Abstract
In our previous study, we defined a semantics of modal μ-calculus based on min-plus algebra N ∞ and developed a model-checking algorithm. N ∞ is the set of all natural numbers and infinity (∞), and has two operations min and plus. In our semantics, disjunctions are interpreted by min and conjunctions by plus. This semantics allows interesting properties of a Kripke structure to be expressed using simple formulae. In this study, we investigate the satisfiability problem in the N ∞ semantics and show decidability and undecidability results: the problem is decidable if the logic does not contain the implication operator, while it becomes undecidable if we allow the implication operator.
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References
Kozen, D.: Results on the propositional μ-calculus. Theoret. Comput. Sci. 27(3), 333–354 (1983)
Gurfinkel, A., Chechik, M.: Multi-valued model checking via classical model checking. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 263–277. Springer, Heidelberg (2003)
Bruns, G., Godefroid, P.: Model checking with multi-valued logics. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 281–293. Springer, Heidelberg (2004)
Kameyama, Y., Kinoshita, Y., Nishizawa, K.: Weighted Kripke structures and refinement of models. In: 23rd Conference of Japan Society for Software Science and Technology (2006)
Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380(1-2), 69–86 (2007)
Meinecke, I.: A weighted μ-calculus on words. In: Developments in Language Theory, 13th International Conference, DLT 2009, pp. 384–395 (2009)
Ikarashi, D., Tanabe, Y., Nishizawa, K., Hagiya, M.: Modal μ-calculus on min-plus algebra N ∞ . In: 10th Workshop on Programming and Programming Languages (PPL 2008), Japanese Society on Software Science and Technology, pp. 216–230 (2008)
Simon, I.: Limited subsets of a free monoid. In: 19th Annual Symposium on Foundations of Computer Science, pp. 143–150 (1978)
Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. John Wiley & Sons, Chichester (1992)
Ikarashi, D., Tanabe, Y., Nishizawa, K., Hagiya, M.: Modal μ-calculus on min-plus algebra N-infinity. In: Computer Software, Japan Society for Software Science and Technology (to appear)
Post, E.L.: A variant of a recursively unsolvable problem. Bull. Amer. Math. Soc. 52(4), 264–268 (1946)
Zappe, J.: Modal μ-calculus and alternating tree automata. In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata, Logics, and Infinite Games. LNCS, vol. 2500, pp. 171–184. Springer, Heidelberg (2002)
Tanabe, Y., Hagiya, M.: Fixed-point computations over functions on integers with operations min, max and plus. In: 6th Workshop on Fixed Points in Computer Science (FICS 2009), pp. 108–115 (2009)
Goyet, A., Hagiya, M., Tanabe, Y.: Decidability and undecidability results of modal μ-calculi with N∞ semantics. In: PRO Workshop, Information Processing Society of Japan (June 2009), http://cent.xii.jp/tanabe.yoshinori/09/06/ninfmu.pdf
Tanabe, Y., Hagiya, M.: Games and natural number-valued semantics of the modal μ-calculus. In: 26th Conference of Japan Society for Software Science and Technology (2009), http://cent.xii.jp/tanabe.yoshinori/09/09/72.pdf
Kupferman, O., Vardi, M.Y.: Weak alternating automata and tree automata emptiness. In: 30th Annual ACM Symposium on the Theory of Computing, pp. 224–233 (1998)
Wilke, T.: Alternating tree automata, parity games, and modal μ-calculus. Bull. Soc. Math. Belg. 8(2) (2001)
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Goyet, A., Hagiya, M., Tanabe, Y. (2010). Decidability and Undecidability Results on the Modal μ-Calculus with a Natural Number-Valued Semantics . In: Dawar, A., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2010. Lecture Notes in Computer Science(), vol 6188. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13824-9_13
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DOI: https://doi.org/10.1007/978-3-642-13824-9_13
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