Completeness for the modal μ-calculus: separating the combinatorics from the dynamics

The modal mu-calculus 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 Kripke-style 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 mu-calculus. Our main contribution is to take the automata-theoretic 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 mu-calculus. Once the connection between automata and derivations is in place, large parts of the completeness proof can be reformulated as purely automata-theoretic theorems. From a conceptual viewpoint, our automata-theoretic approach lets us distinguish two key aspects of the mu-calculus: the one-step 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 semi-disjunctive automata that roughly correspond to the fragment of the mu-calculus 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 semi-disjunctive automata in these games. As our main result we prove that every formula of the modal mu-calculus provably implies the translation of a disjunctive automaton; from this the completeness of Kozen’s axiomatization is immediate. Mathematics Subject Classification (MSC2010): 03B45; 03B70; 68Q60; 91A43.