Abstract
We investigate the structure of the modal μ-calculus L μ with respect to the question of how many different fixed point variables are necessary to define a given property. Most of the logics commonly used in verification, such as CTL, LTL, CTL*, PDL, etc. can in fact be embedded into the two-variable fragment of the μ-calculus. It is also known that the two-variable fragment can express properties that occur at arbitrarily high levels of the alternation hierarchy. However, it is an open problem whether the variable hierarchy is strict.
Here we study this problem with a game-based approach and establish the strictness of the hierarchy for the case of existential (i.e.,□-free) formulae. It is known that these characterize precisely the Lμ-definable properties that are closed under extensions. We also relate the strictness of the variable hierarchy to the question whether the finite variable fragments satisfy the existential preservation theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Andréka, J. Van Benthem, AND I. Németi, Modal languages and bounded fragments of predicate logic, Journal of Philosophical Logic, 27 (1998), 217–274.
A. Arnold, The mu-calculus alternation-depth is strict on binary trees, RAIRO Informatique Théorique et Applications, 33 (1999), 329–339.
A. Arnold AND D. Niwiński, Rudiments of μ-calculus, North Holland, 2001.
D. Berwanger, Game logic is strong enough for parity games, Studia Logica. Special issue on Game Logic and Game Algebra, (2002).
D. Berwanger AND E. Grädel, Games and model checking for guarded logics, in Proceedings of LPAR 2001, Lecture Notes in Computer Science Nr. 2250, Springer, 2001, 70–84.
J. Bradfield, The modal μ-calculus alternation hierarchy is strict, Theoretical Computer Science, 195 (1998), 133–153.
G. D’agostino AND M. Hollenberg, Logical questions concerning the μ-calculus: interpolation, Lyndon, and Los-Tarski, Journal of Symbolic Logic, 65 (2000), 310–332.
A. Emerson AND C. Jutla, Tree automata, mu-calculus and determinacy, in Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, 368–377.
E. Grädel AND E. Rosen, Preservation theorems for two-variable logic, Mathematical Logic Quarterly, 45 (1999), 315–325.
D. Janin AND I. Walukiewicz, Automata for the modal μ-calculus and related results, in Proceedings of MFCS 95, Lecture Notes in Computer Science Nr. 969, Springer-Verlag, 1995, 552–562.
M. Jurdziński, Small progress measures for solving parity games, in STACS 2000, 17th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings, vol. 1770 of Lecture Notes in Computer Science, Springer, 2000, 290–301.
O. Kupferman, M. Vardi, AND P. Wolper, An automata-theoretic approach to branching-time model checking, Journal of the ACM, 47 (2000), 312–360.
G. Lenzi, A hierarchy theorem for the mu-calculus, in Proceedings of the 23rd International Colloquium on Automata, Languages and Programming, ICALP’ 96, F. Meyer auf der Heide and B. Monien, eds., vol. 1099 of Lecture Notes in Computer Science, Springer-Verlag, July 1996, 87–97.
J.. Los, On the extending of models (I), Fundamenta Mathematicae, 42 (1955), 38–54.
R. Parikh, The logic of games and its applications, Annals of Discrete Mathematics, 24 (1985), 111–140.
M. Pauly, Logic for Social Software, PhD thesis, University of Amsterdam, 2001.
C. Stirling, Bisimulation, model checking and other games. Notes for the Mathfit instructional meeting on games and computation. Edinburgh, 1997.
W. W. Tait, A counterexample to a conjecture of Scott and Suppes, Journal of Symbolic Logic, 24 (1959), 15–16.
A. Tarski, Contributions to the theory of models I, II, Indagationes Mathematicae, 16 (1954), 572–588.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berwanger, D., Grädel, E., Lenzi, G. (2002). On the Variable Hierarchy of the Modal μ-Calculus. In: Bradfield, J. (eds) Computer Science Logic. CSL 2002. Lecture Notes in Computer Science, vol 2471. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45793-3_24
Download citation
DOI: https://doi.org/10.1007/3-540-45793-3_24
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44240-0
Online ISBN: 978-3-540-45793-0
eBook Packages: Springer Book Archive