## Abstract

The μ-calculus is an expressive specification language in which modal logic is extended with fixpoint operators, subsuming many dynamic, temporal, and description logics. Formulas of μ-calculus are classified according to their alternation depth, which is the maximal length of a chain of nested alternating least and greatest fixpoint operators. Alternation depth is the major factor in the complexity of μ-calculus model-checking algorithms. A refined classification of μ-calculus formulas distinguishes between formulas in which the outermost fixpoint operator in the nested chain is a least fixpoint operator (Σ_{i} formulas, where i is the alternation depth) and formulas where it is a greatest fixpoint operator (Π_{i} formulas). The alternation-free μ-calculus (AFMC) consists of μ-calculus formulas with no alternation between least and greatest fixpoint operators. Thus, AFMC is a natural closure of Σ_{1} ∪ Π_{1}, which is contained in both Σ_{2} and Π_{2}. In this work we show that Σ_{2} ∩ Π_{1} ≡ AFMC. In other words, if we can express a property ξ both as a least fixpoint nested inside a greatest fixpoint and as a greatest fixpoint nested inside a least fixpoint, then we can express ξ also with no alternation between greatest and least fixpoints. Our result refers to μ-calculus over arbitrary Kripke structures. A similar result, for directed μ-calculus formulas interpreted over trees with a fixed finite branching degree, follows from results by Arnold and Niwinski. Their proofs there cannot be easily extended to Kripke structures, and our extension involves symmetric nondeterministic Büchi tree automata, and new constructions for them.

Original language | American English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Editors | Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, Gerhard J. Woeginger |

Publisher | Springer Verlag |

Pages | 697-713 |

Number of pages | 17 |

ISBN (Print) | 3540404937, 9783540404934 |

DOIs | |

State | Published - 2003 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2719 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

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