The modal μ-calculus hierarchy over restricted classes of transition systems
L. Alberucci and A. Facchini
Source: J. Symbolic Logic Volume 74, Issue 4 (2009), 1367-1400.
Abstract
We study the strictness of the modal μ-calculus hierarchy over some restricted classes of transition systems. First, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment. Finally, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748696
Digital Object Identifier: doi:10.2178/jsl/1254748696
Zentralblatt MATH identifier:
05654335
Mathematical Reviews number (MathSciNet):
MR2583825
References
L. Alberucci, Strictness of the modal $\mu$-calculus hierarchy, In Grädel et al. [gradel?], pp. 185--201.
Mathematical Reviews (MathSciNet):
MR2070742
Zentralblatt MATH:
1021.03012
Digital Object Identifier: doi:10.1007/3-540-36387-4_11
--------, A syntactical treatment of simultaneous fixpoints in the modal $\mu$-calculus, Technical report, University of Berne, 2008.
L. Alberucci and A. Facchini, On modal $\mu$-calculus and Gödel--Löb logic, Studia Logica, vol. 91 (2009), pp. 145--169.
Mathematical Reviews (MathSciNet):
MR2495542
Digital Object Identifier: doi:10.1007/s11225-009-9170-9
L. Alberucci and J. Krähenbühl, Justifying induction on modal $\mu$-formulae, in preparation.
A. Arnold, The $\mu$-calculus alternation-depth hierarchy is strict on binary trees, ITA, vol. 33 (1999), no. 4/5, pp. 329--340.
Mathematical Reviews (MathSciNet):
MR1748659
Digital Object Identifier: doi:10.1051/ita:1999121
Zentralblatt MATH:
0945.68118
P. Blackburn, M. de Rijke, and Y. Venema, Modal logic, Cambridge University Press, 2001.
Mathematical Reviews (MathSciNet):
MR1837791
J. Bradfield, The modal $\mu$-calculus alternation hierarchy is strict, Theoretical Computer Science, vol. 195 (1998), no. 2, pp. 133--153.
Mathematical Reviews (MathSciNet):
MR1609327
Zentralblatt MATH:
0915.03017
Digital Object Identifier: doi:10.1016/S0304-3975(97)00217-X
--------, Simplifying the modal $\mu$-calculus alternation hierarchy, STACS, 1998, 1998, pp. 39--49.
J. Bradfield and C. Stirling, Modal logic and mu-calculi, Handbook of process algebra (J. Bergstra, A. Ponse, and S. Smolka, editors), Elsevier, North-Holland, 2001, pp. 293--332.
Mathematical Reviews (MathSciNet):
MR1843385
Zentralblatt MATH:
1002.03021
Digital Object Identifier: doi:10.1016/B978-044482830-9/50022-9
G. D'Agostino and G. Lenzi, On the $\mu$-calculus over transitive and finite transitive frames, submitted, 2008.
A. Dawar and M. Otto, Modal characterisation theorems over special classes of frames, Annals of Pure and Applied Logic, to appear.
Mathematical Reviews (MathSciNet):
MR2567925
Zentralblatt MATH:
05612542
Digital Object Identifier: doi:10.1016/j.apal.2009.04.002
E. A. Emerson and C. S. Jutla, Tree automata, $\mu$-calculus and determinacy (extended abstract), FOCS, 1991, 1991, pp. 368--377.
E. Grädel, W. Thomas, and T. Wilke (editors), Automata, logics, and infinite games, Lecture Notes in Computer Science, no. 2500, Springer-Verlag, Berlin Heidelberg, 2002.
Mathematical Reviews (MathSciNet):
MR2070731
G. E. Hughes and M. J. Cresswell, A new introduction to modal logic, Routledge, London, 1996.
Mathematical Reviews (MathSciNet):
MR1666445
Zentralblatt MATH:
0855.03002
D. Kirsten, Alternating tree automata and parity games, In Grädel et al. [gradel?], pp. 157--167.
D. Kozen, Results on the propositional mu-calculus, Theoretical Computer Science, vol. 27 (1983), pp. 333--354.
Mathematical Reviews (MathSciNet):
MR731069
Zentralblatt MATH:
0553.03007
Digital Object Identifier: doi:10.1016/0304-3975(82)90125-6
--------, A finite model theorem for the propositional mu-calculus, Studia Logica, vol. 47 (1988), no. 3, pp. 233--241.
Mathematical Reviews (MathSciNet):
MR999779
Digital Object Identifier: doi:10.1007/BF00370554
Zentralblatt MATH:
0667.03019
G. Lenzi, A hierarchy theorem for the $\mu$-calculus, ICALP, 1996, 1996, pp. 87--97.
Mathematical Reviews (MathSciNet):
MR1464442
Zentralblatt MATH:
1045.03516
--------, The transitive $\mu$-calculus is Büchi-definable, WSEAS Transactions on Mathematics, vol. 5 (2006), no. 9, pp. 1021--1026.
Mathematical Reviews (MathSciNet):
MR2245027
--------, Personal communication, February 26t h2008.
R. Mazala, Infinite games, In Grädel et al. [gradel?], pp. 23--38.
Mathematical Reviews (MathSciNet):
MR2070733
Zentralblatt MATH:
1021.91502
Digital Object Identifier: doi:10.1007/3-540-36387-4_2
A. W. Mostowski, Games with forbidden position, Technical Report 78, University of Gdansk, 1991.
D. Scott and J. De Bakker, A theory of programs, unpublished manuscript, 1969, IBM, Vienna.
C. Stirling, Modal and temporal properties of processes, Springer-Verlag, Berlin Heidelberg, 2001.
R. S. Streett and E. A. Emerson, An automata theoretic decision procedure for the propositional $\mu$-calculus, Information and Computation, vol. 81 (1989), no. 3, pp. 249--264.
Mathematical Reviews (MathSciNet):
MR1000065
Zentralblatt MATH:
0671.03023
Digital Object Identifier: doi:10.1016/0890-5401(89)90031-X
A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, vol. 5 (1955), no. 2, pp. 285--309.
Mathematical Reviews (MathSciNet):
MR74376
Zentralblatt MATH:
0064.26004
Project Euclid: euclid.pjm/1103044538
B. ten Cate, G. Fontaine, and T. Litak, Some modal aspects of XPath, Methods for modalities, 2007.
J. van Benthem, Modal frame correspondences and fixed-points, Studia Logica, vol. 83 (2006), pp. 133--155.
Mathematical Reviews (MathSciNet):
MR2250108
Digital Object Identifier: doi:10.1007/s11225-006-8301-9
Zentralblatt MATH:
1106.03017
A. Visser, Uniform interpolation and layered bisimulation, Gödel '96 (Brno, 1996), Lectures Notes in Logic, vol. 6, Springer, Berlin, 1996, pp. 139--164.
Mathematical Reviews (MathSciNet):
MR1441107
Zentralblatt MATH:
0854.03026
--------, Löb's logic meets the $\mu$-calculus, Processes, terms and cycles, steps on the road to infinity, essays dedicated to Jan Willem Klop on the occasion of his 60th birthday (A. Middeldorp, V. van Oostrom, F. van Raamsdonk, and R. de Vrijer, editors), Springer, Berlin, 2005, pp. 14--25.
I. Walukiewicz, Monadic second-order logic on tree-like structures, Theoretical Computer Science, vol. 275 (2002), no. 1--2, pp. 311--346.
Mathematical Reviews (MathSciNet):
MR1902097
Zentralblatt MATH:
1026.68087
Digital Object Identifier: doi:10.1016/S0304-3975(01)00185-2
T. Wilke, Alternating tree automata, parity games, and modal $\mu$-calculus, Bulletin of the Belgian Mathematical Society, vol. 8 (2001), no. 2, pp. 359--391.
Mathematical Reviews (MathSciNet):
MR1838939
Zentralblatt MATH:
0994.68079
Project Euclid: euclid.bbms/1102714178
