Computability theory

Computability theory, also known as recursive function theory, is a branch of mathematical logic which studies the properties of algorithms, and their associated computing devices. It is a foundation of computer science and is closely related to other theories in theoretical computer science such as complexity theory and algorithmic information theory.

The foundation of computability theory is the Church-Turing thesis, which states that a function is computable if and only if it can be computed by a Turing machine. A computable function is any algorithm, formula, or rule which can be written and evaluated on a computer. All the algorithms used in computers are based on computability theory.

The theory was developed by a number of different researchers, beginning with the work of Alan Turing in the 1930s. By the late 1950s, the theory was well developed and was being applied to other areas, such as automata theory, programming languages, and artificial intelligence. As the technology of computing has advanced, so too have the theories and applications of computability theory.

Computability theory is used to analyze the complexity of algorithms and how they can be solved, as well as to understand the limitations of computers. It is also used to prove the correctness of algorithms. Finally, it is used to study the limitations of expression and computation of functions and predicates.