Abstract
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.
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Berwanger, D., Gradel, E. & Lenzi, G. The Variable Hierarchy of the μ-Calculus Is Strict. Theory Comput Syst 40, 437–466 (2007). https://doi.org/10.1007/s00224-006-1317-8
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DOI: https://doi.org/10.1007/s00224-006-1317-8