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.

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.

In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property.

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.

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.

Due to the computability of consciousness and the function of consciousness as a matrix for interpretation, Copies hold the unique position of being the only conscious beings which themselves are not being computed by self-consistent mathematical rules ( existing, of course, in virtual realities held together by heuristics merely for the sake of their experience ).

The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem.

In computability theory, the domain of a function is taken to be the set of all inputs for which the function is defined.

In computability theory two sets A and B are computably isomorphic or recursively isomorphic if there exists a bijective computable function f with f ( A ) = B.

In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem a successively harder decision problem with the property that is not decidable by an oracle machine with an oracle for.

In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion of computability on a set.

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set.

This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes.

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.

However, even simple systems based on this simple logic can be used to represent data that is well beyond the processing capability of current computer systems: see computability for reasons.

Another way of stating Rice's theorem that is more useful in computability theory follows.

In fact, in computability theory it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines.

The field is divided into three major branches: automata theory, computability theory and computational complexity theory.

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.

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.

The first result of computability theory is that it is impossible in general to predict what a Turing-complete program will do over an arbitrarily long time.

Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the " computability " tradition ( see below ).

CA ) is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling.

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.

In computability theory, one of the basic undecidable problems is that of deciding whether a deterministic Turing machine ( DTM ) halts.

In computability theory, the term " Gödel numbering " is used in settings more general than the one described above.

It is also one of the primitive functions used in the characterization of computability by recursive functions.

In computability theory and computational complexity theory, a many-one reduction is a reduction which converts instances of one decision problem into instances of a second decision problem.

The halting problem is one of the most famous problems in computer science, because it has profound implications on the theory of computability and on how we use computers in everyday practice.

In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem.

In computability theory, a truth-table reduction is a reduction from one set of natural numbers to another.