## Abstract

The μ-calculus model-checking problem has been of great interest in the context of concurrent programs. Beyond the need to use symbolic methods in order to cope with the state-explosion problem, which is acute in concurrent settings, several concurrency related problems are naturally solved by evaluation of μ-calculus formulas. The complexity of a naive algorithm for model checking a μ-calculus formula ψ is exponential in the alternation depth d of ψ. Recent studies of the μ-calculus and the related area of parity games have led to algorithms exponential only in d/2. No symbolic version, however, is known for the improved algorithms, sacrificing the main practical attraction of the μ-calculus. The μ-calculus can be viewed as a fragment of first-order fixpoint logic. One of the most fundamental theorems in the theory of fixpoint logic is the Collapse Theorem, which asserts that, unlike the case for the μ-calculus, the fixpoint alternation hierarchy over finite structures collapses at its first level. In this paper we show that the Collapse Theorem of fixpoint logic holds for a measured variant of the μ-calculus, which we call μ^{#}-calculus. While μ-calculus formulas represent characteristic functions, i.e., functions from the state space to {0,1}, formulas of the μ^{#}-calculus represent measure functions, which are functions from the state space to some measure domain. We prove a Measured-Collapse Theorem: every formula in the μ-calculus is equivalent to a least-fixpoint formula in the μ^{#}calculus. We show that the Measured-Collapse Theorem provides a logical recasting of the improved algorithm for μ-calculus model-checking, and describe how it can be implemented symbolically using Algebraic Decision Diagrams. Thus, we describe, for the first time, a symbolic μ-calculus model-checking algorithm whose complexity matches the one of the best known enumerative algorithm.

Original language | English |
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Title of host publication | STACS 2004 |

Subtitle of host publication | 21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings |

Editors | Volker Diekert, Michel Habib |

Publisher | Springer Verlag |

Pages | 522-533 |

Number of pages | 12 |

ISBN (Print) | 3-540-21236-1 |

DOIs | |

State | Published - 2004 |

Event | 21st Annual Symposium on Theoretical Aspects of Computer Science, STACS 2004 - Montpellier, France Duration: 25 Mar 2004 → 27 Mar 2004 Conference number: 21 https://www.lirmm.fr/stacs2004/ |

### Publication series

Name | Lecture Notes in Computer Science |
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Volume | 2996 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st Annual Symposium on Theoretical Aspects of Computer Science, STACS 2004 |
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Country/Territory | France |

City | Montpellier |

Period | 25/03/04 → 27/03/04 |

Internet address |