The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete

TL;DR

This paper proves that adding the temporal neighborhood operator to a certain interval temporal logic increases its complexity from PSPACE-complete to EXPSPACE-complete, highlighting the impact of this extension on computational difficulty.

Contribution

It introduces and analyzes the complexity of the extended logic $ extsf{BDA}_{hom}$, showing it is EXPSPACE-complete, thus advancing understanding of temporal logic expressiveness and complexity.

Findings

Adding the neighborhood modality $A$ increases complexity from PSPACE to EXPSPACE.

The extended logic $ extsf{BDA}_{hom}$ is shown to be EXPSPACE-complete.

Homogeneous models correspond to restricted regular expressions with specific relations.

Abstract

A classic result by Stockmeyer gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of chop under the homogeneity assumption. In this paper, we study the complexity of the satisfiability problem for suitable weakenings of the chop interval temporal logic, that can be equivalently viewed as fragments of Halpern and Shoham interval logic. We first consider the logic $\mathsf{BD}_{hom}$ featuring modalities $B$, for \emph{begins}, corresponding to the prefix relation on pairs of intervals, and $D$, for \emph{during}, corresponding to the infix relation. The homogeneous models of $\mathsf{BD}_{hom}$ naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations. Such a fragment has been recently shown to be PSPACE-complete . In this paper, we study the extension $\mathsf{BD}_{hom}$ with the temporal neighborhood modality $A$ (corresponding to the Allen relation \emph{Meets}), and prove that it increases both its expressiveness and complexity. In particular, we show that the resulting logic $\mathsf{BDA}_{hom}$ is EXPSPACE-complete.