# Rice–Shapiro theorem

Rice–Shapiro theorem In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro.[1] Inhalt 1 Formale Aussage 2 Perspective from effective topology 3 Anmerkungen 4 References Formal statement Let A be a set of partial-recursive unary functions on the domain of natural numbers such that the set {displaystyle Ix(EIN):={nmid varphi _{n}in einem}} is recursively enumerable, wo {Anzeigestil Varphi _{n}} denotes the {Anzeigestil n} -th partial-recursive function in a Gödel numbering.

Then for any unary partial-recursive function {Anzeigestil psi } , wir haben: {displaystyle psi in ALeftrightarrow exists } a finite function {displaystyle theta subseteq psi } so dass {displaystyle theta in A.} In the given statement, a finite function is a function with a finite domain {Anzeigestil x_{1},x_{2},...,x_{m}} und {displaystyle theta subseteq psi } means that for every {Anzeigestil xin {x_{1},x_{2},...,x_{m}}} it holds that {Anzeigestil psi (x)} is defined and equal to {Theta im Display-Stil (x)} .

Perspective from effective topology For any finite unary function {Theta im Display-Stil } on integers, Lassen {Anzeigestil C(Theta )} denote the 'frustum' of all partial-recursive functions that are defined, and agree with {Theta im Display-Stil } , an {Theta im Display-Stil } 's domain.

Equip the set of all partial-recursive functions with the topology generated by these frusta as base. Note that for every frustum {Anzeigestil C} , {displaystyle Ix(C)} is recursively enumerable. More generally it holds for every set {Anzeigestil A} of partial-recursive functions: {displaystyle Ix(EIN)} is recursively enumerable iff {Anzeigestil A} is a recursively enumerable union of frusta.

Notes ^ Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. ISBN 0-262-68052-1. References Cutland, Nigel (1980). Computability: an introduction to recursive function theory. Cambridge University Press.; Satz 7-2.16. Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. p. 482. ISBN 0-262-68052-1. Odifreddi, Piergiorgio (1989). Klassische Rekursionstheorie. North Holland.

This computing article is a stub. Sie können Wikipedia helfen, indem Sie es erweitern.

Kategorien: Theorems in the foundations of mathematicsTheorems in theory of computationComputing stubs

Wenn Sie andere ähnliche Artikel wissen möchten Rice–Shapiro theorem Sie können die Kategorie besuchen Computing stubs.

Geh hinauf

Wir verwenden eigene Cookies und Cookies von Drittanbietern, um die Benutzererfahrung zu verbessern Mehr Informationen