Verification of Gap-Order Constraint Abstractions of Counter Systems

Abstract

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) [27]. GCS extend monotonicity constraint systems [5], integral relation automata [12], and constraint automata in [15]. First, we show that checking the existence of infinite runs in GCS satisfying acceptance conditions à la Büchi (fairness problem) is decidable and Pspace-complete. Next, we consider a constrained branching-time logic, GCCTL*, obtained by enriching CTL* with GC, thus enabling expressive properties and subsuming the setting of [12]. 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 the 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 [12], but for which no elementary upper bounds are known.