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Computable function

Definition

Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation.

Related concepts

2-EXPTIME A. Turing AC0 ACC0 ALL (complexity)APX Abstract logic Ackermann function Ackermann set theory Addition Aleph number Algebraic logic Algorithm Alphabet (formal languages)Arg max Argument Arithmetical hierarchy Arity Arthur–Merlin protocol Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras BPP (complexity)BQP Banach–Tarski paradox Bijection Binary operation Blum axioms Boolean algebra Boolean algebras canonically defined Boolean function Boolean hierarchy Busy beaver Bézout coefficient CC (complexity)Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Chaitin's constant Church encoding Church–Turing thesis Class (set theory)Classical logic Closure (mathematics)Co-NP Co-NP-complete Codomain Compactness theorem Compiler Complement (set theory)Complete theory Complexity class Computability theory Computability theory (computer science)Computable number Computable ordinal Computable set Computably enumerable set Computational complexity theory Concrete category Conservative extension Consistency Constant function Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Countability Countable set DLOGTIME DSPACE DTIME David Hilbert Decidability (logic)Decision problem Deductive system Diagram (mathematical logic)Domain of a function ELEMENTARY EXPSPACE EXPTIME Effective method Effectively calculable Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Entscheidungsproblem Enumeration Equiconsistency Equivalence relation Euclid's Elements Euclidean geometry Existential quantification Exponential growth Exponential hierarchy Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality FL (complexity)FNP (complexity)FP (complexity)Feasible computability Finitary Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Function composition Function problem Functional predicate Fuzzy set General recursive function General set theory Greatest common divisor Grothendieck universe Ground expression Ground formula Grzegorczyk hierarchy Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hartley Rogers Jr.Herbert Enderton Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic Hyperarithmetical theory Hypercomputation IP (complexity)If and only if Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information theory Inhabited set Injective Injective function Interactive proof system Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Isomorphism Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Kőnig's lemma L (complexity)Lambda calculus Large cardinal Lemma (mathematics)Lexicographic order Lindström's theorem List of Hilbert systems List of axioms List of complexity classes List of first-order theories List of formal systems List of mathematical theories List of set identities and relations List of statements independent of ZFC List of undecidable problems Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Material conditional Mathematical logic Mathematical object Mathematician Mathematische Annalen Metalanguage Minimal axioms for Boolean algebra Model complete theory Model of computation Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Multiplication NC (complexity)NEXPTIME NL-complete NL (complexity)NONELEMENTARY NP-completeness NP-hardness NP (complexity)NSPACE NTIME Naive set theory Natural deduction Natural number Natural numbers Negation New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Open formula Operation (mathematics)Oracle (computability)Ordinal analysis Ordinal number P-complete PP (complexity)PR (complexity)PSPACE PSPACE-complete P (complexity)P versus NP problem Paradoxes of set theory Parity P Partial function Partition of a set Peano arithmetic Peano axioms Philosophy of mathematics PolyL Polynomial hierarchy Post–Turing machine Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime factor Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Probabilistically checkable proof Proceedings of the London Mathematical Society Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional variable QMA Quantifier (logic)Quantifier rank Quasi-polynomial time RE (complexity)RL (complexity)RP (complexity)R (complexity)Recursion Recursion theory Recursive definition Recursive set Recursively enumerable Register machine Relation (mathematics)Relative computability Reverse mathematics Robinson arithmetic Rule of inference Russell's paradox Rózsa Péter SC (complexity)SL (complexity)Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Semicomputable function Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Stephen Cole Kleene Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Super-recursive algorithm Supertask Surjective function Syllogism Symbol (formal)Syntax (logic)T-schema TC0 TC (complexity)TFNP Tag system Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic Tetration Theorem Theories of truth Theory (mathematical logic)Theory of computation Three-valued logic Timeline of mathematical logic Total Turing machine Transfer principle Transitive set True arithmetic Truth predicate Truth table Truth value Tuple Turing degree Turing jump Turing machine Type (model theory)Type theory UP (complexity)Ultrafilter (set theory)Ultraproduct Unary operation Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Well-partial-order ZPP (complexity)Zermelo–Fraenkel set theory Μ-recursive function Μ operator ♯P ♯P-complete

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