# A non-uniformly C-productive sequence & non-constructive disjunctions

John Case, Michael Ralston, Yohji Akama

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

Let φ be an acceptable programming system/numbering of the partial computable functions: ℕ = {0; 1; 2; …} →ℕ, where, for p ∈ ℕ N, φp is the partial computable function computed by program p of the φ-system and Wp = domain(φp). Wp is, then, the computably enumerable (c.e.) set accepted by p. The first author taught in a class a recursion theorem proof employing a pair of cases that, for each q, {x | φx = φq} is not computably enumerable. The particular pair of cases employed was: domain(φq) is infinite vs. finite - incidentally, by Rice's Theorem, a non-constructive disjunction. Some student asked why the proof involved an analysis by cases. The answer given straightaway to the student was that his teacher didn't know how else to do it. The present paper provides, among other things, a better answer: any proof that, for each q, {x | φx = φq} is not c.e. provably must involve some such non-constructiveness. Furthermore, completely characterized are all the possible such pairs of (index set based) cases that will work. Along the way, relevantly explored are uniform lifts of the concept completely productive (abbr: c-productive) sets to sequences of sets. The c-productive sets are the effectively non-c.e. sets. A set sequence Sq, q ∈ ℕ, is said to be uniformly c-productive iff there is a computable f so that, for all q, x, f(q, x) is a counterexample to Sq = Wx. Relevance: a completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Relevantly shown, then, is that the sequence {x | φx = φq}, q ∈ ℕ, is not uniformly c-productive | and this even though, of course, the set {‹q, x›| φx = φq} is c-productive. For some results we provide upper and/or lower bounds for them as to position in the Arithmetical Hierarchy of the Law of Excluded Middle needed in addition to Heyting Arithmetic. Results are also obtained for similar problems, including for run-time bounded programming systems and other subrecursive systems. Proof methods include recursion theorems.

Original language English Proceedings of the 13th Asian Logic Conference, ALC 2013 Xishun Zhao, Qi Feng, Byunghan Kim, Liang Yu World Scientific Publishing Co. Pte Ltd 29-52 24 9789814675994 https://doi.org/10.1142/9789814678001_0002 Published - 2013 13th Asian Logic Conference, ALC 2013 - Guangzhou, ChinaDuration: 2013 Sept 16 → 2013 Sept 20

### Publication series

Name Proceedings of the 13th Asian Logic Conference, ALC 2013

### Conference

Conference 13th Asian Logic Conference, ALC 2013 China Guangzhou 13/9/16 → 13/9/20

## Keywords

• Arithmetically limited law of excluded middle
• Clocked programming systems
• Complete productivity
• Constructivity
• Index sets
• Recursion theorems
• Uniformly computable

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