Constraint Satisfaction Problems around Skolem Arithmetic

TL;DR

This paper explores the decidability and complexity of certain constraint satisfaction problems involving Skolem Arithmetic, revealing both decidability results and a diverse complexity landscape including P and NP-complete cases.

Contribution

It proves the decidability of a specific satisfaction problem on circuits and analyzes the complexity of first-order expansions of Skolem Arithmetic without constants.

Findings

Decidability of a satisfaction problem on circuits using Skolem Arithmetic.

Identification of CSPs with both polynomial-time and NP-complete complexities.

Rich landscape of problems in first-order expansions of Skolem Arithmetic.

Abstract

We study interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs). We revisit results of Glass er et al. in the context of CSPs and settle the major open question from that paper, finding a certain satisfaction problem on circuits to be decidable. This we prove using the decidability of Skolem Arithmetic. We continue by studying first-order expansions of Skolem Arithmetic without constants, (N;*), where * indicates multiplication, as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.