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Computability theory
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Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originatedDecision problem (1,272 words) [view diff] no match in snippet view article find links to article
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of theReduction (complexity) (1,657 words) [view diff] no match in snippet view article
In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficientlyMyhill isomorphism theorem (701 words) [view diff] no match in snippet view article find links to article
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notionNormal form (abstract rewriting) (1,285 words) [view diff] no match in snippet view article
rewriting system has the unique normal form property with respect to reduction (UN→) if for every term reducing to normal forms a and b, a is equal toOracle machine (2,014 words) [view diff] no match in snippet view article find links to article
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a TuringList of computability and complexity topics (466 words) [view diff] no match in snippet view article find links to article
list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what canCircuit satisfiability problem (1,183 words) [view diff] no match in snippet view article find links to article
been proven to be CoNP-Complete via a reduction from Circuit UNSAT problem. The gadgets constructed for this reduction are: wire, split, AND and NOT gatesTheory of computation (2,168 words) [view diff] no match in snippet view article find links to article
into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question:Decider (Turing machine) (1,302 words) [view diff] no match in snippet view article
In computability theory, a decider is a Turing machine that halts for every input. A decider is also called a total Turing machine as it represents a totalTuring reduction (1,841 words) [view diff] no match in snippet view article find links to article
In computability theory, a Turing reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyle B} is an oracle machineComputational complexity theory (6,302 words) [view diff] no match in snippet view article find links to article
fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computationalAckermann function (6,780 words) [view diff] no match in snippet view article find links to article
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computableLambda-mu calculus (541 words) [view diff] no match in snippet view article find links to article
(which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operatorComputable isomorphism (180 words) [view diff] no match in snippet view article find links to article
In computability theory two sets A , B {\displaystyle A,B} of natural numbers are computably isomorphic or recursively isomorphic if there exists a totalIndex set (computability) (703 words) [view diff] no match in snippet view article
In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, accordingKrivine machine (1,901 words) [view diff] no match in snippet view article find links to article
normal form reduction of a lambda term using call-by-name reduction. Thanks to its formalism, it tells in details how a kind of reduction works and setsSimple set (539 words) [view diff] no match in snippet view article find links to article
In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement isTruth-table reduction (628 words) [view diff] no match in snippet view article find links to article
In computability theory a truth-table reduction is a type of reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyleRice's theorem (1,686 words) [view diff] no match in snippet view article find links to article
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about theComputability (3,293 words) [view diff] no match in snippet view article find links to article
problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computerMartin measure (386 words) [view diff] no match in snippet view article find links to article
degrees that are "at least as complex" as X {\displaystyle X} under Turing reduction. In order-theoretic terms, the cone of [ X ] {\displaystyle [X]} is theComputation in the limit (1,678 words) [view diff] no match in snippet view article find links to article
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computableApproximation-preserving reduction (1,497 words) [view diff] no match in snippet view article find links to article
In computability theory and computational complexity theory, especially the study of approximation algorithms, an approximation-preserving reduction isK-trivial set (1,837 words) [view diff] no match in snippet view article find links to article
studied in the field of algorithmic randomness, which is a subfield of Computability theory and related to algorithmic information theory in computer scienceMany-one reduction (1,763 words) [view diff] no match in snippet view article find links to article
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instancesLambda calculus (11,553 words) [view diff] no match in snippet view article find links to article
Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In the simplest form of lambda calculus, terms areList of undecidable problems (1,588 words) [view diff] no match in snippet view article find links to article
In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be anyAlpha recursion theory (1,455 words) [view diff] no match in snippet view article find links to article
similar to those of the primitive recursive functions. We say R is a reduction procedure if it is α {\displaystyle \alpha } recursively enumerable andHalting problem (7,334 words) [view diff] no match in snippet view article find links to article
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether thePost correspondence problem (2,521 words) [view diff] no match in snippet view article find links to article
some new problem by reducing from PCP, it often happens that the first reduction one finds is not from PCP itself but from an apparently weaker versionOutline of logic (2,084 words) [view diff] no match in snippet view article find links to article
Non-standard model Normal model Model Semantic consequence Truth value Computability theory – branch of mathematical logic that originated in the 1930s withEmil Leon Post (1,397 words) [view diff] no match in snippet view article find links to article
known for his work in the field that eventually became known as computability theory. Post was born in Augustów, Suwałki Governorate, Congress PolandSelman's theorem (1,188 words) [view diff] no match in snippet view article find links to article
In computability theory, Selman's theorem is a theorem relating enumeration reducibility with enumerability relative to oracles. It is named after AlanMedvedev reducibility (133 words) [view diff] no match in snippet view article find links to article
In computability theory, a set P of functions ℕ → ℕ is said to be Medvedev-reducible to another set Q of functions ℕ → ℕ when there exists an oracle TuringComputability logic (2,560 words) [view diff] no match in snippet view article find links to article
⊓x(p(x)⊔¬p(x))→⊓x(q(x)⊔¬q(x)) does the same but for the stronger version of Turing reduction where the oracle for p can be queried only once. ⊓x⊔y(q(x)↔p(y)) doesDiophantine set (1,425 words) [view diff] no match in snippet view article find links to article
existential quantification merely reflects the usual applications in computability theory and model theory. It does not matter whether natural numbers referKolmogorov complexity (7,151 words) [view diff] no match in snippet view article find links to article
13English 14words". It is also possible to show the non-computability of K by reduction from the non-computability of the halting problem H, since K and H areUniqueness quantification (804 words) [view diff] no match in snippet view article find links to article
principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable ComputableJulia Robinson (2,166 words) [view diff] no match in snippet view article find links to article
American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problemsS2S (mathematics) (4,618 words) [view diff] no match in snippet view article
large grid minors, which can be used to simulate a Turing machine. By reduction to S2S, the MSO theory of countable orders is decidable, as is the MSODecidability (logic) (1,901 words) [view diff] no match in snippet view article
undecidable. This is closely related to the concept of a many-one reduction in computability theory. A property of a theory or logical system weaker than decidabilityAlgorithm characterizations (8,851 words) [view diff] no match in snippet view article find links to article
statements into a form fit for the engine to work with – in this case the reduction of each proposition to its elementary denials", (3) "Thirdly, there isConfluence (abstract rewriting) (1,744 words) [view diff] no match in snippet view article
39:472-482, 1936 Baader & Nipkow 1998, p. 9. Cooper, S. B. (2004). Computability theory. Boca Raton: Chapman & Hall/CRC. p. 184. ISBN 1584882379. BaaderEXPTIME (991 words) [view diff] no match in snippet view article find links to article
be solved in polynomial time, by the time hierarchy theorem. In computability theory, one of the basic undecidable problems is the halting problem: decidingWadge hierarchy (1,336 words) [view diff] no match in snippet view article find links to article
cone lemma from computability theory. The Wadge game is a simple infinite game used to investigate the notion of continuous reduction for subsets of BaireAmalgamation property (817 words) [view diff] no match in snippet view article find links to article
relation,[clarification needed] and in lambda calculus as a manner of reduction having the Church–Rosser property. An amalgam can be formally definedSatisfiability (1,500 words) [view diff] no match in snippet view article find links to article
principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable ComputableInteraction nets (1,880 words) [view diff] no match in snippet view article find links to article
Models of Computation". Models of Computation: An Introduction to Computability Theory. Springer Science & Business Media. pp. 107–130. ISBN 9781848824348Combinatory logic (5,243 words) [view diff] no match in snippet view article find links to article
combinatory logic is used as a simplified model of computation, used in computability theory and proof theory. Despite its simplicity, combinatory logic capturesSubstitution (logic) (1,650 words) [view diff] no match in snippet view article
but not identical to, function composition; it is closely related to β-reduction in lambda calculus. In contrast to these notions, however, the accentTheory (4,353 words) [view diff] no match in snippet view article find links to article
theory — Choquet theory — Coding theory — Combinatorial game theory — Computability theory — Computational complexity theory — Deformation theory — DimensionProof theory (2,641 words) [view diff] no match in snippet view article find links to article
the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. This provides the foundation for the intuitionisticNorman Shapiro (806 words) [view diff] no match in snippet view article find links to article
coined the phrase "strong reducibility" for a computability theory currently called the many-one reduction. His thesis was titled Degrees of ComputabilityHistory of logic (13,240 words) [view diff] no match in snippet view article find links to article
inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory, and its ideas and methods began to influence philosophyGentzen's consistency proof (1,959 words) [view diff] no match in snippet view article find links to article
false, then there is a least such ordinal. Gentzen defines a notion of "reduction procedure" for proofs in Peano arithmetic. For a given proof, such a procedureAxiomatic system (1,936 words) [view diff] no match in snippet view article find links to article
theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms underliesCompactness theorem (1,948 words) [view diff] no match in snippet view article find links to article
{\displaystyle \Sigma .} Barwise compactness theorem Herbrand's theorem – reduction of first-order mathematical logic to propositional logicPages displayingYuri Matiyasevich (1,054 words) [view diff] no match in snippet view article find links to article
ISSN 0090-4104. S2CID 121919479. Yuri Matiyasevich, Julia Robinson (1975). "Reduction of an arbitrary Diophantine equation to one in 13 unknowns". Acta ArithmeticaLogic (16,841 words) [view diff] no match in snippet view article find links to article
Major subareas include model theory, proof theory, set theory, and computability theory. Research in mathematical logic commonly addresses the mathematicalFunction (mathematics) (11,236 words) [view diff] no match in snippet view article
the major open problems in mathematics, the Riemann hypothesis. In computability theory, a general recursive function is a partial function from the integersType theory (7,866 words) [view diff] no match in snippet view article find links to article
inference rules known as β {\displaystyle \beta } -reduction and η {\displaystyle \eta } -reduction. They generalize the notion of function applicationModel checking (2,717 words) [view diff] no match in snippet view article find links to article
to hardware designs. For software, because of undecidability (see computability theory) the approach cannot be fully algorithmic, apply to all systems,Propositional formula (11,097 words) [view diff] no match in snippet view article find links to article
they have designed using synthesis techniques and then apply various reduction and minimization techniques to simplify their designs. Synthesis: EngineersSecond-order logic (4,321 words) [view diff] no match in snippet view article find links to article
formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary predicates thatMathematical proof (4,598 words) [view diff] no match in snippet view article find links to article
contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, aProof of impossibility (3,907 words) [view diff] no match in snippet view article find links to article
later than Turing) was a short paper by Emil Post that discussed the reduction of an algorithm to a simple machine-like "method" very similar to Turing'sLogicism (11,833 words) [view diff] no match in snippet view article find links to article
reduction of relation to the ordered pair of sets. Grattan-Guinness observes that in the second edition of Principia Russell ignored this reduction thatAlgorithm (7,341 words) [view diff] no match in snippet view article find links to article
algorithm general topics Regulation of algorithms Theory of computation Computability theory Computational complexity theory Computational mathematics "DefinitionNeurophilosophy (5,785 words) [view diff] no match in snippet view article find links to article
Church–Turing thesis, which formalized the mathematics underlying computability theory. The second group consisted of Warren McCulloch and Walter PittsHilary Putnam (8,854 words) [view diff] no match in snippet view article find links to article
previous research by Putnam, Julia Robinson and Martin Davis. In computability theory, Putnam investigated the structure of the ramified analytical hierarchyComplexity class (10,382 words) [view diff] no match in snippet view article find links to article
breakdown into decidable and undecidable pertains more to the study of computability theory, but is useful for putting the complexity classes in perspectiveNatural deduction (6,770 words) [view diff] no match in snippet view article find links to article
}}\wedge _{I}\end{aligned}}} These notions correspond exactly to β-reduction (beta reduction) and η-conversion (eta conversion) in the lambda calculus, usingFoundations of mathematics (7,100 words) [view diff] no match in snippet view article find links to article
smooth graph. At this point, the program of arithmetization of analysis (reduction of mathematical analysis to arithmetic and algebraic operations) advocatedComputer (13,744 words) [view diff] no match in snippet view article find links to article
organizations, clubs and societies of both a formal and informal nature. Computability theory Computer security Glossary of computer hardware terms History ofChurch encoding (6,538 words) [view diff] no match in snippet view article find links to article
0} Calculating n − m {\displaystyle n-m} takes many beta reductions. Unless doing the reduction by hand, this doesn't matter that much, but it is preferableSatisfiability modulo theories (4,370 words) [view diff] no match in snippet view article find links to article
arithmetic (often implemented in SMT solvers via bit-blasting, i.e., reduction to bitvectors), strings, (co)-datatypes, sequences (used to model dynamicHistory of mathematics (16,090 words) [view diff] no match in snippet view article find links to article
areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Derrick Henry Lehmer's use of ENIAC to furtherPhilosophy of mathematics (11,935 words) [view diff] no match in snippet view article find links to article
requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantificationTuring's proof (7,109 words) [view diff] no match in snippet view article find links to article
Hodges p. 125) — mainly because he had arrived simultaneously at a similar reduction of "algorithm" to primitive machine-like actions, so he took a personalConstant-recursive sequence (4,615 words) [view diff] no match in snippet view article find links to article
constant-recursive sequence can also be investigated from the perspective of computability theory. To do so, the description of the sequence s n {\displaystyle s_{n}}Glossary of artificial intelligence (27,506 words) [view diff] no match in snippet view article find links to article
divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question:List of women in mathematics (22,485 words) [view diff] no match in snippet view article find links to article
educator Julia F. Knight, American specialist in model theory and computability theory Sarah Koch (born 1979), American complex analyst and complex dynamicistJanuary–March 2021 in science (13,103 words) [view diff] no match in snippet view article find links to article
January 2021). "Superintelligence Cannot be Contained: Lessons from Computability Theory". Journal of Artificial Intelligence Research. 70: 65–76. doi:10Glossary of logic (29,963 words) [view diff] no match in snippet view article find links to article
logical and linguistic intuitions. busy beaver problem A problem in computability theory that seeks the Turing machine with the largest possible behavior