Taming Past LTL and Flat Counter Systems

Stéphane Demri (1, 2) , Amit K Kumar Dhar (3) , Arnaud Sangnier (3)

1 CNRS - Centre National de la Recherche Scientifique
2 LSV - Laboratoire Spécification et Vérification [Cachan]
3 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

Reachability and LTL model-checking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that this problem is only NP-complete even if LTL admits past-time operators and arithmetical constraints on counters. As far as past-time operators are concerned, their addition to LTL immediately leads to complications and hence an NP upper bound cannot be deduced by translating formulae into LTL and studying the problem only for this latter logic. Actually, the NP upper bound is shown by adequately combining a new stuttering theorem for Past LTL and the property of small integer solutions for quantifier-free Presburger formulae. This latter complexity bound extends known and recent results on modelchecking weak Kripke structures with LTL formulae as well as reachability problems for flat counter systems. We also provide other complexity results obtained by restricting further the class of flat counter systems.

Mots clés: linear-time temporal Logic LTL; stuttering; model-checking; counter system; flatness; complexity; system of equations; small solution; Presburger arithmetic; linear-time temporal logic; path schema

Domaines: Informatique [cs]; Logique en informatique [cs.LO]

Stéphane Demri, Amit K Kumar Dhar, Arnaud Sangnier. Taming Past LTL and Flat Counter Systems. Information and Computation, 2015, 242, pp.306-339. ⟨10.1016/j.ic.2015.03.007⟩. ⟨hal-03192199⟩