On the Complexity of Verifying Regular Properties on Flat Counter Systems,

Abstract

Abstract. Among the approximation methods for the verification of counter systems, one of them consists in model-checking their flat unfoldings. Unfortunately, the complexity characterization of model-checking problems for such operational models is not always well studied except for reachability queries or for Past LTL. In this paper, we characterize the complexity of model-checking problems on flat counter systems for the specification languages including first-order logic, linear mu-calculus, infinite automata, and related formalisms. Our results span different complexity classes (mainly from PTime to PSpace) and they apply to languages in which arithmetical constraints on counter values are systematically allowed. As far as the proof techniques are concerned, we provide a uniform approach that focuses on the main issues.

How Hard is It to Verify Flat Affine Counter Systems with the Finite Monoid Property?

“…For a FO formula φ, its quantifier height qh(φ) is the maximal nesting depth of its quantifiers, and the size of φ is its number of subformulae. Similarly, as for the PLTL case, in [8,Theorem 6], a stuttering theorem for FO is provided, which says that that two ω-words w = w 1 w M 2 w 3 and w = w 1 w M ′ 2 w 3 with w = ǫ are indistinguishable by a FO formula φ if M and M ′ are strictly bigger than 2 qh(φ)+2 . The main difference with PLTL is that this provides an exponential bound in the maximum number of times an infix of an ω-word needs to be repeated to satisfy a FO formula.…”

“…In the sequel we consider a flat counter system S = Q, X n , ∆, Λ with the finite monoid property and we reuse the notations introduced in the previous section. The results of [8] can be restated as follows. As for the PLTL case, this allows us to deduce a NPSPACE algorithm for the model-checking problem of flat counter system with the finite monoid property with FO formulae.…”

“…For instance, in the case of Kripke structures, flatness reduces the complexity of the model-checking of Linear Temporal Logic (LTL) from PSPACE to NP [19]. When considering flat counter systems whose updates are described by translations, the complexity of these problems drops from undecidable to NP-complete [10], while model checking for First Order Logic (FO) is coined to PSPACE-complete [8]. For branching time temporal logics, flatness yields decidable problems, but with less remarkable complexity bounds [9].…”

Equivalence Between Model-Checking Flat Counter Systems and Presburger Arithmetic

“…[3,6,14,4], but it is unclear how this can be extended to model-checking problems with temporal logics, which indeed, is done in [10] for flat counter systems. A complexity characterization for model-checking linear-time properties is provided in [9]. In the present paper, we study flat counter systems and branching-time temporal logics, more specifically with a variant of CTL * [12], already known to be difficult to mechanize in the propositional case with labelled transition systems.…”

“…[9,4]) and in this paper we wish to understand the computational complexity for branching-time temporal logics such as CTL or CTL * (see e.g. [12]).…”