Optimal decision procedures for MPNL over finite structures, the natural numbers, and the integers

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reasoning over interval structures, where the truth of formulas is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability problem for Metric Propositional Neighborhood 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 has been recently shown to be decidable over finite linear orders and the natural numbers by a doubly exponential procedure, leaving the tightness of the complexity bound as an open problem. We improve such a result by proving that the satisfiability problem for MPNL over finite linear orders and the natural numbers, as well as over the integers, is actually EXPSPACE-complete, even when length constraints are encoded in binary.