)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime p such that the p-component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which case any prime p would work.

For example, they showed that ::On any discrete data type, functions are definable as the unique solutions of small finite systems of equations if, and only if, they are computable by algorithms. The results combined techniques of universal algebra and recursion theory, including term rewriting and Matiyasevich's theorem. For the other problems, he and his co-workers have developed two independent disparate generalisations of classical computability/recursion theory, which are equivalent for many continuous data types. The first generalisation, created with Jeffrey Zucker, focuses on imperative programming with abstract data types and covers specifications and verification using Hoare logic.

Although there are infinitely many halting probabilities, one for each method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction instead of Chaitin's constant when not referring to any specific encoding. Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits. Indeed, each halting probability is Martin-Löf random, meaning there is not even any algorithm which can reliably guess its digits.

This also works when a time in 24-hour format is included after the date, as long as all times are understood to be in the same time zone. ISO 8601 is used widely where concise, human readable yet easily computable and unambiguous dates are required, although many applications store dates internally as UNIX time and only convert to ISO 8601 for display. It is worth noting that all modern computer Operating Systems retain date information of files outside of their titles, allowing the user to choose which format they prefer and have them sorted thus, irrespective of the files' names.

The rank of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X. There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,John R. Stallings. Problems about free quotients of groups. Geometric group theory (Columbus, OH, 1992), pp.

This leads to computable variants of AC and AP, and Universal "Levin" Search (US) solves all inversion problems in optimal time (apart from some unrealistically large multiplicative constant). AC and AP also allow a formal and rigorous definition of randomness of individual strings to not depend on physical or philosophical intuitions about non-determinism or likelihood. Roughly, a string is Algorithmic "Martin-Löf" Random (AR) if it is incompressible in the sense that its algorithmic complexity is equal to its length. AC, AP, and AR are the core sub-disciplines of AIT, but AIT spawns into many other areas.

The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing. The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious. This article provides detail of that debate and discovery from Peano's axioms in 1889 through recent discussion of the meaning of "axiom".

A closely related methodology that pre-dates DSGE modeling is computable general equilibrium (CGE) modeling. Like DSGE models, CGE models are often microfounded on assumptions about preferences, technology, and budget constraints. However, CGE models focus mostly on long-run relationships, making them most suited to studying the long-run impact of permanent policies like the tax system or the openness of the economy to international trade. DSGE models instead emphasize the dynamics of the economy over time (often at a quarterly frequency), making them suited for studying business cycles and the cyclical effects of monetary and fiscal policy.

In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in the following way: no efficient algorithm can distinguish (with significant advantage) between a function chosen randomly from the PRF family and a random oracle (a function whose outputs are fixed completely at random). Pseudorandom functions are vital tools in the construction of cryptographic primitives, especially secure encryption schemes. Pseudorandom functions are not to be confused with pseudorandom generators (PRGs). The guarantee of a PRG is that a single output appears random if the input was chosen at random.

The Karp–Lipton theorem can be restated as a result about Boolean formulas with polynomially-bounded quantifiers. Problems in \Pi_2 are described by formulas of this type, with the syntax :\phi = \forall x \exists y \; \psi(x, y) where \psi is a polynomial-time computable predicate. The Karp–Lipton theorem states that this type of formula can be transformed in polynomial time into an equivalent formula in which the quantifiers appear in the opposite order; such a formula belongs to \Sigma_2. Note that the subformula :s(x)=\exists y \; \psi(x, y) is an instance of SAT.

A cognitive robot that needs at least 1 liter of gasoline per hour interacts with a partially unknown environment, trying to find hidden, limited gasoline depots to occasionally refuel its tank. It is rewarded in proportion to its lifetime, and dies after at most 100 years or as soon as its tank is empty or it falls off a cliff, and so on. The probabilistic environmental reactions are initially unknown but assumed to be sampled from the axiomatized Speed Prior, according to which hard-to-compute environmental reactions are unlikely. This permits a computable strategy for making near-optimal predictions.

He has also led the focus on interactive publishing technologyMaking Science Leap From the Page The New York Times, 17 December 2011 with the stated aim of "making new applications as everyday as new documents"The day that documents and applications merged, Wolfram Research. claiming that "If a picture is worth a thousand words, an interactive document is worth a thousand pictures."Interactive deployment, Wolfram Research. These technologies converged to form the Computable Document FormatWolfram Alpha Creator plans to delete the PDF The Telegraph (UK) which Wolfram says can "transfer knowledge in a much higher-bandwidth way".

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).

171 Von Mises never totally formalized his rules for sub-sequence selection, but in his 1940 paper "On the concept of random sequence", Alonzo Church suggested that the functions used for place settings in the formalism of von Mises be computable functions rather than arbitrary functions of the initial segments of the sequence, appealing to the Church–Turing thesis on effectiveness.Alonzo Church, "On the concept of random sequence," Bull. Amer. Math. Soc., 46 (1940), 254–260J. Alberto Coffa, "Randomness and knowledge," in PSA 1972: proceedings of the 1972 Biennial Meeting Philosophy of Science Association, Volume 20, Springer, 1974 p.

In June 2009, Nick Smith released an economic modelling report on the NZ ETS by economic consultants NZIER and Infometrics which had been prepared for the Emissions Trading Scheme Review Committee. Smith stated that the report supported the Government's intention to modify the NZ ETS. The report "Economic modelling of New Zealand climate change policy" created static computable general equilibrium (CGE) models using 2008 emissions projections. The terms of reference set the policy options to look at as the 2008 NZETS vs the least-cost option for meeting the Kyoto liability vs a revenue-neutral tax on carbon equivalents.

A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology involves analysis of homology at different resolutions, registering homology classes (holes) that persist as the resolution is changed.

The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s.

This complexity measure is closely related to the diameter of the network. Let D be the diameter of the network. On the one hand, any computable problem can be solved trivially in a synchronous distributed system in approximately 2D communication rounds: simply gather all information in one location (D rounds), solve the problem, and inform each node about the solution (D rounds). On the other hand, if the running time of the algorithm is much smaller than D communication rounds, then the nodes in the network must produce their output without having the possibility to obtain information about distant parts of the network.

Every decision problem can be converted into the function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the associated decision problem is decidable. However, this reduction is more liberal than the standard reduction used in computational complexity (sometimes called polynomial-time many-one reduction); for example, the complexity of the characteristic functions of an NP-complete problem and its co-NP-complete complement is exactly the same even though the underlying decision problems may not be considered equivalent in some typical models of computation.

He obtained his bachelor's degree at the University of Lund, earned an MA in Economics from the University of Pittsburgh and a Ph.D. in Economics from Northwestern University in 1967. He accepted a position as Acting Assistant Professor in the Economics Department at UCLA in 1964, was promoted to Associate Professor in 1967, and to Full Professor in 1971. In 1991, he started the Center for Computable Economics at UCLA and remained its Director until 1997. Leijonhufvud was awarded an honoris causa doctoral degree by the University of Lund in 1983 and by the University Nice - Sophia Antipolis in 1996.

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let F be a formula in the language with one free variable, then: Intuitively, \psi is a self-referential sentence saying that \psi has the property F. The sentence \psi can also be viewed as a fixed point of the operation assigning to each formula \theta the sentence F(^\circ \\#(\theta)). The sentence \psi constructed in the proof is not literally the same as F(^\circ \\#(\psi)), but is provably equivalent to it in the theory T.

Quantum computers offer a search advantage over classical computers by searching many database elements at once as a result of quantum superpositions. A sufficiently advanced quantum computer would break current encryption methods by factorizing large numbers several orders of magnitude faster than any existing classical computer. Any computable problem may be expressed as a general quantum search algorithm although classical computers may have an advantage over quantum search when using more efficient tailored classical algorithms. The issue with quantum computers is that a measurement must be made to determine if the problem is solved which collapses the superposition.

She was very isolated and lonely at Harvard, with few friends and, initially, no other students even willing to sit next to her in her classes. The nearest restroom to her classes was in a different building, and one of the few buildings with air conditioning in the summers was off-limits to women, even when she was assigned as an instructor to a class in that building. Because there were no logicians at Harvard at that time, she spent five years of her time as a visiting student at the University of California, Berkeley. Her doctoral dissertation was Computable Functions.

The project was managed by John R. Womersley, superintendent of the Mathematics Division of the National Physical Laboratory (NPL). The use of the word Engine was in homage to Charles Babbage and his Difference Engine and Analytical Engine. Turing's technical design Proposed Electronic Calculator was the product of his theoretical work in 1936 "On Computable Numbers" (and ) and his wartime experience at Bletchley Park where the Colossus computers had been successful in breaking German military codes. In his 1936 paper, Turing described his idea as a "universal computing machine", but it is now known as the Universal Turing machine.

However, a detrimental aspect of such ratio optimizations is that, once the achieved ratio in some state is high, the optimization might select states leading to a low ratio because they bear a high probability of termination, so that the process is likely to terminate before the ratio drops significantly. A problem setting to prevent such early terminations consists of defining the optimization as maximization of the future ratio seen by each state. An indexation is conjectured to exist for this problem, be computable as simple variation on existing restart-in-state or state elimination algorithms and evaluated to work well in practice.

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.

An FPGA can be used to solve any problem which is computable. This is trivially proven by the fact that FPGAs can be used to implement a soft microprocessor, such as the Xilinx MicroBlaze or Altera Nios II. Their advantage lies in that they are significantly faster for some applications because of their parallel nature and optimality in terms of the number of gates used for certain processes. FPGAs originally began as competitors to CPLDs to implement glue logic for printed circuit boards. As their size, capabilities, and speed increased, FPGAs took over additional functions to the point where some are now marketed as full systems on chips (SoCs).

Since an FML program realizes only a static view of a fuzzy system, the so-called eXtensible Stylesheet Language Translator (XSLT) is provided to change this static view to a computable version. In particular, the XSLT technology is used convert a fuzzy controller description into a general-purpose computer language to be computed on several hardware platforms. Currently, a XSLT converting FML program in runnable Java code has been implemented. In this way, thanks to the transparency capabilities provided by Java virtual machines, it is possible to obtain a fuzzy controller modeled in high level way by means of FML and runnable on a plethora of hardware architectures through Java technologies.

More abstract semantics are then derived; for instance, one may consider only the set of reachable states in the executions (which amounts to considering the last states in finite traces). The goal of static analysis is to derive a computable semantic interpretation at some point. For instance, one may choose to represent the state of a program manipulating integer variables by forgetting the actual values of the variables and only keeping their signs (+, − or 0). For some elementary operations, such as multiplication, such an abstraction does not lose any precision: to get the sign of a product, it is sufficient to know the sign of the operands.

He attended the Bronx High School of Science and City College of New York, where he (still in his teens) developed the theory that led to his independent discovery of algorithmic complexity. Chaitin has defined Chaitin's constant Ω, a real number whose digits are equidistributed and which is sometimes informally described as an expression of the probability that a random program will halt. Ω has the mathematical property that it is definable, with asymptotic approximations from below (but not from above), but not computable. Chaitin is also the originator of using graph coloring to do register allocation in compiling, a process known as Chaitin's algorithm.

Free, authorized version viewable at: Despite the prominence of Stiglitz' 2001 Nobel prize lecture, the use of arguably misleading neoclassical models persisted in 2007, according to these authors: The working paper, "Debunking the Myths of Computable General Equilibrium Models", SCEPA Working Paper 01-2008. provides both a history, and a readable theoretical analysis of what CGE models are, and are not. In particular, despite their name, CGE models use neither the Walrass general equilibrium, nor the Arrow-Debreus General Equilibrium frameworks. Thus, CGE models are highly distorted simplifications of theoretical frameworks—collectively called "the neoclassical economic paradigm"—which—themselves—were largely discredited by Joseph Stiglitz.

It is straightforward to express the monochromatic triangle problem in the monadic second-order logic of graphs (MSO2), by a logical formula that asserts the existence of a partition of the edges into two subsets such that there do not exist three mutually adjacent vertices whose edges all belong to the same side of the partition. It follows from Courcelle's theorem that the monochromatic triangle problem is fixed-parameter tractable on graphs of bounded treewidth. More precisely, there is an algorithm for solving the problem whose running time is the number of vertices of the input graph multiplied by a quickly-growing but computable function of the treewidth..

An analogous statement has been used to show that humans are subject to the same limits as machines., , under "The Argument from Mathematics" where he writes "although it is established that there are limitations to the powers of any particular machine, it has only been stated, without sort of proof, that no such limitations apply to the human intellect." Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as formal proof systems, and are therefore running a non-computable algorithm.

According to the Church–Turing thesis, no function computable by a finite algorithm can implement a true random oracle (which by definition requires an infinite description because it has infinitely many possible inputs, and its outputs are all independent from each other and need to be individually specified by any description). In fact, certain artificial signature and encryption schemes are known which are proven secure in the random oracle model, but which are trivially insecure when any real function is substituted for the random oracle.Ran Canetti, Oded Goldreich and Shai Halevi, The Random Oracle Methodology Revisited, STOC 1998, pp. 209–218 (PS and PDF).

Church's thesis. One of the main objectives of this and the next chapter is to present the evidence for Church's thesis (Thesis I §60)." Kleene 1952 in (Davis 1965:317) About Turing's "formulation", Kleene says: :"Turing's formulation hence constitutes an independent statement of Church's thesis (in equivalent terms). Post 1936 gave a similar formulation."Post 1936:321 Kleene proposes that what Turing showed: "Turing's computable functions (1936-1937) are those which can be computed by a machine of a kind which is designed, according to his analysis, to reproduce all the sorts of operations which a human computer could perform, working according to preassigned instructions.

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.

A polynomial-time counting reduction is usually used to transform instances of a known-hard problem X into instances of another problem Y that is to be proven hard. It consists of two functions f and g, both of which must be computable in polynomial time. The function f transforms inputs for X into inputs for Y, and the function g transforms outputs for Y into outputs for X. These two functions must preserve the correctness of the output. That is, suppose that one transforms an input x for problem X to an input y=f(x) for problem Y, and then one solves y to produce an output z.

Let p and q be two unary predicates. Then ⊓x(p(x)⊔¬p(x))⟜⊓x(q(x)⊔¬q(x)) expresses the problem of Turing-reducing q to p (in the sense that q is Turing reducible to p if and only if the interactive problem ⊓x(p(x)⊔¬p(x))⟜⊓x(q(x)⊔¬q(x)) is computable). ⊓x(p(x)⊔¬p(x))→⊓x(q(x)⊔¬q(x)) does the same but for the stronger version of Turing reduction where the oracle for p can be queried only once. ⊓x⊔y(q(x)↔p(y)) does the same for the problem of many-one reducing q to p.

His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics.

This classification can be achieved by noticing that, to be testable, for a functionality of the system under test "S", which takes input "I", a computable functional predicate "V" must exists such that V(S,I) is true when S, given input I, produce a valid output, false otherwise. This function "V" is known as the verification function for the system with input I. Many software systems are untestable, or not immediately testable. For example, Google's ReCAPTCHA, without having any metadata about the images is not a testable system. Recaptcha, however, can be immediately tested if for each image shown, there is a tag stored elsewhere.

In computability theory the '' theorem', (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the computable functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name ' comes from the occurrence of an S with subscript n and superscript m in the original formulation of the theorem (see below). In practical terms, the theorem says that for a given programming language and positive integers m and n, there exists a particular algorithm that accepts as input the source code of a program with m + n free variables, together with m values.

Roughly speaking, Jeff Paris and Leo Harrington (1977) showed that the strengthened finite Ramsey theorem is unprovable in Peano arithmetic by showing that in Peano arithmetic it implies the consistency of Peano arithmetic itself. Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem. The smallest number N that satisfies the strengthened finite Ramsey theorem is a computable function of n, m, k, but grows extremely fast. In particular it is not primitive recursive, but it is also far larger than standard examples of non-primitive recursive functions such as the Ackermann function.

Arbitrary-precision arithmetic in most computer software is implemented by calling an external library that provides data types and subroutines to store numbers with the requested precision and to perform computations. Different libraries have different ways of representing arbitrary-precision numbers, some libraries work only with integer numbers, others store floating point numbers in a variety of bases (decimal or binary powers). Rather than representing a number as single value, some store numbers as a numerator/denominator pair (rationals) and some can fully represent computable numbers, though only up to some storage limit. Fundamentally, Turing machines cannot represent all real numbers, as the cardinality of exceeds the cardinality of .

Even though the graph isomorphism problem is polynomial time reducible to crystal net topological equivalence (making topological equivalence a candidate for being "computationally intractable" in the sense of not being polynomial time computable), a crystal net is generally regarded as novel if and only if no topologically equivalent net is known. This has focused attention on topological invariants. One invariant is the array of minimal cycles (often called rings in the chemistry literature) arrayed about generic vertices and represented in a Schlafli symbol. The cycles of a crystal net are related to another invariant, that of the coordination sequence (or shell map in topology), which is defined as follows.

For some of these computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable). Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom"—an idea that is described more formally by flow of control.

A blue plaque at the college was unveiled on the centenary of his birth on 23 June 2012 and is now installed at the college's Keynes Building on King's Parade. In 1936, Turing published his paper "On Computable Numbers, with an Application to the Entscheidungsproblem". It was published in the Proceedings of the London Mathematical Society journal in two parts, the first on 30 November and the second on 23 December. In this paper, Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines.

It is important to disambiguate algorithmic randomness with stochastic randomness. Unlike algorithmic randomness, which is defined for computable (and thus deterministic) processes, stochastic randomness is usually said to be a property of a sequence that is a priori known to be generated (or is the outcome of) by an independent identically distributed equiprobable stochastic process. Because infinite sequences of binary digits can be identified with real numbers in the unit interval, random binary sequences are often called (algorithmically) random real numbers. Additionally, infinite binary sequences correspond to characteristic functions of sets of natural numbers; therefore those sequences might be seen as sets of natural numbers.

The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.

It is an interdisciplinary field encompassing techniques from computer science, data mining, machine learning, social network analysis, network science, sociology, ethnography, statistics, optimization, and mathematics. Social media mining faces grand challenges such as the big data paradox, obtaining sufficient samples, the noise removal fallacy, and evaluation dilemma. Social media mining represents the virtual world of social media in a computable way, measures it, and designs models that can help us understand its interactions. In addition, social media mining provides necessary tools to mine this world for interesting patterns, analyze information diffusion, study influence and homophily, provide effective recommendations, and analyze novel social behavior in social media.

Church proved that there is no computable function which decides for two given λ-calculus expressions whether they are equivalent or not. He relied heavily on earlier work by Stephen Kleene. Turing reduced the question of the existence of a 'general method' which decides whether any given Turing Machine halts or not (the halting problem) to the question of the existence of an 'algorithm' or 'general method' able to solve the '. If 'Algorithm' is understood as being equivalent to a Turing Machine, and with the answer to the latter question negative (in general), the question about the existence of an Algorithm for the ' also must be negative (in general).

Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis.

A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (see ). This notion cannot be expressed as a formula in the language of set theory. All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, π, e, et cetera, along with all algebraic numbers.

The difficulty of a computation can be useful: modern protocols for encrypting messages (for example, RSA) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number- theoretical problems. Some things may not be computable at all; in fact, this can be proven in some instances.

In computability theory, the T predicate, first studied by mathematician Stephen Cole Kleene, is a particular set of triples of natural numbers that is used to represent computable functions within formal theories of arithmetic. Informally, the T predicate tells whether a particular computer program will halt when run with a particular input, and the corresponding U function is used to obtain the results of the computation if the program does halt. As with the smn theorem, the original notation used by Kleene has become standard terminology for the concept.The predicate described here was presented in (Kleene 1943) and (Kleene 1952), and this is what is usually called "Kleene's T predicate".

Harberger's seminal paper on the corporate income tax pioneered the use of general-equilibrium modeling by recasting the classic Heckscher-Ohlin model of international trade as a model of one country with two sectors, one comprising incorporated firms subject to a tax on their net incomes and the other made up of unincorporated firms. Harberger's approach laid the groundwork for the later use of computable general equilibrium analysis of the impact of taxes on an entire economy. In Harberger's model, the output of both sectors is produced under conditions of constant returns to scale, using homogeneous labor and capital. Labor is perfectly mobile, so wages are equalized between the two sectors.

There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a computable function, with μ-recursive functions, Turing machines and the lambda calculus possibly being the best-known examples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable into each other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all are most readily understood as models of sequential computation. The subsequent consolidation of computer science required a more subtle formulation of the notion of computation, in particular explicit representations of concurrency and communication.

Let f: N→N be the function defined by: :f(#(θ)) = #(θ(°#(θ))) for each T-formula θ in one free variable, and f(n) = 0 otherwise. The function f is computable, so there is a formula Γf representing f in T. Thus for each formula θ, T proves :(∀y) [Γf(°#(θ),y) ↔ y = °f(#(θ))], which is to say :(∀y) [Γf(°#(θ),y) ↔ y = °#(θ(°#(θ)))]. Now define the formula β(z) as: :β(z) = (∀y) [Γf(z,y) → F(y)]. Then T proves :β(°#(θ)) ↔ (∀y) [ y = °#(θ(°#(θ))) → F(y)], which is to say :β(°#(θ)) ↔ F(°#(θ(°#(θ)))).

The class is formally defined by specifying one of its complete problems, known as End-Of-The-Line: :G is a (possibly exponentially large) directed graph with no isolated vertices, and with every vertex having at most one predecessor and one successor. G is specified by giving a polynomial-time computable function f(v) (polynomial in the size of v) that returns the predecessor and successor (if they exist) of the vertex v. Given a vertex s in G with no predecessor, find a vertex t≠s with no predecessor or no successor. (The input to the problem is the source vertex s and the function f(v)).

A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. A simple extension is to assign cardinal numbers to physical objects according to the choice of some base of reference and of measurement units for counting or measuring these objects within a given precision. In such case, numbering is a kind of classification, i.e. assigning a numeric property to each object of the set to subdivide this set into related subsets forming a partition of the initial set, possibly infinite and not enumeratable using a single natural number for each class of the partition.

The completeness of first-order logic is a corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic clearly articulated it. In any event, Kurt Gödel first proved this completeness in 1930. Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions, as a means of avoiding the so-called paradoxes of the infinite.

The problem of computing the number of 3-colorings of a given graph is a canonical example of a #P-complete problem, so the problem of computing the coefficients of the chromatic polynomial is #P-hard. Similarly, evaluating P(G, 3) for given G is #P-complete. On the other hand, for k=0,1,2 it is easy to compute P(G, k), so the corresponding problems are polynomial-time computable. For integers k>3 the problem is #P-hard, which is established similar to the case k=3. In fact, it is known that P(G, x) is #P-hard for all x (including negative integers and even all complex numbers) except for the three “easy points”.

Zeno machines would allow some functions to be computed that are not Turing-computable. For example, the halting problem for Turing machines can be solved by a Zeno machine (using the following pseudocode algorithm): begin program write 0 on the first position of the output tape; begin loop simulate 1 successive step of the given Turing machine on the given input; if the Turing machine has halted then write 1 on the first position of the output tape and break out of loop; end loop end program Computing of this kind that goes beyond the Turing Limit is called hypercomputation, in this case hypercomputation through a supertask – see there for further discussion and literature.

Finding a spanning caterpillar in a graph is NP- complete. A related optimization problem is the Minimum Spanning Caterpillar Problem (MSCP), where a graph has dual costs over its edges and the goal is to find a caterpillar tree that spans the input graph and has the smallest overall cost. Here the cost of the caterpillar is defined as the sum of the costs of its edges, where each edge takes one of the two costs based on its role as a leaf edge or an internal one. There is no f(n)-approximation algorithm for the MSCP unless P = NP. Here f(n) is any polynomial-time computable function of n, the number of vertices of a graph.

As Peter Landin noted, the language Algol was the first language to combine seamlessly imperative effects with the (call-by-name) lambda calculus. Perhaps the most elegant formulation of the language is due to John C. Reynolds, and it best exhibits its syntactic and semantic purity. Reynolds's idealized Algol also made a convincing methodological argument regarding the suitability of local effects in the context of call-by-name languages, to be contrasted with the global effects used by call-by-value languages such as ML. The conceptual integrity of the language made it one of the main objects of semantic research, along with Programming Computable Functions (PCF) and ML.Peter O'Hearn and Robert D. Tennent. 1996. Algol-Like Languages.

In 2001, he was awarded a Monash graduate school scholarship to study in Australia for three years. Professor Narayan completed his PhD in 18 months. His doctorate thesis, An Econometric Model of Tourism Demand and a Computable General Equilibrium Analysis of the Impact of Tourism: The Case of Fiji Islands was assessed to be the most outstanding work, earning him the Mollie Hollman Medal in 2004 at Monash University. In addition, he received a Monash University postgraduate travel grant in 2002; the Australian Population Association Borrie Prize for an essay titled, Determinants of Female Fertility in Taiwan, 1966–2001: Evidence from Cointegration and Variance Decomposition Analysis, a postgraduate Publications Award in 2003 and numerous research grants.

As the computation of a resultant may be reduced to computing determinants and polynomial greatest common divisors, there are algorithms for computing resultants in a finite number of steps. However, the generic resultant is a polynomial of very high degree (exponential in ) depending on a huge number of indeterminates. It follows that, except for very small and very small degrees of input polynomials, the generic resultant is, in practice, impossible to compute, even with modern computers. Moreover, the number of monomials of the generic resultant is so high, that, if it would be computable, the result could not be stored on available memory devices, even for rather small values of and of the degrees of the input polynomials.

In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies on the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive.

To identify the real numbers with the computable numbers would then be a contradiction. And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number that doesn't fit, and thereby proves a contradiction. We can indeed enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non- terminating (T is a partial function), so this fails to produce the required bijection.

In March 2014, at the annual South by Southwest (SXSW) event, Wolfram officially announced the Wolfram Language as a new general multi-paradigm programming languageWolfram Language reference page Retrieved on 14 May 2014 and currently better known as a multi-paradigm computational communication language. The documentation for the language was pre-released in October 2013 to coincide with the bundling of Mathematica and the Wolfram Language on every Raspberry Pi computer. While the Wolfram Language has existed for over 25 years as the primary programming language used in Mathematica, it was not officially named until 2014.Slate's article Stephen Wolfram's New Programming Language: He Can Make The World Computable, 6 March 2014.

The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well. It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd–Coxeter algorithmJ.A. Todd and H.S.M. Coxeter.

Determining if a graph can be colored with 2 colors is equivalent to determining whether or not the graph is bipartite, and thus computable in linear time using breadth- first search or depth-first search. More generally, the chromatic number and a corresponding coloring of perfect graphs can be computed in polynomial time using semidefinite programming. Closed formulas for chromatic polynomial are known for many classes of graphs, such as forests, chordal graphs, cycles, wheels, and ladders, so these can be evaluated in polynomial time. If the graph is planar and has low branch-width (or is nonplanar but with a known branch decomposition), then it can be solved in polynomial time using dynamic programming.

An algorithm is said to run in sub-linear time (often spelled sublinear time) if T(n) = o(n). In particular this includes algorithms with the time complexities defined above. Typical algorithms that are exact and yet run in sub-linear time use parallel processing (as the NC1 matrix determinant calculation does), or alternatively have guaranteed assumptions on the input structure (as the logarithmic time binary search and many tree maintenance algorithms do). However, formal languages such as the set of all strings that have a 1-bit in the position indicated by the first log(n) bits of the string may depend on every bit of the input and yet be computable in sub-linear time.

Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction. R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point, q, on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from q to a fixed point.

The study of computability came to be known as recursion theory or computability theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.A detailed study of this terminology is given by Soare (1996). When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept - the computable function - had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper.

A Malament–Hogarth (M-H) spacetime, named after David B. Malament and Mark Hogarth, is a relativistic spacetime that possesses the following property: there exists a worldline \lambda and an event p such that all events along \lambda are a finite interval in the past of p, but the proper time along \lambda is infinite. The event p is known as an M-H event. The significance of M-H spacetimes is that they allow for the implementation of certain non-Turing computable tasks (hypercomputation). The idea is for an observer at some event in p's past to set a computer (Turing machine) to work on some task and then have the Turing machine travel on \lambda, computing for all eternity.

Wiley-Interscience New York, N.Y. The process of model design begins with a specification of the problem to be solved, and the objectives for the model. Ecological systems are composed of an enormous number of biotic and abiotic factors that interact with each other in ways that are often unpredictable, or so complex as to be impossible to incorporate into a computable model. Because of this complexity, ecosystem models typically simplify the systems they are studying to a limited number of components that are well understood, and deemed relevant to the problem that the model is intended to solve. The process of simplification typically reduces an ecosystem to a small number of state variables and mathematical functions that describe the nature of the relationships between them.

Gold also showed that if the learner is given only positive examples (that is, only grammatical sentences appear in the input, not ungrammatical sentences), then the language can only be guaranteed to be learned in the limit if there are only a finite number of possible sentences in the language (this is possible if, for example, sentences are known to be of limited length). Language identification in the limit is a highly abstract model. It does not allow for limits of runtime or computer memory which can occur in practice, and the enumeration method may fail if there are errors in the input. However the framework is very powerful, because if these strict conditions are maintained, it allows the learning of any program known to be computable.

Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore anything else.

These are the types of reduction used to prove ♯P-completeness. In parameterized complexity, FPT parsimonious reductions are used; these are parsimonious reductions whose transformation is a fixed-parameter tractable algorithm and that map bounded parameter values to bounded parameter values by a computable function. Polynomial-time parsimonious reductions are a special case of a more general class of reductions for counting problems, the polynomial-time counting reductions.. See in particular pp. 634–635. One common technique used in proving that a reduction R is parsimonious is to show that there is a bijection between the set of solutions to x and the set of solutions to R(x) which guarantees that the number of solutions to both problems is the same.

The dimensions of spaces of cusp forms are, in principle, computable via the Riemann–Roch theorem. For example, the Ramanujan tau function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a1 = 1\. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell's proof of Ramanujan's identities). Explicitly it is the modular discriminant :\Delta(z,q), which represents (up to a normalizing constant) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function.

Equivalently, the vertices correspond to variables, and two variables form an edge if they share an inequality. The sparsity measure d of A is the minimum between the tree-depth of the graph of A and the tree-depth of the graph of the transpose of A. Let a be the numeric measure of A defined as the maximum absolute value of any entry of A. Let n be the number of variables of the integer program. Then it was shown in 2018 that integer programming can be solved in strongly polynomial and fixed-parameter tractable time parameterized by a and d. That is, for some computable function f and some constant k, integer programming can be solved in time f(a,d)n^k.

The role of reduction in computer science can be thought as a (precise and unambiguous) mathematical formalization of the philosophical idea of "theory reductionism". In a general sense, a problem (or set) is said to be reducible to another problem (or set), if there is a computable/feasible method to translate the questions of the former into the latter, so that, if one knows how to computably/feasibly solve the latter problem, then one can computably/feasibly solve the former. Thus, the latter can only be at least as "hard" to solve as the former. Reduction in theoretical computer science is pervasive in both: the mathematical abstract foundations of computation; and in real-world performance or capability analysis of algorithms.

On the other hand, Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is , which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.

The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

The book is self-contained, and targeted at researchers in mathematical analysis and computability; reviewers Douglas Bridges and Robin Gandy disagree over which of these two groups it is better aimed at. Although co-author Marian Pour-El came from a background in mathematical logic, and the two series in which the book was published both have logic in their title, readers are not expected to be familiar with logic. Despite complaining about the formality of the presentation and that the authors did not aim to include all recent developments in computable analysis, reviewer Rod Downey writes that this book "is clearly a must for anybody whose research is in this area", and Gandy calls it "an interesting, readable and very well written book".

With Theodore Slaman, Groszek showed that (if they exist at all) non-constructible real numbers must be widespread, in the sense that every perfect set contains one of them, and they asked analogous questions of the non-computable real numbers. With Slaman, she has also shown that the existence of a maximally independent set of Turing degrees, of cardinality less than the cardinality of the continuum, is independent of ZFC. In the theory of ordinal definable sets, an unordered pair of sets is said to be a Groszek–Laver pair if the pair is ordinal definable but neither of its two elements is; this concept is named for Groszek and Richard Laver, who observed the existence of such pairs in certain models of set theory.

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".

In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo- randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal TOEs describable by very few bits of information. Related critique was offered by Solomon Feferman, among others. Douglas S. Robertson offers Conway's game of life as an example: The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.

The notion of solid modeling as practised today relies on the specific need for informational completeness in mechanical geometric modeling systems, in the sense that any computer model should support all geometric queries that may be asked of its corresponding physical object. The requirement implicitly recognizes the possibility of several computer representations of the same physical object as long as any two such representations are consistent. It is impossible to computationally verify informational completeness of a representation unless the notion of a physical object is defined in terms of computable mathematical properties and independent of any particular representation. Such reasoning led to the development of the modeling paradigm that has shaped the field of solid modeling as we know it today.

Commander's added value as a quality system over the years was thus improved and adjusted through proprietary use, as well as from projects for customers or industry standards. Commander was later used against the company itself in court proceedings in 2002 when the company laid off workers in the Netherlands using legislation regarding temporary personnel.Dutch article in Computable magazine how CMG's own method CMG:Commander was used to prove company responsibility for employee work There was a lawsuit brought against CMG using Commander as evidence to show that employees were the responsibility of CMG, and therefore could not be seen as temporary personnel. Despite protests that Commander was proprietary information, the entire Commander cd was released to the court proceedings and became public record.

In his chapter XIII Computable Functions, Kleene adopts the Post model; Kleene's model uses a blank and one symbol "tally mark ¤" (Kleene p. 358), a "treatment closer in some respects to Post 1936. Post 1936 considered computation with a 2-way infinite tape and only 1 symbol" (Kleene p. 361). Kleene observes that Post's treatment provided a further reduction to "atomic acts" (Kleene p. 357) of "the Turing act" (Kleene p. 379). As described by Kleene "The Turing act" is the combined 3 (time-sequential) actions specified on a line in a Turing table: (i) print-symbol/erase/do- nothing followed by (ii) move-tape-left/move-tape-right/do-nothing followed by (iii) test-tape-go-to-next-instruction: e.g.

Parameterized complexity is the complexity-theoretic study of problems that are naturally equipped with a small integer parameter and for which the problem becomes more difficult as increases, such as finding -cliques in graphs. A problem is said to be fixed- parameter tractable if there is an algorithm for solving it on inputs of size , and a function , such that the algorithm runs in time . That is, it is fixed-parameter tractable if it can be solved in polynomial time for any fixed value of and moreover if the exponent of the polynomial does not depend on .. Technically, there is usually an additional requirement that be a computable function. For finding -vertex cliques, the brute force search algorithm has running time .

In data storage and retrieval applications, use of a hash function is a trade off between search time and data storage space. If search time were unbounded, a very compact unordered linear list would be the best medium; if storage space were unbounded, a randomly accessible structure indexable by the key value would be very large, very sparse, but very fast. A hash function takes a finite amount of time to map a potentially large key space to a feasible amount of storage space searchable in a bounded amount of time regardless of the number of keys. In most applications, it is highly desirable that the hash function be computable with minimum latency and secondarily in a minimum number of instructions.

CoPS and Dixon then moved to Victoria University. Dixon is mainly known for developing, with his collaborators, computable general equilibrium (CGE) models. A further development of Leif Johansen's multi-sectoral model, Dixon's ORANI model (1977, 1982), named after his wife, built on work by Paul Armington and Wassily Leontief"Johansen's contribution to CGE modelling: originator and guiding light for 50 years" by Peter B. Dixon and Maureen T. Rimmer, Centre of Policy Studies, Monash University, May 2010, , its dynamic further development, the MONASH/VU-National model, and the US Applied General Equilibrium (USAGE) model which is widely used by the US government and the Global Trade Analysis Project."2015 Reflections" by Alan Powell, Academy of the Social Sciences in Australia These models are available in CoPS's Gempack/RunGEM software implementation.

A finite set of relations S over the Boolean domain defines a polynomial time computable satisfiability problem if any one of the following conditions holds: # all relations which are not constantly false are true when all its arguments are true; # all relations which are not constantly false are true when all its arguments are false; # all relations are equivalent to a conjunction of binary clauses; # all relations are equivalent to a conjunction of Horn clauses; # all relations are equivalent to a conjunction of dual-Horn clauses; # all relations are equivalent to a conjunction of affine formulae. Schaefer (1978, p.218 left) defines an affine formula to be of the form x1 ⊕ ... ⊕ xn = c, where each xi is a variable, c is a constant, i.e. true or false, and "⊕" denotes XOR, i.e.

Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem." It was the second proof (after Church's theorem) of the conjecture that some purely mathematical yes-no questions can never be answered by computation; more technically, that some decision problems are "undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "...what I shall prove is quite different from the well- known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]..." (Undecidable, p. 145). Turing followed this proof with two others.

NS diagrams (blue) and flow charts (green). The structured program theorem, also called the Böhm–Jacopini theorem, is a result in programming language theory. It states that a class of control flow graphs (historically called flowcharts in this context) can compute any computable function if it combines subprograms in only three specific ways (control structures). These are #Executing one subprogram, and then another subprogram (sequence) #Executing one of two subprograms according to the value of a boolean expression (selection) #Repeatedly executing a subprogram as long as a boolean expression is true (iteration) The structured chart subject to these constraints may however use additional variables in the form of bits (stored in an extra integer variable in the original proof) in order to keep track of information that the original program represents by the program location.

A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π but still excludes most of the real numbers, because most cannot be described by a finite program. Traditional Turing machines with a write-only output tape cannot edit their previous outputs; generalized Turing machines, according to Jürgen Schmidhuber, can edit their output tape as well as their work tape. He defines the constructively describable symbol sequences as those that have a finite, non- halting program running on a generalized Turing machine, such that any output symbol eventually converges, that is, it does not change any more after some finite initial time interval.

This branch of recursion theory analyzed the following question: For fixed m and n with 0 < m < n, for which functions A is it possible to compute for any different n inputs x1, x2, ..., xn a tuple of n numbers y1,y2,...,yn such that at least m of the equations A(xk) = yk are true. Such sets are known as (m, n)-recursive sets. The first major result in this branch of Recursion Theory is Trakhtenbrot's result that a set is computable if it is (m, n)-recursive for some m, n with 2m > n. On the other hand, Jockusch's semirecursive sets (which were already known informally before Jockusch introduced them 1968) are examples of a set which is (m, n)-recursive if and only if 2m < n + 1\.

The first known PSPACE-complete problem was the word problem for deterministic context-sensitive grammars. In the word problem for context-sensitive grammars, one is given a set of grammatical transformations which can increase, but cannot decrease, the length of a sentence, and wishes to determine if a given sentence could be produced by these transformations. The technical condition of "determinism" (implying roughly that each transformation makes it obvious that it was used) ensures that this process can be solved in polynomial space, and showed that every (possibly non-deterministic) program computable in linear space could be converted into the parsing of a context-sensitive grammar, in a way which preserves determinism. In 1970, Savitch's theorem showed that PSPACE is closed under nondeterminism, implying that even non-deterministic context-sensitive grammars are in PSPACE.

In his original proof Turing formalized the concept of algorithm by introducing Turing machines. However, the result is in no way specific to them; it applies equally to any other model of computation that is equivalent in its computational power to Turing machines, such as Markov algorithms, Lambda calculus, Post systems, register machines, or tag systems. What is important is that the formalization allows a straightforward mapping of algorithms to some data type that the algorithm can operate upon. For example, if the formalism lets algorithms define functions over strings (such as Turing machines) then there should be a mapping of these algorithms to strings, and if the formalism lets algorithms define functions over natural numbers (such as computable functions) then there should be a mapping of algorithms to natural numbers.

He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable: in general, it is not possible to decide algorithmically whether a given Turing machine will ever halt. He also introduced the notion of a "universal machine" (now known as a universal Turing machine), with the idea that such a machine could perform the tasks of any other machine, or in other words, it is provably capable of computing anything that is computable by executing a program stored on tape, allowing the machine to be programmable. Von Neumann acknowledged that the central concept of the modern computer was due to this paper.

The mathematician Alan Turing, who had been alerted to a problem of mathematical logic by the lectures of Max Newman at the University of Cambridge, wrote a paper in 1936 entitled On Computable Numbers, with an Application to the Entscheidungsproblem, which was published in the Proceedings of the London Mathematical Society. (and ) In it he described a hypothetical machine he called a universal computing machine, now known as the "Universal Turing machine". The hypothetical machine had an infinite store (memory in today's terminology) that contained both instructions and data. John von Neumann became acquainted with Turing while he was a visiting professor at Cambridge in 1935, and also during Turing's PhD year at the Institute for Advanced Study in Princeton, New Jersey during 1936 – 1937. Whether he knew of Turing's paper of 1936 at that time is not clear.

Uchionnye Zapiski Penzenskogo Pedinstituta (Transactions of the Penza Pedagogoical Institute) 4, 75–87 (1956) (in Russian) As he remembers: In 1967, Manuel Blum formulated a set of axioms (now known as Blum axioms) specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called speed-up theorem. The field began to flourish in 1971 when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete. In the 1980s, much work was done on the average difficulty of solving NP-complete problems—both exactly and approximately.

The triples of numbers (e,i,x) that belong to the relation (the ones for which T1(e,i,x) is true) are defined to be exactly the triples in which x encodes a computation history of the computable function with index e when run with input i, and the program halts as the last step of this computation history. That is, T1 first asks whether x is the Gödel number of a finite sequence ⟨xj⟩ of complete configurations of the Turing machine with index e, running a computation on input i. If so, T1 then asks if this sequence begins with the starting state of the computation and each successive element of the sequence corresponds to a single step of the Turing machine. If it does, T1 finally asks whether the sequence ⟨xj⟩ ends with the machine in a halting state.

Gu et al. presented a class of physical systems that exhibits non-computable macroscopic properties. More precisely, if one could compute certain macroscopic properties of these systems from the microscopic description of these systems, then one would be able to solve computational problems known to be undecidable in computer science. Gu et al. concluded that > Although macroscopic concepts are essential for understanding our world, > much of fundamental physics has been devoted to the search for a 'theory of > everything', a set of equations that perfectly describe the behavior of all > fundamental particles. The view that this is the goal of science rests in > part on the rationale that such a theory would allow us to derive the > behavior of all macroscopic concepts, at least in principle. The evidence we > have presented suggests that this view may be overly optimistic.

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 information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel. This result was presented by Claude Shannon in 1948 and was based in part on earlier work and ideas of Harry Nyquist and Ralph Hartley. The Shannon limit or Shannon capacity of a communication channel refers to the maximum rate of error-free data that can theoretically be transferred over the channel if the link is subject to random data transmission errors, for a particular noise level. It was first described by Shannon (1948), and shortly after published in a book by Claude Elwood Shannon and Warren Weaver in 1949 entitled The Mathematical Theory of Communication. ().

REAL's capabilities revolve around comprehensive state and metropolitan models that integrate econometric and input-output analysis to provide for both impact and forecasting analyses. Current activities focus on interstate trade, forecasts for the Chicago, Illinois and other Midwest economies, housing market analysis and forecasts, demographic-economic modeling (especially the role of aging and immigration) and the development of alternative computable general equilibrium models created on the same data base. Research at REAL always attempts to provide a range of exposure to new curricula materials, methods of conducting interdisciplinary and international collaborative research and guidance in the preparation of material for dissemination in the public policy arena. The latter component is extremely important and undervalued, yet academic research over the next decade is likely to be more project-driven with funding from stakeholders whose interest lie in the policy arena and less in the academic or scholarly content.

To see that the theorem is true, it suffices to notice that if there were no such number n_0, one could algorithmically test membership of a number n in this non-computable set by simultaneously running the algorithm A to see whether n is output while also checking all possible k-tuples of natural numbers seeking a solution of the equation :p(n,x_1,\ldots,x_k)=0. We may associate an algorithm A with any of the usual formal systems such as Peano arithmetic or ZFC by letting it systematically generate consequences of the axioms and then output a number n whenever a sentence of the form : eg \exists x_1,\ldots , x_k [p(n,x_1,\ldots,x_k)=0] is generated. Then the theorem tells us that either a false statement of this form is proved or a true one remains unproved in the system in question.

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.

Primatologists have noted that, due to their highly social nature, primates must maintain personal contact with the other members of their social group, usually through social grooming. Such social groups function as protective cliques within the physical groups in which the primates live. The number of social group members a primate can track appears to be limited by the volume of the neocortex. This suggests that there is a species-specific index of the social group size, computable from the species' mean neocortical volume. In 1992, Dunbar used the correlation observed for non-human primates to predict a social group size for humans. Using a regression equation on data for 38 primate genera, Dunbar predicted a human "mean group size" of 148 (casually rounded to 150), a result he considered exploratory due to the large error measure (a 95% confidence interval of 100 to 230).

In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1: :"A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape]." (Minsky 1967:159) The key notions in the definition are (1) that some n is specified at the start, (2) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates. An alternate form of (2) – the machine successively prints all n of the digits on its tape, halting after printing the nth – emphasizes Minsky's observation: (3) That by use of a Turing machine, a finite definition – in the form of the machine's table – is being used to define what is a potentially-infinite string of decimal digits.

The general graph Steiner tree problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices. The metric closure of a graph G is the complete graph in which each edge is weighted by the shortest path distance between the nodes in G. This algorithm produces a tree whose weight is within a 2 − 2/t factor of the weight of the optimal Steiner tree where t is the number of leaves in the optimal Steiner tree; this can be proven by considering a traveling salesperson tour on the optimal Steiner tree. The approximate solution is computable in polynomial time by first solving the all-pairs shortest paths problem to compute the metric closure, then by solving the minimum spanning tree problem. A series of papers provided approximation algorithms for the minimum Steiner tree problem with approximation ratios that improved upon the 2 − 2/t ratio.

In a sound proof system, every provably total function is indeed total, but the converse is not true: in every first-order proof system that is strong enough and sound (including Peano arithmetic), one can prove (in another proof system) the existence of total functions that cannot be proven total in the proof system. If the total computable functions are enumerated via the Turing machines that produces them, then the above statement can be shown, if the proof system is sound, by a similar diagonalization argument to that used above, using the enumeration of provably total functions given earlier. One uses a Turing machine that enumerates the relevant proofs, and for every input n calls fn(n) (where fn is n-th function by this enumeration) by invoking the Turing machine that computes it according to the n-th proof. Such a Turing machine is guaranteed to halt if the proof system is sound.

As a politically induced strategy, the question virtual water trade can be implemented in a sustainable way, whether the implementation can be managed in a social, economical, and ecological fashion, and for which countries the concept offers a meaningful option. The data that underlie the concept of virtual water can readily be used to construct water satellite accounts, and brought into economic models of international trade such as the GTAP Computable General Equilibrium Model. Such a model can be used to study the economic implications of changes in the water supply or water policy, as well as the water resource implications of economic development and trade liberalization. In sum, virtual water trade allows a new, amplified perspective on water problems: In the framework of recent developments from a supply-oriented to demand-oriented management of water resources, it opens up new fields of governance and facilitates differentiation and balancing of different perspectives, basic conditions, and interests.

Post strongly disagreed with Church's "identification" of effective computability with the λ-calculus and recursion, stating: Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by inductive reasoning to a "natural law" rather than by "a definition or an axiom".Post 1936 in Davis 1952:291. This idea was "sharply" criticized by Church.Sieg 1997:171 and 176–177. Thus Post in his 1936 paper was also discounting Kurt Gödel's suggestion to Church in 1934–35 that the thesis might be expressed as an axiom or set of axioms.Sieg 1997:160. Turing adds another definition, Rosser equates all three: Within just a short time, Turing's 1936–37 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" appeared. In it he stated another notion of "effective computability" with the introduction of his a-machines (now known as the Turing machine abstract computational model). And in a proof-sketch added as an "Appendix" to his 1936–37 paper, Turing showed that the classes of functions defined by λ-calculus and Turing machines coincided.Turing 1936–37 in .

Positive Results: # Every model that satisfies Linear Stochastic Transitivity must also satisfy Strong Stochastic Transitivity, which in turn must satisfy Weak Stochastic Transitivity. This is represented as: LST \implies SST\impliesWST ; # Since the Bradeley-Terry models and the are LST models, they also satisfy SST and WST; # Due to the convenience of , a few authors have identified axiomatic of linear stochastic transitivity (and other models), most notably Gérard Debreu showed that : + \implies LST (see also Debreu Theorems); # Two LST models given by invertible comparison functions F(x) and G(x) are if and only if F(x) = G(\kappa x)for some \kappa \geq 0. Negative Results: # Stochastic transitivity models are empirically , however, they may be falsifiable; # between LST comparison functions F(x) and G(x) can be impossible even if an infinite amount of data is provided over a finite number of ; # The for WST, SST and LST models are in general NP-Hard, however, near optimal polynomially computable estimation procedures are known for SST and LST models.

For this reason, the consortium membership (those contributing to the database) includes prominent global governance and policy research institutions like the World Bank, European Commission, World Trade Organization, International Monetary Fund, the Massachusetts Institute of Technology Joint Program on the Science and Policy of Global Change, United Nations Conference on Trade and Development, and the Organisation for Economic Co-operation and Development. There are currently three "Consortium Members At Large" – Joseph Francois, Mark Horridge, and Brian O'Neill, who represent the broader scientific community. The primary database is essentially a multi-year form of a multi-region input output (MRIO) database supplemented by national macroeconomic data, though extensive satellite datasets cover other measures that are linked to the economic flows in the core database, including trade policies, greenhouse gas emissions, energy use, migration flows, and land use patterns. While from its inception the database was closely tied to the computable general equilibrium (CGE) research community, in recent years the database has also been at the center of greenhouse gas emissions accounting exercises, and related assessments of resource use.

The impacts of the various options were estimated as the differences between a carbon tax and a reference 'business as usual' scenario assuming New Zealand had not signed the Kyoto Protocol. The variables changed in the model runs were the NZ carbon price ($0, $10, $25 or $100), the world carbon price, the duration (short term to 2012, long term to 2025), the level of free allocation and whether the Government assumed all Kyoto liabilities. The results of each model run was reported as the percent difference from the 'no-Kyoto' 'business as usual' scenario in 2012 or 2025.Stroombergen et al 2009, p 22 The report noted several limitations of computable general equilibrium models that should be kept in mind in interpreting the results: CGE models are only an approximation of highly complex real economies, results can only ever be indicative,Stroombergen et al 2009, p 16 and are highly dependent on the structure of the models and the input assumptions and on the assumption that other variables remain constant.

On 13 October 2018, the Bank of England announced that the next £50 note will be printed on polymer, rather than cotton paper. Members of the public were invited to nominate a scientist to feature on it. It was announced on 15 July 2019 that scientist and mathematician Alan Turing would be featured on the note issued in late 2021. The note's reverse will have an image of Turing based on a photograph taken by the Elliott & Fry photographic studio in 1951, a table of formulae from Turing's 1936 work On Computable Numbers, with an application to the Entscheidungsproblem, an image of the Automatic Computing Engine Pilot machine, technical drawings of the British bombe machine, the quote "This is only a foretaste of what is to come, and only the shadow of what is going to be" from an interview Turing gave to The Times on 11 June 1949, a ticker tape showing Turing's date of birth in binary code, and a copy of Turing's signature from the visitors book at Bletchley Park in 1947.

A well-structured systemParosh Aziz Abdulla, Kārlis Čerāns, Bengt Jonsson, Yih-Kuen Tsay: Algorithmic Analysis of Programs with Well Quasi-ordered Domains (2000), Information and Computation, Vol. 160 issues 1-2, pp. 109--127 is a transition system (S,\to) with state set S = Q \times D made up from a finite control state set Q, a data values set D, furnished with a decidable pre-order \leq \subseteq D \times D which is extended to states by (q,d)\le(q',d') \Leftrightarrow q=q' \wedge d\le d', which is well-structured as defined above (\to is monotonic, i.e. upward compatible, with respect to \le) and in addition has a computable set of minima for the set of predecessors of any upward closed subset of S. Well- structured systems adapt the theory of well-structured transition systems for modelling certain classes of systems encountered in computer science and provide the basis for decision procedures to analyse such systems, hence the supplementary requirements: the definition of a WSTS itself says nothing about the computability of the relations \le, \to.

An algorithm is said to be fixed-parameter tractable if the number of elementary operations it performs has a bound of the form O(n^c)+f(k), where n is some measure of the input size (such as the number of vertices in a graph), k is a parameter describing the complexity of the input (such as the treewidth of the graph), c is a constant that does not depend on n or k, and f is a computable function. Given a time bound of this form, the klam value of the algorithm (or more properly of the time bound) is defined to be the largest value of k such that f(k) does not exceed "some reasonable absolute bound on the maximum number of steps of any computation". More precisely both and use the number 1020 as this bound, and this has been followed by later researchers. To prevent artificially improving the klam value of an algorithm by putting more of its complexity into the O(n^c) part of the time bound, also limit c to be at most three, valid for many known fixed-parameter tractable algorithms.

Cobham's thesis, also known as Cobham–Edmonds thesis (named after Alan Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they can be computed in polynomial time; that is, if they lie in the complexity class P. In modern terms, it identifies tractable problems with the complexity class P. Formally, to say that a problem can be solved in polynomial time is to say that there exists an algorithm that, given an n-bit instance of the problem as input, can produce a solution in time O(nc), the letter O is big-O notation and c is a constant that depends on the problem but not the particular instance of the problem. Alan Cobham's 1965 paper entitled "The intrinsic computational difficulty of functions" is one of the earliest mentions of the concept of the complexity class P, consisting of problems decidable in polynomial time. Cobham theorized that this complexity class was a good way to describe the set of feasibly computable problems. Jack Edmonds's 1965 paper "Paths, trees, and flowers" is also credited with identifying P with tractable problems.

In computability theory, a Turing reduction (also known as a Cook reduction) from a problem A to a problem B, is a reduction which solves A, assuming the solution to B is already known (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to solve A if it had available to it a subroutine for solving B. More formally, a Turing reduction is a function computable by an oracle machine with an oracle for B. Turing reductions can be applied to both decision problems and function problems. If a Turing reduction of A to B exists then every algorithm for B can be used to produce an algorithm for A, by inserting the algorithm for B at each place where the oracle machine computing A queries the oracle for B. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for B or the oracle machine computing A, and may require as much space as both together. The first formal definition of relative computability, then called relative reducibility, was given by Alan Turing in 1939 in terms of oracle machines.

The Series F version was introduced in 2011, while its predecessor was in circulation for twenty years, so there was some consideration to withdrawing the £50 note entirely as a way of combatting tax evasion, and the fact that cash transactions using such a high value note are becoming increasingly rare. However, in October 2018 the Bank of England announced plans to introduce a Series G polymer £50 note, following the review by the government. The new £50 note is planned for introduction once the Series G £20 note featuring JMW Turner is released. In July 2019, it was announced that the new note would feature computer pioneer Alan Turing, from a photograph taken by the Elliott & Fry photographic studio in 1951, a table of formulae from Turing's 1936 work On Computable Numbers, with an application to the Entscheidungsproblem, an image of the Automatic Computing Engine Pilot machine, technical drawings of the British bombe machine, the quote "This is only a foretaste of what is to come, and only the shadow of what is going to be" from an interview Turing gave to The Times on 11 June 1949, and a ticker tape showing Turing's date of birth in binary code.