Bisimulation and Modal Logics

Modal logic is a family of logics for reasoning about modalities. For example, given a sentence φ, one may wish to express statements such as “φ is necessarily true” or “φ is possibly true”. Modalities are expressed using modal operators. In the most basic modal logic, called propositional modal logic, one may use the operators 2 and 3 to formalize the above two statements as 2φ and 3φ. In this paper, we are interested in one important notion called bisimulation. In modal logic, bisimulation is the fundamental concept of equivalence between structures, originally introduced by van Benthem [1]. A model-theoretic viewpoint of bisimulation investigates the preservation of the truth of a modal formula under a class of structures; this notion of preservation is also referred as bisimulation invariance. Outside of the mathematics community, bisimulation has also had significant impact on theoretical computer science, especially in areas such as automata theory and verification of finite-state systems. We begin our discussion by formalizing bisimulation and its related concepts. Then, we introduce three different types of modal logics—in the order of increasing expressiveness, the propositional modal logic (ML), linear and branching time logics (LTL, CTL, and CTL∗) and, the fixed point logic (μ-calculus). All of these logics share a feature that makes them suitable for expressing properties of a finitestate system; that is, every property definable in each of the logics is bisimulationinvariant. Another way to characterize these modal logics is that they express the observational behaviours of a finite-state system. Given two systems M and M ′, their internal configurations may differ, but from an observer’s point of view, M and M ′ are distinguishable only by their external behaviours. This characterization of bisimulation as a concept of behavioural equivalence has sustained much interest in computer science, especially among researchers who are interested in studying abstract state machines as models of computation.