An Optimal Decision Procedure for MPNL over the Integers

TL;DR

This paper establishes that the satisfiability problem for Metric Propositional Neighborhood Logic (MPNL) over the integers is EXPSPACE-complete, providing an optimal decision procedure and advancing understanding of its computational complexity.

Contribution

The paper introduces an EXPSPACE decision procedure for MPNL over the integers, improving previous doubly exponential bounds and applicable to various structures.

Findings

MPNL is EXPSPACE-complete over the integers.

Developed an optimal EXPSPACE decision procedure for MPNL.

Results extend to finite structures and the natural numbers.

Abstract

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reason- ing over interval structures, where the truth of formulae is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability problem for Metric Propositional Neigh- borhood Logic (MPNL). MPNL features two modalities to access intervals "to the left" and "to the right" of the current one, respectively, plus an infinite set of length constraints. MPNL, interpreted over the naturals, has been recently shown to be decidable by a doubly exponential procedure. We improve such a result by proving that MPNL is actually EXPSPACE-complete (even when length constraints are encoded in binary), when interpreted over finite structures, the naturals, and the in- tegers, by developing an EXPSPACE decision procedure for MPNL over the integers, which can be easily tailored to finite linear orders and the naturals (EXPSPACE-hardness was already known).