Yakoub Salhi
ABSTRACT
This article aims at studying different aspect of the satisfiability problem in the modal logic S5. We first introduce a new resolution method that is syntactically pure in the sense that does not use explicitly semantic information. Then, we propose simplification rules that can be applied during preprocessing and solving. Some of these rules can be seen as adaptations of existing simplification rules for CNF formulas in classical propositional logic. Finally, we argue in favor of modeling in S5 to solve NP-complete problems. Indeed, we provide encodings that allow us to solve three different well-known NP-complete problems: graph coloring, Hamiltonian path, and closest string. Our models in S5 show in particular that the possible-worlds semantics allows solving NP-complete problems with fewer propositional variables than in classical propositional logic.
- Olivier Bailleux, Yacine Boufkhad, and Olivier Roussel. 2006. A Translation of Pseudo Boolean Constraints to SAT. JSAT 2, 1-4 (2006), 191--200.Google Scholar
- Meghyn Bienvenu, Hélène Fargier, and Pierre Marquis. 2010. Knowledge Compilation in the Modal Logic S5. In Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2010, 2010. AAAI Press, Atlanta, Georgia, USA. http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/view/1696Google ScholarCross Ref
- Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh (Eds.). 2009. Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, Vol. 185. IOS Press.Google ScholarDigital Library
- Patrick Blackburn, Maarten de Rijke, and Yde Venema. 2001. Modal Logic. Cambridge University Press.Google Scholar
- Kai Brünnler. 2009. Deep sequent systems for modal logic. Archive for Mathematical Logic 48, 6 (2009), 551--577.Google ScholarCross Ref
- James M. Crawford and Larry D. Auton. 1993. Experimental Results on the Crossover Point in Satisfiability Problems. In Proceedings of the 11th National Conference on Artificial Intelligence (AAAI-93). AAAI Press / The MIT Press, Washington, DC, USA, 21--27.Google Scholar
- Marcello D'Agostino, Dov M. Gabbay, Reiner Hähnle, and Joachim Posegga (Eds.). 1999. Handbook of Tableau Methods. Kluwer Academic Publishers.Google Scholar
- Luis Fariñas del Cerro and Martti Penttonen. 1987. A Note of the Complexity of the Satisfiability of Modal Horn Clauses. The Journal of Logic Programming. 4, 1 (1987), 1--10.Google ScholarDigital Library
- Patrice Enjalbert and Luis Fariñas del Cerro. 1989. Modal Resolution in Clausal Form. Theoretical Computer Science 65, 1 (1989), 1 -- 33.Google ScholarDigital Library
- Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. 2003. Reasoning About Knowledge. The MIT Press.Google Scholar
- Melvin Fitting. 1983. Proof Methods for Modal and Intuitionistic Logics. Synthese Library, Vol. 169. Kluwer.Google Scholar
- Didier Galmiche and Yakoub Salhi. 2010. Label-free natural deduction systems for intuitionistic and classical modal logics. Journal of Applied Non-Classical Logics 20, 4 (2010), 373--421.Google ScholarCross Ref
- Andreas Herzig, Jérôme Lang, and Pierre Marquis. 2003. Action representation and partially observable planning using epistemic logic. In IJCAI-03, Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, 2003. Morgan Kaufmann, Acapulco, Mexico, 1067--1072.Google Scholar
- Richard E. Ladner. 1977. The Computational Complexity of Provability in Systems of Modal Propositional Logic. SIAM J. Comput. 6, 3 (1977), 467--480.Google ScholarDigital Library
- Jérôme Lang and Bruno Zanuttini. 2013. Knowledge-Based Programs as Plans: Succinctness and the Complexity of Plan Existence. In Proceedings of the 14th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2013), 2013. Chennai, India. http://www.tark.org/proceedings/tark_jan7_13/p138-lang.pdfGoogle Scholar
- Cláudia Nalon and Clare Dixon. 2007. Clausal resolution for normal modal logics. Journal of Algorithms 62, 3-4 (2007), 117--134.Google ScholarDigital Library
- Sara Negri. 2005. Proof Analysis in Modal Logic. Journal of Philosophical Logic 34, 5-6 (2005), 507--544.Google ScholarCross Ref
- Dag Prawitz. 1965. Natural deduction: a proof-theoretical study. Ph.D. Dissertation. Almqvist & Wiksell.Google Scholar
- Yakoub Salhi and Michael Sioutis. 2015. A Resolution Method for Modal Logic S5. In Global Conference on Artificial Intelligence, GCAI 2015. EasyChair, Tbilisi, Georgia, 252--262.Google Scholar
- Carsten Sinz. 2005. Towards an Optimal CNF Encoding of Boolean Cardinality Constraints. In 11th International Conference on Principles and Practice of Constraint Programming - CP 2005. Springer, Sitges (Barcelona), Spain, 827--831.Google ScholarDigital Library
- G. S. Tseitin. 1968. On the Complexity of Derivations in the Propositional Calculus. In Studies in Constructives Mathematics and Mathematical Logic, Part II, H.A.O. Slesenko (Ed.). Steklov Mathematical Institute, 115--125.Google Scholar
- Heinrich Wansing. 1994. Sequent Calculi for Normal Modal Proposisional Logics. Journal of Logic and Computation 4, 2 (1994), 125--142.Google ScholarCross Ref
Index Terms
- On satisfiability problem in modal logic S5
Recommendations
Ground Nonmonotonic Modal Logic S5: New Results
We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T ...
Falsification-Aware Calculi and Semantics for Normal Modal Logics Including S4 and S5
AbstractFalsification-aware (hyper)sequent calculi and Kripke semantics for normal modal logics including S4 and S5 are introduced and investigated in this study. These calculi and semantics are constructed based on the idea of a falsification-aware ...
Natural Deduction for Full S5 Modal Logic with Weak Normalization
Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [Prawitz, D., ''Natural Deduction: A Proof-theoretical Study'', Stockholm Studies in Philosophy 3, Almqvist and Wiksell, Stockholm, 1965] from a ...
Comments