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]
Formal statement
Let A be a set of partial-recursive unary functions on the domain of natural numbers such that the set is recursively enumerable, where denotes the -th partial-recursive function in a Gödel numbering.
Then for any unary partial-recursive function , we have:
- a finite function such that
In the given statement, a finite function is a function with a finite domain and means that for every it holds that is defined and equal to .
In general, one can obtain the following statement: The set is recursively enumerable iff the following two conditions hold:
(a) is recursively enumerable;
(b) iff a finite function such that extends where is the canonical index of .
Notes
References
- Cutland, Nigel (1980). Computability: an introduction to recursive function theory. Cambridge University Press.; Theorem 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). Classical Recursion Theory. North Holland.