Taming past {LTL} and flat counter systems

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. We also provide other complexity results obtained by restricting further the class of flat counter systems.