TY - GEN

T1 - Complexity of metric temporal logics with counting and the pnueli modalities

AU - Rabinovich, Alexander

PY - 2008

Y1 - 2008

N2 - 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 ( ) is the extension of until-since temporal logic by "counting modalities" C n (X) and 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 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 , "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 . In this paper we investigate the complexity of the satisfiability problem for 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.

AB - 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 ( ) is the extension of until-since temporal logic by "counting modalities" C n (X) and 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 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 , "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 . In this paper we investigate the complexity of the satisfiability problem for 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.

UR - http://www.scopus.com/inward/record.url?scp=53049101175&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-85778-5_8

DO - 10.1007/978-3-540-85778-5_8

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AN - SCOPUS:53049101175

SN - 354085777X

SN - 9783540857778

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 93

EP - 108

BT - Formal Modeling and Analysis of Timed Systems - 6th International Conference, FORMATS 2008, Proceedings

T2 - 6th International Conference on Formal Modeling and Analysis of Timed Systems, FORMATS 2008

Y2 - 15 September 2008 through 17 September 2008

ER -