Mathscinet searches for the titles like "computably enumerable" and "c.e." show that many papers have been published with this terminology as well as with the other one. These researchers also use terminology such as partial computable function and computably enumerable (c.e.) set instead of partial recursive function and recursively enumerable (r.
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We build an uncountably categorical but not countably categorical theory whose only computably presentable model is the saturated one.
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We shall take for granted the extension of these ideas to computably convergent complex sequences, and the natural definitions of computable continuity.
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The basic idea of Chomsky's work is to divide language into four levels, or the language hierarchy. The four levels are: regular, context-free, context-sensitive, and computably enumerable or unrestricted. Regular is the least complex while computably enumerable is the most complex. While his work grew out of combinatorics on words, it drastically affected other disciplines, especially computer science.
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Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: :Every computably enumerable set is Diophantine. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0. Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0.
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The question "Does any arbitrary "Diophantine equation" have an integer solution?" is undecidable.That is, it is impossible to answer the question for all cases. Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich- Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can decide whether or not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an example of a computably enumerable but not decidable set, and the set of unsolvable Diophantine equations is not computably enumerable" (p. 71).
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A language is called computable (synonyms: recursive, decidable) if there is a computable function such that for each word over the alphabet, if the word is in the language and if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language. A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function such that is defined if and only if the word is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers.
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Yuri Matiyasevich utilized a method involving Fibonacci numbers, which grow exponentially, in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam – hence, MRDP – had shown that this suffices to show that every computably enumerable set is Diophantine.
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In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable.
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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.
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Here, the difference between the two notions of program mentioned in the last paragraph becomes clear; one is easily recognized by some grammar, while the other requires arbitrary computation to recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent to the halting problem.
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He was among the first to claim that Gödel's incompleteness theorem is relevant for theories of everything (TOE) in theoretical physics.Cf. Jaki's "A Late Awakening to Gödel in Physics" Gödel's theorem states that any theory that includes certain basic facts of number theory and is computably enumerable will be either incomplete or inconsistent. Since any 'theory of everything' must be consistent, it also must be incomplete.
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Kleene's original realizability interpretation has received much attention among those who study connections between computability and logic. It was extended to full higher-order intuitionistic logic by Martin Hyland in 1982 who constructed the effective topos. In 2002, Steve Awodey, Lars Birkedal, and Dana Scott formulated a modal logic for computability which extended the usual realizability interpretation with two modal operators expressing the notion of being "computably true".
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He and Richard Friedberg independently introduced the priority method which gave an affirmative answer to Post's Problem regarding the existence of recursively enumerable Turing degrees between 0 and 0' . This result, now known as the Friedberg-Muchnik Theorem,Robert I. Soare, Recursively Enumberable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer-Verlag, 1999, ; p. 118Nikolai Vereshchagin, Alexander Shen, Computable functions. American Mathematical Society, 2003, ; p.
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In total functional programming languages, such as Charity and Epigram, all functions are total and must terminate. Charity uses a type system and control constructs based on category theory, whereas Epigram uses dependent types. The LOOP language is designed so that it computes only the functions that are primitive recursive. All of these compute proper subsets of the total computable functions, since the full set of total computable functions is not computably enumerable.
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Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system.
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For instance, in 1970, it was proven, as a solution to Hilbert's 10th problem, that there is no Turing machine which can solve all Diophantine equations. Reprinted in The Collected Works of Julia Robinson, Solomon Feferman, editor, pp. 269–378, American Mathematical Society 1996. In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions.
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Assigning a Gödel number to each Turing machine definition produces a subset S of the natural numbers corresponding to the computable numbers and identifies a surjection from S to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable. The set S of these Gödel numbers, however, is not computably enumerable (and consequently, neither are subsets of S that are defined in terms of it). This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals.
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He also showed that the K-trivials are computable in the halting problem. This class of sets is commonly known as \Delta_2^0 sets in arithmetical hierarchy. Robert M. Solovay was the first to construct a noncomputable K-trivial set, while construction of a computably enumerable such A was attempted by Calude, Coles Cristian Calude, Richard J. Coles, Program-Size Complexity of Initial Segments and Domination Reducibility, (1999), proceeding of: Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa and other unpublished constructions by Kummer of a K-trivial, and Muchnik junior of a low for K set.
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George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388).
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Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as strings or any other data structure. In other words, it is shown within algorithmic information theory that computational incompressibility "mimics" (except for a constant that only depends on the chosen universal programming language) the relations or inequalities found in information theory. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously."Algorithmic Information Theory Algorithmic information theory principally studies measures of irreducible information content of strings (or other data structures).
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The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but according to the Cantor's theorem, there are uncountably many sets of natural numbers. Although the halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerable and semidecidable). Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.
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