PP-2017-04: Completeness for mu-calculi: a coalgebraic approach

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 [Pre-print] (Unpublished)

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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
Report Nr: PP-2017-04
Series Name: Prepublication (PP) Series
Year: 2017
Subjects: Computation
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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