Computability theory

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For the concept of computability, see Computability.

Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.
The basic questions addressed by recursion theory are "What does it mean for a function on the natural numbers to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched. The field has since grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.
Arguably, a child of recursion theory is complexity theory that concentrates on the complexity of decidable problems rather than proves undecidable problems. Both theories share the same tool, namely a Turing machine. Remarkably, the invention of the central combinatorial object of recursion theory, namely the Universal Turing Machine, pre-dates and pre-determines the invention of modern computers.
Another reason for the area being quite active these days is the celebrated Higman's embedding theorem that provides a link between recursion theory and group theory.
Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article. This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science. There is considerable overlap in knowledge and methods between these two research communities, however, and no firm line can be drawn between them.