Model Checking for Graded CTL

Recently, complexity issues related to the decidability of the $\mu$-calculus, when the universal and existential quantifiers are augmented with `graded modalities`, have been investigated by Kupfermann, Sattler and Vardi ([KSV02]). Graded modalities refer to the use of the universal and existential quantifiers with the added capability to express the concept of `at least` $k$ or `all but` $k$, for a non-negative integer $k$. In this paper we study the Computational Tree Logic CTL, a branching time extension of classical modal logic, augmented with graded modalities and investigate the complexity issues with respect to the model-checking problem. We consider a system model represented by a Kripke structure $K$ and give an algorithm to solve the model-checking problem running in time $O(|K|\cdot |\varphi|)$ which is hence tight for the problem (here $|\varphi|$ 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 to a specification $\varphi$ 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 graded-\ctl, and have proved that the model-checking problem is solvable in polynomial time.