Verification of Gap-Order Constraint Abstractions of Counter Systems

We investigate verification problems for gap-order constraint systems (GCS), an (infinitely-branching) abstract model of counter machines, in which constraints (over Z) between the variables of the source state and the target state of a transition are gap-order constraints (GC) [21]. GCS extend monotonicity constraint systems [3], integral relation automata [8], and constraint automata in [11]. As specification language, we consider a branching-time logic, GCCTL∗, obtained by enriching CTL∗ with GC, thus enabling expressive properties and subsuming the setting of [8]. We establish that, while model-checking GCS against the universal fragment of GCCTL∗ is undecidable, model-checking against the existential fragment, and satisfiability of both universal and existential fragments are instead decidable and PSPACE-complete (note that the two fragments are not dual since GC are not closed under negation). Moreover, our results imply PSPACE-completeness of the verification problems investigated and shown to be decidable in [8], but for which no elementary upper bounds are known.