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In [[recursion theory|computability theory]], the '''Rice–Shapiro theorem''' is a generalization of [[Rice's theorem]], and is named after [[Henry Gordon Rice]] and [[Norman Shapiro]].<ref name=rogers_1987>{{ cite book | title=Theory of Recursive Functions and Effective Computability | author=Rogers Jr., Hartley  | publisher=MIT Press|isbn=0-262-68052-1 | year=1987}}</ref>


==Informal statement==
The statement can be expressed informally as follows: given that a [[computable function]] (viewed as a black-box) is an infinite object{{clarification needed|date=September 2013|Requiring infinite information to specify? Some infinite set?}}, and given that we cannot hope to develop a general algorithm for studying a property of the function on infinite input-output pairs, there is in general no truly better way than to apply the function on some inputs and hope to get their outputs.{{citation needed|The interpretation of an authority is needed here|date=January 2014}}


==Formal statement==
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Let ''A'' be a set of partial-recursive unary [[Function (mathematics)|functions]] on the domain of [[natural numbers]] such that the set <math>\{ n \mid \varphi_n \in A \}</math> is [[recursively enumerable]], where <math>\varphi_n</math> denotes the <math>n</math>-th partial-recursive function in a [[Gödel number]]ing.
 
Then for any unary partial-recursive function ''<math>\psi</math>'', we have:
 
:<math>\psi \in A \Leftrightarrow \exists\mbox{ a finite function }\theta \subseteq \psi\mbox{ such that }\theta \in A.</math>
 
In the given statement, a finite function is a function with a finite domain <math>x_1,x_2,...,x_n</math> and <math>\theta \subseteq \psi</math> means that for every <math>x \in \{x_1,x_2,...,x_n\}</math> it holds that <math>\psi(x)</math> is defined and equal to <math>\theta(x)</math>.
 
In general, one can obtain the following statement: The set <math>\{n: \varphi_n \in A\}</math> is recursively enumerable iff the following two conditions hold:
 
(a) <math>\{\theta: \theta = \varphi_n \mbox{ for an } n \in A\}</math> is recursively enumerable;
 
(b) <math>n \in A \mbox{ iff there is a finite function } \theta \mbox{ such that } \varphi_n \mbox{ extends } \theta \wedge c(\theta) \in A</math> where <math>c(\theta) \mbox{ is the canonical index of } \theta</math>.
 
==Notes==
{{reflist}}
 
==References==
*{{cite book | title=Computability: an introduction to recursive function theory| author=Cutland, Nigel | publisher=Cambridge University Press | year=1980 }}; Theorem 7-2.16.
*{{cite book | title=Theory of Recursive Functions and Effective Computability | author=Rogers Jr., Hartley  | publisher=MIT Press|isbn=0-262-68052-1 | pages=482 | year=1987}}
*{{cite book | title=Classical Recursion Theory| author = Odifreddi, Piergiorgio | publisher = North Holland | year=1989}}
 
{{DEFAULTSORT:Rice-Shapiro theorem}}
[[Category:Theorems in the foundations of mathematics]]
[[Category:Theory of computation]]
 
 
{{comp-stub}}

Latest revision as of 20:56, 29 November 2014


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