Independence Results for n-Ary Recursion Theorems

Abstract

The n-ary first and second recursion theorems formalize two distinct, yet similar, notions of self-reference. Roughly, the n-ary first recursion theorem says that, for any n algorithmic tasks (of an appropriate type), there exist n partial computable functions that use their own graphs in the manner prescribed by those tasks; the n-ary second recursion theorem says that, for any n algorithmic tasks (of an appropriate type), there exist n programs that use their own source code in the manner prescribed by those tasks.

Results include the following. The constructive 1-ary form of the first recursion theorem is independent of either 1-ary form of the second recursion theorem. The constructive 1-ary form of the first recursion theorem does not imply the constructive 2-ary form; however, the constructive 2-ary form does imply the constructive n-ary form, for each n ≥ 1. For each n ≥ 1, the not-necessarily-constructive n-ary form of the second recursion theorem does not imply the presence of the (n + 1)-ary form.

This paper received some support from NSF Grant CCR-0208616.