Specification and Automatic Verification of Computational Reductions

TL;DR

This paper introduces cookbook reductions, a graphical specification method for computational reductions, and demonstrates that the validation problem is decidable for certain subclasses, balancing expressiveness and verifiability.

Contribution

It proposes cookbook reductions as an expressive, easy-to-use framework and proves the decidability of the validation problem for specific subclasses.

Findings

Cookbook reductions cover many classical graph reductions.

Validation problem is decidable for natural subclasses.

SAT remains NP-complete with cookbook reductions.

Abstract

We are interested in the following validation problem for computational reductions: for algorithmic problems $P$ and $P^\star$, is a given candidate reduction indeed a reduction from $P$ to $P^\star$? Unsurprisingly, this problem is undecidable even for very restricted classes of reductions. This leads to the question: Is there a natural, expressive class of reductions for which the validation problem can be attacked algorithmically? We answer this question positively by introducing an easy-to-use graphical specification mechanism for computational reductions, called cookbook reductions. We show that cookbook reductions are sufficiently expressive to cover many classical graph reductions and expressive enough so that SAT remains NP-complete (in the presence of a linear order). Surprisingly, the validation problem is decidable for natural and expressive subclasses of cookbook reductions.