On approximate decidability of minimal programs

TL;DR

This paper explores the approximate decidability of minimal programs, investigating whether algorithms can reliably identify minimal indices or produce candidate lists, and presents negative results with open questions for future research.

Contribution

It introduces the problem of approximate decidability for minimal programs and provides initial negative results, highlighting open questions in the field.

Findings

No effective procedure can always correctly identify minimal indices.

Algorithms cannot reliably produce short lists containing minimal indices.

The paper leaves open the possibility of positive approximate solutions.

Abstract

An index $e$ in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from $e$. Since the 1960's it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We investigate whether the task of determining minimal indices can be solved in an approximate sense. Our first question, regarding the set of minimal indices, is whether there exists an algorithm which can correctly label 1 out of $k$ indices as either minimal or non-minimal. Our second question, regarding the function which computes minimal indices, is whether one can compute a short list of candidate indices which includes a minimal index for a given program. We give some negative results and leave the possibility of positive results as open questions.