Automatic verification of finite-state concurrent systems using temporal logic specifications
ACM Transactions on Programming Languages and Systems (TOPLAS)
IEEE Transactions on Software Engineering - Special issue on formal methods in software practice
Model checking
Complexity Results for First-Order Two-Variable Logic with Counting
SIAM Journal on Computing
Introduction to algorithms
Model checking of hierarchical state machines
ACM Transactions on Programming Languages and Systems (TOPLAS)
Specification and verification of concurrent systems in CESAR
Proceedings of the 5th Colloquium on International Symposium on Programming
The Complexity of the Graded µ-Calculus
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
Two-variable logic with counting is decidable
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
The Two-Variable Guarded Fragment with Transitive Relations
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Packing cycles in undirected graphs
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
A framework for counterexample generation and exploration
International Journal on Software Tools for Technology Transfer (STTT)
Verification of scope-dependent hierarchical state machines
Information and Computation
Graded alternating-time temporal logic
LPAR'10 Proceedings of the 16th international conference on Logic for programming, artificial intelligence, and reasoning
A NuSMV extension for Graded-CTL model checking
CAV'10 Proceedings of the 22nd international conference on Computer Aided Verification
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Recently, 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 ([19]). 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| · |ϕ|) which is hence tight for the problem (here |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 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 graded-CTL, and have proved that the model-checking problem is solvable in polynomial time.