Complexity of Metric Temporal Logics with Counting and the Pnueli Modalities

Abstract

The common metric temporal logics for continuous time were shown to be insufficient, when it was proved in [7, 12] that they cannot express a modality suggested by Pnueli. Moreover no temporal logic with a finite set of modalities can express all the natural generalizations of this modality. The temporal logic with counting modalities ($\mathit{TLC}$) is the extension of until-since temporal logic $\mathit{TL}(\mbox{\bf{U}}, \mbox{{\bf{S}}})$ by “counting modalities” C _n (X) and $\overleftarrow{C}_n$ (n ∈ ℕ); for each n the modality C _n (X) says that X will be true at least at n points in the next unit of time, and its dual $\overleftarrow{C}_n(X)$ says that X has happened n times in the last unit of time. In [11] it was proved that this temporal logic is expressively complete for a natural decidable metric predicate logic. In particular the Pnueli modalities $\mathit{Pn}_k(X_1,\ldots,X_k)$, “there is an increasing sequence t ₁,...,t _k of points in the unit interval ahead such that X _i holds at t _i ”, are definable in $\mathit{TLC}$.

In this paper we investigate the complexity of the satisfiability problem for $\mathit{TLC}$ and show that the problem is PSPACE complete when the index of C _n is coded in unary, and EXPSPACE complete when the index is coded in binary. We also show that the satisfiability problem for the until-since temporal logic extended by Pnueli’s modalities is PSPACE complete.