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 (n+ 1)-ary form.

DOI: 10.1007/978-3-642-03409-1_5

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Cite this paper

@inproceedings{Case2009IndependenceRF, title={Independence Results for n-Ary Recursion Theorems}, author={John Case and Samuel E. Moelius}, booktitle={FCT}, year={2009} }