CTL Model-Checking with Graded Quantifiers

Abstract

The use of the universal and existential quantifiers with the capability to express the concept of at least k or all but k, for a non-negative integer k, has been thoroughly studied in various kinds of logics. In classical logic there are counting quantifiers, in modal logics graded modalities, in description logics number restrictions.\ud Recently, the complexity issues related to the decidability of the μ-calculus, when the universal and existential quantifiers are augmented with graded modalities, have been investigated by Kupfermann, Sattler and Vardi. They have shown that this problem is ExpTime-complete.\ud In this paper we consider another extension of modal logic, the Computational Tree Logic CTL, augmented with graded modalities generalizing standard quantifiers and investigate the complexity issues, with respect to the model-checking problem. We consider a system model represented by a pointed Kripke structure and give an algorithm to solve the model-checking problem running in time O() which is hence tight for the problem (where |ϕ| is the number of temporal and boolean operators and does not include the values occurring in the graded modalities).\ud In this framework, the graded modalities express the ability to generate a user-defined number of counterexamples (or evidences) to a specification ϕ given in CTL. However these multiple counterexamples can partially overlap, that is they may share some behavior. We have hence investigated the case when all of them are completely disjoint. In this case we prove that the model-checking problem is both NP-hard and coNP-hard and give an algorithm for solving it running in polynomial space. We have thus studied a fragment of this graded-CTL logic, and have proved that the model-checking problem is solvable in polynomial time

Graded computation tree logic

“…The work continues in Ferrante et al [2009], by showing a symbolic model-checking algorithm for the binary coding and, limited to the unary case, a satisfiability procedure. Regarding the comparison between GCTL and graded CTL with overlapping paths studied in Ferrante et al [2008], it can be shown that they are equivalent by using an exponential reduction in both ways, whereas we do not know whether any of the two blow-ups can be avoided. However, it is important to note that our general technique can be also adapted to obtain an EXPTIME satisfiability procedure for the binary graded CTL under the semantics proposed in Ferrante et al [2008].…”

“…Regarding the comparison between GCTL and graded CTL with overlapping paths studied in Ferrante et al [2008], it can be shown that they are equivalent by using an exponential reduction in both ways, whereas we do not know whether any of the two blow-ups can be avoided. However, it is important to note that our general technique can be also adapted to obtain an EXPTIME satisfiability procedure for the binary graded CTL under the semantics proposed in Ferrante et al [2008]. Indeed, it is needed only to slightly modify the transition function of the main automaton (with respect to until and release formulas), without changing the structure of the whole satellite.…”

“…Graded modalities along with CTL have been also studied in Ferrante et al [2008Ferrante et al [ , 2009, but under a different semantics. There, the authors consider overlapping paths (as we do) as well as disjoint paths, but they neither consider the general framework of equivalence classes over paths nor the particular concepts of minimality and conservativeness, which we deeply analyze in our paper.…”

“…This property can be easily expressed in GCTL. There are also several other practical examples that show the usefulness of GCTL and we refer to Ferrante et al [2008Ferrante et al [ , 2009 for a list of them.…”

“…There, the authors consider overlapping paths (as we do) as well as disjoint paths, but they neither consider the general framework of equivalence classes over paths nor the particular concepts of minimality and conservativeness, which we deeply analyze in our paper. In Ferrante et al [2008] the model-checking problem for nonminimal and nonconservative unary GCTL has been investigated. In particular, by opportunely extending the classical algorithm for CTL [Clarke and Emerson 1981], they show that, in the case of overlapping paths, the model-checking problem is PTIME-COMPLETE (thus not harder than CTL), while in the case of disjoint paths, it is in PSPACE and both NPTIME-HARD and CONPTIME-HARD.…”