TY - GEN

T1 - Succinct graph representations of µ-calculus formulas

AU - Kupke, Clemens

AU - Marti, Johannes

AU - Venema, Yde

PY - 2022/2/19

Y1 - 2022/2/19

N2 - 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.

AB - 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.

KW - alternating tree automata

KW - hierachical equation systems

KW - modal mu-calculus

KW - model checking

UR - https://arxiv.org/abs/2010.14430

UR - http://www.scopus.com/inward/record.url?scp=85124241076&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CSL.2022.29

DO - 10.4230/LIPIcs.CSL.2022.29

M3 - Conference contribution book

AN - SCOPUS:85124241076

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 30th EACSL Annual Conference on Computer Science Logic, CSL 2022

A2 - Manea, Florin

A2 - Simpson, Alex

CY - Dagstuhl, Germany

T2 - 30th EACSL Annual Conference on Computer Science Logic, CSL 2022

Y2 - 14 February 2022 through 19 February 2022

ER -