Program Self-reference in Constructive Scott Subdomains

Abstract

Intuitively, a recursion theorem asserts the existence of self-referential programs. Two well-known recursion theorems are Kleene’s Recursion Theorem (krt) and Rogers’ Fixpoint Recursion Theorem (fprt). Does one of these two theorems better capture the notion of program self-reference than the other? In the context of the partial computable functions over the natural numbers (), fprt is strictly weaker than krt, in that fprt holds in any effective numbering of in which krt holds, but not vice versa. It is shown that, in this context, the existence of self-reproducing programs (a.k.a. quines) is assured by krt, but not by fprt. Most would surely agree that a self-reproducing program is self-referential. Thus, this result suggests that krt is better than fprt at capturing the notion of program self-reference in .

A generalization of krt to arbitrary constructive Scott subdomains is then given. (For fprt, a similar generalization was already known.) Surprisingly, for some such subdomains, the two theorems turn out to be equivalent. A precise characterization is given of those constructive Scott subdomains in which this occurs. For such subdomains, the two theorems capture the notion of program self-reference equally well.