Verification of Qualitative ℤ Constraints

Abstract

We introduce an LTL-like logic with atomic formulae built over a constraint language interpreting variables in Z. The constraint language includes periodicity constraints, comparison constraints of the form x = y and x < y, it is closed under Boolean operations and it admits a restricted form of existential quantification. This is the largest set of qualitative constraints over Z known so far, shown to admit a decidable LTL extension. Such constraints are those used for instance in calendar formalisms or in abstractions of counter automata by using congruences modulo some power of two. Indeed, various programming languages perform arithmetic operators modulo some integer. We show that the satisfiability and model-checking problems (with respect to an appropriate class of constraint automata) for this logic are decidable in polynomial space improving significantly known results about its strict fragments. As a by-product, LTL model-checking over integral relational automata is proved complete for polynomial space which contrasts with the known undecidability of its CTL counterpart.

Branching-Time Temporal Logic Extended with Qualitative Presburger Constraints

“…However, full Presburger LTL is undecidable, and to regain decidability, one can either restrict the underlying constraint language, see e.g. [DD03,DG05], or restrict the logical language, see e.g. [BEH95,CC00].…”

“…[BEH95,CC00]. In [DG05], full LTL extended with a wide set of qualitative constraints, including comparison and periodicity constraints, has been shown to have PSPACE-complete satisfiability and model-checking problems (over constraint automata mentioned above). Similar extensions have also been considered for description logics where models are Kripke structures, see for instance [Lut04].…”

“…In this paper, we introduce the logic CCTL * as an extension of the branching-time temporal logic CTL * with a wide set of qualitative constraints including periodicity constraints of the form x ≡ k y + c, comparison constraints of the form x < y and a restricted form of quantification. This logic is the branching-time counterpart of the constraint LTL defined in [DG05] and extends the logic from [Čer93] by introducing richer constraints and the possibility to compare counters at different states of the model. The operational models on which we check temporal properties expressed in this logic are extensions of Integral Relational Automata (IRA, for short) [BBK77,Čer93,ACJT96] introduced in [BBK77] as a model for studying possibilities of automated complete test set generation for data processing programs.…”

“…For this fragment, we show that both satisfiability and model checking of the proposed extension of IRA are decidable. The existential and universal fragments of CCTL * are strictly more expressive than the constraint LTL defined in [DG05]. Moreover, the symbolic algorithm we describe builds a finite representation of the set of states satisfying a given formula, a very substantial information compared to the symbolic representation used in [DG05].…”

“…For instance, Petri nets represent the subclass of counter systems obtained by removing the ability to test a counter for zero. Other examples include reversal-bounded counter machines [Iba78], flat counter systems [Boi98,BFLP03,LS04] and constraint automata with qualitative constraints on Z between the states of variables at different steps of the execution [DG05]. "Qualitative" means that the relationship between the constrained variables is not sharp, like x < y.…”