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Completeness for mu-calculi: A coalgebraic approach
Stockholm University, Faculty of Humanities, Department of Philosophy.
Number of Authors: 32019 (English)In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 170, no 5, p. 578-641Article in journal (Refereed) Published
Abstract [en]

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 transfer to the level of the full coalgebraic modal mu-calculus.

Place, publisher, year, edition, pages
2019. Vol. 170, no 5, p. 578-641
Keywords [en]
Modal fixpoint logic, Completeness, Coalgebra, Automata, Graded modal mu-calculus, Monotone modal mu-calculus
National Category
Mathematics Computer and Information Sciences
Identifiers
URN: urn:nbn:se:su:diva-168323DOI: 10.1016/j.apal.2018.12.004ISI: 000462810400003OAI: oai:DiVA.org:su-168323DiVA, id: diva2:1313470
Available from: 2019-05-03 Created: 2019-05-03 Last updated: 2019-05-03Bibliographically approved

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