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| computability | |
|---|---|
| Name | Computability |
| Field | Theoretical computer science, Mathematical logic |
| Introduced | 1930s |
| Notable figures | Alan Turing, Alonzo Church, Emil Post, Kurt Gödel, Stephen Kleene, John von Neumann, Alan M. Turing |
computability Computability is the study of which problems can be solved by precise mechanical procedures and which cannot, connecting models of calculation, limits of algorithmic methods, and formal systems in mathematics and computer science. It unites work from logicians, mathematicians, and early computer scientists—such as Alan Turing, Alonzo Church, Kurt Gödel, Emil Post, and Stephen Kleene—and informs modern theory used by researchers at institutions like Bell Labs, Princeton University, Massachusetts Institute of Technology, Harvard University, and University of Cambridge. Results in the field interact with foundational theorems exemplified by the Gödel–Rosser theorem, the Church–Turing thesis, and constructions related to the Turing machine and lambda calculus.
Computability delineates which decision problems, function computations, and formal derivations are solvable using finite, rule-based procedures, relating to formal systems such as first-order logic, Peano arithmetic, Zermelo–Fraenkel set theory, and frameworks developed by Hilbert, David Hilbert, Emil Post, and Alonzo Church. The scope covers algorithmic solvability questions arising in contexts studied at Stanford University, University of California, Berkeley, Carnegie Mellon University, University of Oxford, and École Normale Supérieure, and ties into formal languages examined in venues like Association for Computing Machinery conferences and publications by Institute of Electrical and Electronics Engineers. The field interacts with decision procedures used in systems influenced by work at IBM Research, Microsoft Research, Google Research, Bell Labs Innovations, and legal or philosophical analyses influenced by scholars associated with Princeton Theological Seminary and Oxford University Press.
Canonical models include the Turing machine introduced by Alan Turing, the lambda calculus of Alonzo Church, the register machine variants inspired by John von Neumann, Post machines from Emil Post, and automata models such as finite automaton, pushdown automaton, and linear bounded automaton examined by researchers at Bell Labs and in texts from MIT Press. Other formal systems studied include Combinatory logic by Haskell Curry, μ-recursive functions from Stephen Kleene, Markov algorithms from Anatoly Markov, and computational models analyzed in seminars at École Polytechnique and Max Planck Institute for Informatics. Studied equivalences link models used in work by Alonzo Church, John McCarthy, Peter Landin, and Dana Scott.
Central notions distinguish decidable problems, semi-decidable (recursively enumerable) sets, and non-recursive problems, with canonical examples like the Halting problem shown undecidable by Alan Turing, the Entscheidungsproblem framed by David Hilbert and resolved by Alonzo Church, and decision properties explored in contexts influenced by Emil Post and Kurt Gödel. Proof techniques draw on reductions used in works at Princeton University and University of Chicago, diagonalization methods pioneered in discussions by Georg Cantor and Kurt Gödel, and completeness results paralleling themes from the Cook–Levin theorem and reductions studied by Stephen Cook, Leonid Levin, and Richard Karp.
The arithmetical hierarchy and analytic hierarchy classify sets via quantifier alternation and relativization, topics explored by Stephen Kleene, Hermann Weyl, Moschovakis, and researchers at University of California, Berkeley and University of Pennsylvania. Turing degrees quantify noncomputability levels developed by Emil Post and formalized in extensive work by Richard Shore, Robert Soare, S. Barry Cooper, and contributors at University of Illinois at Urbana–Champaign. The study of ordinal analyses connects to research by Gerald Sacks, Anil Nerode, Dana Scott, and institutions such as Institute for Advanced Study.
Although distinct from computability, complexity theory constrains computability with resource bounds like time and space, central results include the P versus NP problem articulated by Stephen Cook and debated across communities at Clay Mathematics Institute, Association for Computing Machinery, Institute of Mathematical Statistics, and Simons Foundation. Classes such as P (complexity), NP (complexity), PSPACE, EXPTIME, and probabilistic classes like BPP and RP arise in literature by Leonard Adleman, Michael Rabin, Richard Lipton, and institutions including Stanford University and Carnegie Mellon University. Reductions, completeness, and hierarchies connect to work by Richard Karp, Juraj Hromkovič, Valentina Borokhovich, and research programs at Microsoft Research and Google DeepMind.
Computability informs programming language semantics developed at Massachusetts Institute of Technology and Carnegie Mellon University, verification tools used at NASA, European Space Agency, National Institute of Standards and Technology, and influences cryptography foundations explored by Whitfield Diffie, Martin Hellman, Ron Rivest, Adi Shamir, and Leonard Adleman. Limits on automated reasoning affect automated theorem provers like Coq, Isabelle, HOL, and software verification projects at Microsoft Research and Intel Corporation. Philosophical and foundational implications resonate in debates involving Bertrand Russell, Ludwig Wittgenstein, Saul Kripke, and policy discussions in forums hosted by Royal Society and National Academy of Sciences.
Key milestones include Gödel's incompleteness theorems by Kurt Gödel, the negative solution to the Entscheidungsproblem by Alonzo Church and Alan Turing, Post's problems by Emil Post, Kleene's recursion theory developed at Princeton University and Harvard University, and the classification of NP-complete problems by Stephen Cook and Richard Karp. Subsequent advances occurred in research groups at Bell Labs, AT&T Labs Research, IBM Research, Microsoft Research, and universities like University of California, Berkeley, Stanford University, and University of Cambridge, shaping modern theoretical computer science and ongoing work sponsored by institutions including the National Science Foundation and European Research Council.