KnowraHalting problemLinked fromLinked fromThe 32 pages that link to Halting problem, each with the reason it gives.All 32Broader topic 12Related 18Compared with 2Turing machineBroader topic: Turing machines make the impossibility of a general halting test precise.Gödel's incompleteness theoremsRelated: The computational translation of incompleteness into a concrete algorithmic impossibility.Alan TuringBroader topic: Turing proved that no algorithm can solve it for every possible program and input.Church–Turing thesisRelated: The thesis concerns what algorithms can compute, including limits proved within the model.Computability theoryBroader topic: It is the canonical example of a well-defined problem that no algorithm decides.Gödel numberingRelated: Numerical codes for programs make it possible to formulate and prove undecidability results arithmetically.Decision problemBroader topic: It is a canonical decision problem proved impossible to solve in full generality.EntscheidungsproblemRelated: A hypothetical solution to the Entscheidungsproblem would decide halting through a logical encoding.Rice's theoremRelated: The contradiction in Rice's theorem's proof would decide this problem.Kolmogorov complexityRelated: Deciding exact complexity would solve halting questions for programs within a length bound.Computable functionRelated: Halting determines whether a procedure successfully returns a function value.Many-one reductionRelated: Many-one reductions transfer its undecidability to other decision problems.DecidabilityBroader topic: Its undecidability shows that no general algorithm decides every computation's termination.Universal Turing machineRelated: Universality enables a machine to represent arbitrary computations, including those whose termination cannot be decided in general.DiagonalizationBroader topic: Its undecidability proof builds a program that contradicts any proposed universal decider.Deterministic algorithmRelated: Deterministic execution can still continue forever rather than produce an output.Diagonal argumentRelated: A diagonal argument shows that no algorithm solves this decision problem for all programs.Undecidable problemBroader topic: Turing proved that no algorithm decides this particular problem for every program and input.Hilbert's tenth problemRelated: An undecidable halting question can be translated into a question about integer solutions.Automata theoryRelated: Its undecidability demonstrates a fundamental limit of Turing-machine computation.Arithmetical hierarchyBroader topic: Its set of halting instances is Σ⁰₁-complete.Kleene's recursion theoremRelated: Self-reference from the theorem supports diagonal arguments related to halting undecidability.Computable setCompared with: Its undecidability shows why a membership procedure must halt by design, not merely when it finds a witness.Computably enumerable setBroader topic: Its positive instances form a computably enumerable set, but its negative instances do not.Post's theoremBroader topic: The first Turing jump is the halting problem, matching the first noncomputable level.Computable numberRelated: Undecidability of halting underlies prominent constructions of noncomputable real numbers.General recursive functionRelated: A minimization search may fail to terminate, reflecting undecidable halting behavior.Theory of computationBroader topic: Its undecidability demonstrates that some well-defined computational questions have no general algorithm.Turing degreeRelated: Its degree is the first Turing jump above the computable degree.Blum axiomsCompared with: The axioms do not make halting decidable; they require agreement between two possibly partial domains.Trakhtenbrot's theoremRelated: A reduction from halting explains why no algorithm decides finite satisfiability.Turing's proofBroader topic: Turing’s construction proves that no algorithm solves this problem for all machines and inputs.
KnowraHalting problemLinked fromLinked fromThe 32 pages that link to Halting problem, each with the reason it gives.All 32Broader topic 12Related 18Compared with 2Turing machineBroader topic: Turing machines make the impossibility of a general halting test precise.Gödel's incompleteness theoremsRelated: The computational translation of incompleteness into a concrete algorithmic impossibility.Alan TuringBroader topic: Turing proved that no algorithm can solve it for every possible program and input.Church–Turing thesisRelated: The thesis concerns what algorithms can compute, including limits proved within the model.Computability theoryBroader topic: It is the canonical example of a well-defined problem that no algorithm decides.Gödel numberingRelated: Numerical codes for programs make it possible to formulate and prove undecidability results arithmetically.Decision problemBroader topic: It is a canonical decision problem proved impossible to solve in full generality.EntscheidungsproblemRelated: A hypothetical solution to the Entscheidungsproblem would decide halting through a logical encoding.Rice's theoremRelated: The contradiction in Rice's theorem's proof would decide this problem.Kolmogorov complexityRelated: Deciding exact complexity would solve halting questions for programs within a length bound.Computable functionRelated: Halting determines whether a procedure successfully returns a function value.Many-one reductionRelated: Many-one reductions transfer its undecidability to other decision problems.DecidabilityBroader topic: Its undecidability shows that no general algorithm decides every computation's termination.Universal Turing machineRelated: Universality enables a machine to represent arbitrary computations, including those whose termination cannot be decided in general.DiagonalizationBroader topic: Its undecidability proof builds a program that contradicts any proposed universal decider.Deterministic algorithmRelated: Deterministic execution can still continue forever rather than produce an output.Diagonal argumentRelated: A diagonal argument shows that no algorithm solves this decision problem for all programs.Undecidable problemBroader topic: Turing proved that no algorithm decides this particular problem for every program and input.Hilbert's tenth problemRelated: An undecidable halting question can be translated into a question about integer solutions.Automata theoryRelated: Its undecidability demonstrates a fundamental limit of Turing-machine computation.Arithmetical hierarchyBroader topic: Its set of halting instances is Σ⁰₁-complete.Kleene's recursion theoremRelated: Self-reference from the theorem supports diagonal arguments related to halting undecidability.Computable setCompared with: Its undecidability shows why a membership procedure must halt by design, not merely when it finds a witness.Computably enumerable setBroader topic: Its positive instances form a computably enumerable set, but its negative instances do not.Post's theoremBroader topic: The first Turing jump is the halting problem, matching the first noncomputable level.Computable numberRelated: Undecidability of halting underlies prominent constructions of noncomputable real numbers.General recursive functionRelated: A minimization search may fail to terminate, reflecting undecidable halting behavior.Theory of computationBroader topic: Its undecidability demonstrates that some well-defined computational questions have no general algorithm.Turing degreeRelated: Its degree is the first Turing jump above the computable degree.Blum axiomsCompared with: The axioms do not make halting decidable; they require agreement between two possibly partial domains.Trakhtenbrot's theoremRelated: A reduction from halting explains why no algorithm decides finite satisfiability.Turing's proofBroader topic: Turing’s construction proves that no algorithm solves this problem for all machines and inputs.