10 Facts About Computability theory

Computability theory, known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

Computability theory originated in the 1930s, with work of Kurt Godel, Alonzo Church, Rozsa Peter, Alan Turing, Stephen Kleene, and Emil Post.

Computability theory includes the study of generalized notions of this field such as arithmetic reducibility, hyperarithmetical reducibility and a-recursion theory, as described by Sacks in 1990.

Computability theory is less well developed for analog computation that occurs in analog computers, analog signal processing, analog electronics, neural networks and continuous-time control theory, modelled by differential equations and continuous dynamical systems.

Godel's proofs show that the set of logical consequences of an effective first-order Computability theory is a computably enumerable set, and that if the Computability theory is strong enough this set will be uncomputable.

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Computability theory is linked to second-order arithmetic, a formal theory of natural numbers and sets of natural numbers.

Field of proof Computability theory includes the study of second-order arithmetic and Peano arithmetic, as well as formal theories of the natural numbers weaker than Peano arithmetic.

Computability theory 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.

In 1967, Rogers has suggested that a key property of computability theory is that its results and structures should be invariant under computable bijections on the natural numbers.

Main professional organization for computability theory is the Association for Symbolic Logic, which holds several research conferences each year.