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
Efficient Debugging in a Formal Verification Environment
CHARME '01 Proceedings of the 11th IFIP WG 10.5 Advanced Research Working Conference on Correct Hardware Design and Verification Methods
Specification and verification of concurrent systems in CESAR
Proceedings of the 5th Colloquium on International Symposium on Programming
NUSMV: A New Symbolic Model Verifier
CAV '99 Proceedings of the 11th International Conference on Computer Aided Verification
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)
Graded-CTL: Satisfiability and Symbolic Model Checking
ICFEM '09 Proceedings of the 11th International Conference on Formal Engineering Methods: Formal Methods and Software Engineering
Graded computation tree logic with binary coding
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
Graded alternating-time temporal logic
LPAR'10 Proceedings of the 16th international conference on Logic for programming, artificial intelligence, and reasoning
Graded CTL model checking for test generation
Proceedings of the 2011 Symposium on Theory of Modeling & Simulation: DEVS Integrative M&S Symposium
A NuSMV extension for Graded-CTL model checking
CAV'10 Proceedings of the 22nd international conference on Computer Aided Verification
ACM Transactions on Computational Logic (TOCL)
Fundamenta Informaticae - Advances in Computational Logic (CIL C08)
| Hi-index | 0.00 |
The use of the universal and existential quantifiers with the capability to express the concept of at leastkor all butk, 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 $\mathcal{K}$ and give an algorithm to solve the model-checking problem running in time $O(|\mathcal{K}|\cdot |\varphi|)$ 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 φgiven 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-CTLlogic, and have proved that the model-checking problem is solvable in polynomial time.