Independence results for n-ary recursion theorems

  • Authors:

  • John Case;Samuel E. Moelius, III.

  • Affiliations:

  • Department of Computer & Information Sciences, University of Delaware, Newark, DE;Department of Computer & Information Sciences, University of Delaware, Newark, DE

  • Venue:
  • FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
  • Year:
  • 2009

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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.