Complexity of metric temporal logics with counting and the pnueli modalities

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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 ( ) 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.

Original languageEnglish
Title of host publicationFormal Modeling and Analysis of Timed Systems - 6th International Conference, FORMATS 2008, Proceedings
Pages93-108
Number of pages16
DOIs
StatePublished - 2008
Event6th International Conference on Formal Modeling and Analysis of Timed Systems, FORMATS 2008 - Saint Malo, France
Duration: 15 Sep 200817 Sep 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5215 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference6th International Conference on Formal Modeling and Analysis of Timed Systems, FORMATS 2008
Country/TerritoryFrance
CitySaint Malo
Period15/09/0817/09/08

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