PP-2017-04:
Enqvist, Sebastian and Seifan, Fatemeh and Venema, Yde
(2017)
*Completeness for mu-calculi: a coalgebraic approach.*
[Pre-print]
(Unpublished)

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## Abstract

We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.

Besides these main applications, our result covers the Kozen-Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.

Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfers to the level of the full coalgebraic modal mu-calculus.

Item Type: | Pre-print |
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Report Nr: | PP-2017-04 |

Series Name: | Prepublication (PP) Series |

Year: | 2017 |

Subjects: | Computation Logic |

Depositing User: | Yde Venema |

Date Deposited: | 21 Mar 2017 12:12 |

Last Modified: | 23 Mar 2017 17:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/1530 |

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