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In computability theory, the Church – Turing thesis ( also known as the Turing-Church thesis, the Church – Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis ) is a combined hypothesis (" thesis ") about the nature of functions whose values are effectively calculable ; or, in more modern terms, functions whose values are algorithmically computable.
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computability and theory
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive.
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.
Closely related fields in theoretical computer science are analysis of algorithms and computability theory.
In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically.
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters.
Research in computability theory has typically focused on decision problems.
Automata theory and formal language theory are closely related to computability.
Although not very successful in that respect, the lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem.
It is still used in the area of computability theory, although Turing machines are arguably the preferred model for computation.
* Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, Oxford University Press, 2004, ISBN 0-19-852981-3.
Covers logics in close relation with computability theory and complexity theory
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems.
* Post correspondence problem, an important problem in computability theory
In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.
In computability theory, primitive recursive functions are a class of functions that form an important building block on the way to a full formalization of computability.
computability and Church
This topic was further developed in the 1930s by Alonso Church and Alan Turing, who on the one hand gave two independent but equivalent definitions of computability, and on the other gave concrete examples for undecidable questions.
Because all these different attempts at formalizing the concept of " effective calculability / computability " have yielded equivalent results, it is now generally assumed that the Church – Turing thesis is correct.
Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability.
This work culminated in the theoretical development of so-called Turing machines and the Church – Turing thesis, which formalized the mathematics underlying computability theory.
computability and –
The classical concept of truth is a special case of computability, – computability restricted to problems of zero interactivity degree.
In computability theory and mathematical logic the Tarski – Kuratowski algorithm is a non-deterministic algorithm which provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy.
computability and Turing
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 ).
An attempt to understand the notion of " effective computability " better led Robin Gandy ( Turing's student and friend ) in 1980 to analyze machine computation ( as opposed to human-computation acted out by a Turing machine ).
Other formalisms ( besides recursion, the λ-calculus, and the Turing machine ) have been proposed for describing effective calculability / computability.
In fact, in computability theory it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines.
In the equivalence of models of computability, a parallel is drawn between Turing machines which do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function.
The statement that the halting problem cannot be solved by a Turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine.
Another important step in computability theory was Rice's theorem, which states that for all non-trivial properties of partial functions, it is undecidable whether a Turing machine computes a partial function with that property.
In computability theory, a system of data-manipulation rules ( such as a computer's instruction set, a programming language, or a cellular automaton ) is said to be Turing complete or computationally universal if it can be used to simulate any single-taped Turing machine.
* Turing machine ( limits of computability )
In computability theory, one of the basic undecidable problems is that of deciding whether a deterministic Turing machine ( DTM ) halts.
In computability theory, a busy beaver ( from the colloquial expression for an industrious person ) is a Turing machine that attains the maximum " operational busyness " ( such as measured by the number of steps performed, or the number of nonblank symbols finally on the tape ) among all the Turing machines in a certain class.
The fundamental results the researchers obtained established Turing computability as the correct formalization of the informal idea of effective calculation.
The main form of computability studied in recursion theory was introduced by Turing ( 1936 ).
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 ).
computability and thesis
Defining what such game-playing machines mean, computability logic provides a generalization of the Church-Turing thesis to the interactive level.
The famous Church-Turing thesis attempts to define computation and computability in terms of Turing machines.
Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method ( Enderton 2001, pp. 206ff.
computability and also
Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the " computability " tradition ( see below ).
It was the development of computability theory ( also known as recursion theory ) that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous.
It is also one of the primitive functions used in the characterization of computability by recursive functions.
Besides classical logic, linear logic ( understood in a relaxed sense ) and intuitionistic logic also turn out to be natural fragments of computability logic.
* Recursion theory, also known as computability theory
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 computability theory, a machine that always halts — also called a decider ( Sipser, 1996 ) or a total Turing machine ( Kozen, 1997 )— is a Turing machine that halts for every input.
In computability theory the s < sub > mn </ sub > theorem, ( also called the translation lemma, parameter theorem, or parameterization theorem ) is a basic result about programming languages ( and, more generally, Gödel numberings of the computable functions ) ( Soare 1987, Rogers 1967 ).
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