CTL Model-Checking with Graded Quantifiers

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. 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. 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). 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.