CTLModel-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 nonnegative 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 μcalculus, 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 K and give an algorithm to solve the model-checking problem running in time O(|K| · |φ|) 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