Complexity of metric temporal logics with counting and the Pnueli modalities

The common metric temporal logics for continuous time were shown to be insufficient, when it was proved in Hirshfeld and Rabinovich (1999, 2007) [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 (TLC) is the extension of until-since temporal logic TL(U. S) by "counting modalities" C-n(X) and (C) over right arrow (n) (n is an element of 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 (C) over left arrow (n)(X) says that X has happened n times in the last unit of time. In Hirshfeld and Rabinovich (2006) [11] it was proved that this temporal logic is expressively complete for a natural decidable metric predicate logic. In particular the Pnueli modalities Pn(k) (X-1, ..., 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 TLC. In this paper we investigate the complexity of the satisfiability problem for 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 the Pnueli modalities is PSPACE complete. (C) 2010 Elsevier B.V. All rights reserved.