Completeness for μ-calculi: A coalgebraic approach |
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Authors: | Sebastian Enqvist Fatemeh Seifan Yde Venema |
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Institution: | 1. Department of Philosophy, Stockholm University, Universitetsvägen 10D, Frescati, Stockholm, Sweden;2. Department of Computer Science, University of Erlangen–Nürnberg, Martensstraat 3, 91058 Erlangen, Germany;3. Institute for Logic, Language and Computation, University of Amsterdam, Science Park 105, 1098 XG Amsterdam, Netherlands |
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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 transfer to the level of the full coalgebraic modal mu-calculus. |
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Keywords: | 03B45 03B70 68Q60 91A43 Modal fixpoint logic Completeness Coalgebra Automata Graded modal mu-calculus Monotone modal mu-calculus |
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