Describing the Wadge Hierarchy for the Alternation Free Fragment of μ-Calculus (I)
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
The Variable Hierarchy for the Lattice μ-Calculus
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
ICLA '09 Proceedings of the 3rd Indian Conference on Logic and Its Applications
Electronic Notes in Theoretical Computer Science (ENTCS)
Directed graphs of entanglement two
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Efficient Algorithms for Games Played on Trees with Back-edges
Fundamenta Informaticae
Entanglement and the complexity of directed graphs
Theoretical Computer Science
Solving infinite games on trees with back-edges
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
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Most of the logics commonly used in verification, such as LTL, CTL, CTL*, and PDL can be embedded into the two-variable fragment of the μ-calculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question of whether the number of fixed-point variables in μ-formulae can be bounded in general. We answer this question negatively and prove that the variable-hierarchy of the μ-calculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikh's Game Logic is less expressive than the μ-calculus, thus resolving an open issue raised by Parikh in~1983.