## Abstract

One source of complexity in the μ-calculus is its ability to specify an unbounded number of switches between universal (AX) and existential (EX) branching modes. We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the μ-calculus, in which only one branching mode is allowed. The universal fragment is rich enough to express most specifications of interest, and therefore improved algorithms are of practical importance. We show that while the satisfiability and validity problems become indeed simpler for the existential and universal fragments, this is, unfortunately, not the case for model checking and implication. We also show the corresponding results for the alternation-free fragment of the μ-calculus, where no alternations between least and greatest fixed points are allowed. Our results imply that efforts to find a polynomial-time model-checking algorithm for the μ-calculus can be replaced by efforts to find such an algorithm for the universal or existential fragment.

Original language | American English |
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Pages (from-to) | 173-186 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 354 |

Issue number | 2 |

DOIs | |

State | Published - 28 Mar 2006 |

### Bibliographical note

Funding Information:A preliminary version of this paper appeared in the Proceedings of the 9th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2003), Lecture Notes in Computer Science, vol. 2619, Springer, Berlin, 2003, pp. 49–64. This work was supported in part by NSF Grant CCR-9988172, the AFOSR MURI Grant F49620-00-1-0327, and a Microsoft Research Fellowship. ∗Corresponding author. E-mail address: [email protected] (R. Majumdar).

## Keywords

- Automata and logic
- Model checking
- Satisfiability
- μ-Calculus