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