Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective

Abstract

One of van Benthem’s seminal results is the Bisimulation Theorem characterizing modal logic as the bisimulation-invariant fragment of first-order logic. Janin and Walukiewicz extended this theorem to include fixpoint operators, showing that the modal ${\mu }$-calculus ${\mu }$ML is the bisimulation-invariant fragment of monadic second-order logic MSO. Their proof uses parity automata that operate on Kripke models, and feature a transition map defined in terms of certain fragments of monadic first-order logic. In this paper we decompose their proof in three parts: (1) two automata-theoretic characterizations, of MSO and $\mu$ML respectively, (2) a simple model-theoretic characterization of the identity-free fragment of monadic first-order logic, and (3) an automata-theoretic result, stating that (a strong version of) the second result somehow propagates to the level of full fixpoint logics. Our main contribution shows that the third result is an instance of a more general phenomenon that is essentially coalgebraic in nature. We prove that if one set $\varLambda$ of predicate liftings (or modalities) for a certain set functor $T$ uniformly corresponds to the $T$-natural fragment of another such set $\varLambda ^{\prime }$, then the fixpoint logic associated with $\varLambda$ is the bisimulation-invariant logic of the fixpoint logic associated with $\varLambda ^{\prime }$.