Computability theory, also 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. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory.

Basic questions addressed by computability theory include:

  • What does it mean for a function on the natural numbers to be computable?
  • How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?

Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics.

Further reading

; Undergraduate level texts

; Advanced texts

; Survey papers and collections

; Research papers and collections

  • [https://web.archive.org/web/20090125120159/http://www.isrl.uiuc.edu/~amag/langev/paper/gold67limit.html]
  • Reprinted in .
  • Association for Symbolic Logic homepage
  • Computability in Europe homepage
  • Webpage on Recursion Theory Course at Graduate Level with approximately 100 pages of lecture notes
  • German language lecture notes on inductive inference