Tree automata, Mu-Calculus and determinacy
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
The modal mu-calculus alternation hierarchy is strict
Theoretical Computer Science
An automata-theoretic approach to branching-time model checking
Journal of the ACM (JACM)
Games and Model Checking for Guarded Logics
LPAR '01 Proceedings of the Artificial Intelligence on Logic for Programming
Automata for the Modal mu-Calculus and related Results
MFCS '95 Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science
A Hierarchy Theorem for the µ-Calculus
ICALP '96 Proceedings of the 23rd International Colloquium on Automata, Languages and Programming
Small Progress Measures for Solving Parity Games
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
Ambiguous classes in µ-calculi hierarchies
Theoretical Computer Science - Foundations of software science and computation structures
Undirected graphs of entanglement 2
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
The variable hierarchy of the µ-calculus is strict
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Entanglement and the complexity of directed graphs
Theoretical Computer Science
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We investigate the structure of the modal 碌-calculus L碌 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 碌-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., 驴-free) formulae. It is known that these characterize precisely the L碌-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.