PP201633: Enqvist, Sebastian and Seifan, Fatemeh and Venema, Yde (2016) Completeness for the modal mucalculus: separating the combinatorics from the dynamics. [Report]
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Abstract
The modal mucalculus is a very expressive formalism extending basic modal logic with least and greatest fixpoint operators. In the seminal paper introducing the formalism in the shape known today, Dexter Kozen also proposed an elegant axiom system, and he proved a partial completeness result with respect to the Kripkestyle semantics of the logic.The problem of proving Kozen's axiom system complete for the full language remained open for about a decade, until it was finally resolved by Igor Walukiewicz. Walukiewicz' proof is notoriously difficult however, and the result has remained somewhat isolated from the standard theory of completeness for modal (fixpoint) logics. Our aim in this paper is to develop a framework that will let us clarify and simplify parts of Walukiewicz's proof. We hope that this will also help to facilitate future research into completeness of modal fixpoint logics, including fragments, variants and extensions of the modal mucalculus. Our main contribution is to take the automatatheoretic viewpoint, already implicit in Walukiewicz's proof, much more seriously by bringing automata explicitly into the proof theory. Thus we further develop the theory of modal parity automata as a mathematical framework for proving results about the modal mucalculus. Once the connection between automata and derivations is in place, large parts of the completeness proof can be reformulated as purely automatatheoretic theorems. From a conceptual viewpoint, our automatatheoretic approach lets us distinguish two key aspects of the mucalculus: the onestep dynamics encoded by the modal operators, and the combinatorics involved in dealing with nested fixpoints. This ``deconstruction'' allows us to work with these two features in a largely independent manner. More in detail, prominent roles in our proof are played by two classes of modal automata: next to the disjunctive automata that are known from the work of Janin & Walukiewicz, we introduce here the class of semidisjunctive automata that roughly correspond to the fragment of the mucalculus for which Kozen proved completeness. We will establish a connection between the proof theory of Kozen's system, and two kinds of games involving modal automata: a satisfiability game involving a single modal automaton, and a consequence game relating two such automata. In the key observations on these games we bring the dynamics and combinatorics of parity automata together again, by proving some results that witness the nice behaviour of disjunctive and semidisjunctive automata in these games. As our main result we prove that every formula of the modal mucalculus provably implies the translation of a disjunctive automaton; from this the completeness of Kozen's axiomatization is immediate.
Item Type:  Report 

Report Nr:  PP201633 
Series Name:  Prepublication (PP) Series 
Year:  2016 
Uncontrolled Keywords:  modal mucalculus, modal fixpoint logic, completeness, parity automata, infinite games 
Subjects:  Logic 
Depositing User:  Yde Venema 
Date Deposited:  17 Dec 2016 22:56 
Last Modified:  17 Dec 2016 22:56 
URI:  https://eprints.illc.uva.nl/id/eprint/1427 
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