Succinct Graph Representations of μ-Calculus Formulas

Authors Clemens Kupke, Johannes Marti, Yde Venema

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Author Details

Clemens Kupke
  • University of Strathclyde, UK
Johannes Marti
  • University of Amsterdam, The Netherlands
Yde Venema
  • University of Amsterdam, The Netherlands

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Clemens Kupke, Johannes Marti, and Yde Venema. Succinct Graph Representations of μ-Calculus Formulas. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Many algorithmic results on the modal mu-calculus use representations of formulas such as alternating tree automata or hierarchical equation systems. At closer inspection, these results are not always optimal, since the exact relation between the formula and its representation is not clearly understood. In particular, there has been confusion about the definition of the fundamental notion of the size of a mu-calculus formula. We propose the notion of a parity formula as a natural way of representing a mu-calculus formula, and as a yardstick for measuring its complexity. We discuss the close connection of this concept with alternating tree automata, hierarchical equation systems and parity games. We show that well-known size measures for mu-calculus formulas correspond to a parity formula representation of the formula using its syntax tree, subformula graph or closure graph, respectively. Building on work by Bruse, Friedmann & Lange we argue that for optimal complexity results one needs to work with the closure graph, and thus define the size of a formula in terms of its Fischer-Ladner closure. As a new observation, we show that the common assumption of a formula being clean, that is, with every variable bound in at most one subformula, incurs an exponential blow-up of the size of the closure. To realise the optimal upper complexity bound of model checking for all formulas, our main result is to provide a construction of a parity formula that (a) is based on the closure graph of a given formula, (b) preserves the alternation-depth but (c) does not assume the input formula to be clean.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Logic and verification
  • modal mu-calculus
  • model checking
  • alternating tree automata
  • hierachical equation systems


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