The Modal-Calculus Caught Off Guard

Oliver Friedmann, Martin Lange

Abstract: The modal μ-calculus extends basic modal logic with second-order quantification in terms of arbitrarily nested fixpoint operators. Its satisfiability problem is EXPTIME-complete. Decision procedures for the modal μ-calculus are not easy to obtain though since the arbitrary nesting of fixpoint constructs requires some combinatorial arguments for showing the well-foundedness of least fixpoint unfoldings. The tableau-based decision procedures so far also make assumptions on the unfoldings of fixpoint formulas, e.g. explicitly require formulas to be in guarded normal form. In this paper we present a tableau calculus for deciding satisfiability of arbitrary, i.e. not necessarily guarded μ-calculus formulas. The novel contribution is a new unfolding rule for greatest fixpoint formulas which shows how to handle unguardedness without an explicit transformation into guarded form, thus avoiding a (seemingly) exponential blow-up in formula size. We prove soundness and completeness of the calculus, and discuss its advantages over existing approaches.