On Polynomial-Time Decidability of k-Negations Fragments of First-Order Theories

TL;DR

This paper presents a framework that identifies conditions under which fixed-negation fragments of first-order theories are decidable in polynomial time, expanding the understanding of tractability in logical theories.

Contribution

It introduces a generic framework for polynomial-time decidability of fixed-negation fragments of first-order theories, including new decidability results for weak Presburger arithmetic and related theories.

Findings

Decidable in polynomial time: fixed negation fragment of weak Presburger arithmetic.

Decidable in polynomial time: fixed negation fragments of weak linear real arithmetic.

Decidable in polynomial time: restriction of Presburger arithmetic with at most one variable per inequality.

Abstract

This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1--31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time. We give two further examples of instantiations of our framework, showing polynomial-time decidability of the fixed negation fragments of weak linear real arithmetic and of the restriction of Presburger arithmetic in which each inequality contains at most one variable.