Taming Past LTL and Flat Counter Systems

TL;DR

This paper proves that reachability and LTL model-checking for flat counter systems with past operators are NP-complete, significantly improving previous complexity bounds by combining a new stuttering theorem and small solution properties.

Contribution

It introduces a new stuttering theorem for Past LTL and demonstrates NP-completeness for these problems, refining the understanding of their computational complexity.

Findings

Reachability and LTL model-checking are NP-complete for flat counter systems with past operators.

A new stuttering theorem for Past LTL is developed.

Small integer solutions for Presburger formulas are utilized to establish complexity bounds.

Abstract

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 the problem is only NP-complete even if LTL admits past-time operators and arithmetical constraints on counters. 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. Other complexity results are proved, for instance for restricted classes of flat counter systems.