Model Checking the Logic of Allen's Relations Meets and Started-by is $P^NP$-Complete

TL;DR

This paper investigates the computational complexity of model checking for the Allen's relations AB logic, showing it is P^NP-complete under certain conditions and that adding the Met-by modality does not increase complexity.

Contribution

It establishes the P^NP-completeness of model checking for the AB logic of Allen's relations under the homogeneity assumption and demonstrates that adding Met-by does not increase complexity.

Findings

Model checking for AB is P^NP-complete under homogeneity.

Adding Met-by to AB does not increase the complexity.

Full HS logic has a higher complexity, being EXPSPACE-hard.

Abstract

In the plethora of fragments of Halpern and Shoham's modal logic of time intervals (HS), the logic AB of Allen's relations Meets and Started-by is at a central position. Statements that may be true at certain intervals, but at no sub-interval of them, such as accomplishments, as well as metric constraints about the length of intervals, that force, for instance, an interval to be at least (resp., at most, exactly) k points long, can be expressed in AB. Moreover, over the linear order of the natural numbers N, it subsumes the (point-based) logic LTL, as it can easily encode the next and until modalities. Finally, it is expressive enough to capture the {\omega}-regular languages, that is, for each {\omega}-regular expression R there exists an AB formula {\phi} such that the language defined by R coincides with the set of models of {\phi} over N. It has been shown that the satisfiability problem for AB over N is EXPSPACE-complete. Here we prove that, under the homogeneity assumption, its model checking problem is {\Delta}^p_2 = P^NP-complete (for the sake of comparison, the model checking problem for full HS is EXPSPACE-hard, and the only known decision procedure is nonelementary). Moreover, we show that the modality for the Allen relation Met-by can be added to AB at no extra cost (AA'B is P^NP-complete as well).