On the Modal μ-Calculus Over Finite Symmetric Graphs
-
Giovanna D’Agostino
and
Giacomo Lenzi
Abstract
In this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.
References
[1] ALBERUCCI, L.-FACCHINI, A.: The modal μ-calculus hierarchy over restricted classes of transition systems, J. Symbolic Logic 74 (2009), 1367-1400.10.2178/jsl/1254748696Search in Google Scholar
[2] ARNOLD, A.: The μ-calculus alternation-depth hierarchy is strict on binary trees, Theoret. Inform. Appl., 33 (1999), 329-339.10.1051/ita:1999121Search in Google Scholar
[3] BARANY, V.-BOJANCZYK, M.: Finite satisfiability for guarded fixpoint logic, Inform. Process. Lett. 112 (2012), 371-375.10.1016/j.ipl.2012.02.005Search in Google Scholar
[4] BRADFIELD, J.: The modal μ-calculus alternation hierarchy is strict, Theoret. Comput. Sci. 195 (1998), 133-153.10.1016/S0304-3975(97)00217-XSearch in Google Scholar
[5] BRADFIELD, J.-STIRLING, C.: Modal μ-calculi. In: The Handbook of Modal Logic (P. Blackburn, J. van Benthem, F. Wolter, eds.), Elsevier, Amsterdam, 2006, pp. 721-756.Search in Google Scholar
[6] D’AGOSTINO, G.-LENZI, G.: On Modal μ-calculus over reflexive symmetric graphs, J. Logic Comput. doi: 10.1093/logcom/exs028.10.1093/logcom/exs028Search in Google Scholar
[7] D’AGOSTINO, G.-LENZI, G.: On the μ-calculus over transitive and finite transitive frames, Theoret. Comput. Sci. 411 (2010), 4273-4290.10.1016/j.tcs.2010.09.002Search in Google Scholar
[8] GRAEDEL, G.-WALUKIEWICZ, I.: Guarded fixed point logic. In: Proc. LICS99, 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999, pp. 45-54.Search in Google Scholar
[9] DAWAR, A.-OTTO,M.: Modal characterisation theorems over special classes of frames, Ann. Pure Appl. Logic 161 (2009), 1-42.10.1016/j.apal.2009.04.002Search in Google Scholar
[10] JANIN, D.-WALUKIEWICZ, I.: On the expressive completeness of the propositional μ-calculus w.r.t. monadic second-order logic. In: CONCUR ’96. Lecture Notes in Comput. Sci. 1119, Springer, Heidelberg, 1996, pp. 263-277.Search in Google Scholar
[11] KOZEN, D.: Results on the propositional μ-calculus, Theoret. Comput. Sci. 27 (1983), 333-354.10.1016/0304-3975(82)90125-6Search in Google Scholar
© 2015
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
