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• Computability_theory abstract "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. Remarkably, the invention of the central combinatorial object of recursion theory, namely the Universal Turing Machine, predates and predetermines the invention of modern computers. Historically, the study of algorithmically undecidable sets and functions was motivated by various problems in mathematics that turned to be undecidable; for example, word problem for groups and the like. There are several applications of the theory to other branches of mathematics that do not necessarily concentrate on undecidability. The early applications include the celebrated Higman's embedding theorem that provides a link between recursion theory and group theory, results of Michael O. Rabin and Anatoly Maltsev on algorithmic presentations of algebras, and the negative solution to Hilbert's Tenth Problem. The more recent applications include algorithmic randomness, results of Soare et al. who applied recursion-theoretic methods to solve a problem in algebraic geometry, and the very recent work of Slaman et al. on normal numbers that solves a problem in analytic number theory.Recursion theory overlaps with proof theory, effective descriptive set theory, model theory, and abstract algebra.Arguably, computational complexity theory is a child of recursion theory; both theories share the same technical tool, namely the Turing Machine. 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 a considerable overlap in knowledge and methods between these two research communities; however, no firm line can be drawn between them. For instance, parametrized complexity was invented by a complexity theorist Michael Fellows and a recursion theorist Rod Downey.".
• Computability_theory wikiPageExternalLink slaman86definability.pdf.
• Computability_theory wikiPageExternalLink tp2-ie.pdf.
• Computability_theory wikiPageExternalLink gold67limit.pdf.
• Computability_theory wikiPageExternalLink www.aslonline.org.
• Computability_theory wikiPageExternalLink learning.ps.
• Computability_theory wikiPageExternalLink recursiontheory.html.
• Computability_theory wikiPageExternalLink History_of_Degrees.pdf.
• Computability_theory wikiPageExternalLink gold67limit.html.
• Computability_theory wikiPageExternalLink jumpmrl.pdf.
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• Computability_theory wikiPageID "155414".
• Computability_theory wikiPageRevisionID "601041671".
• Computability_theory hasPhotoCollection Computability_theory.
• Computability_theory subject Category:Computability_theory.
• Computability_theory subject Category:Mathematical_logic.
• Computability_theory comment "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?".".
• Computability_theory label "Berechenbarkeitstheorie".
• Computability_theory label "Computability theory".
• Computability_theory label "Teoria da computabilidade".
• Computability_theory label "Teoria della calcolabilità".
• Computability_theory label "Teoria obliczalności".
• Computability_theory label "Teoría de la computabilidad".
• Computability_theory label "Теория вычислимости".
• Computability_theory label "نظرية الحسبانية".
• Computability_theory label "可计算性理论".
• Computability_theory label "計算可能性理論".
• Computability_theory sameAs Teorie_vyčíslitelnosti.
• Computability_theory sameAs Berechenbarkeitstheorie.
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• Computability_theory sameAs Teoría_de_la_computabilidad.
• Computability_theory sameAs Teoria_della_calcolabilità.
• Computability_theory sameAs 計算可能性理論.
• Computability_theory sameAs 계산_가능성_이론.
• Computability_theory sameAs Teoria_obliczalności.
• Computability_theory sameAs Teoria_da_computabilidade.
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• Computability_theory wasDerivedFrom Computability_theory?oldid=601041671.
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