12 Modal mu-calculi

Abstract

Modal mu-calculus is a logic used extensively in certain areas of computer science and is of considerable intrinsic mathematical and logical interest. Its defining feature is the addition of inductive definitions to modal logic; thereby it achieves a great increase in expressive power and an equally great increase in difficulty of understanding. It includes many of the logics used in systems verification, and is quite straightforward to evaluate. It also provides one of the strongest examples of the connections between modal and temporal logics, automata theory, and the theory of games. It provides second-order expressive power sufficient to generalize the most common temporal logics, but is still decidable and has the finite model property. It raises many intriguing issues about the interface between modal logic, complexity theory, and automata theory. This chapter surveys a range of the questions and results about the modal mu-calculus and related logics. The logic is defined, some approaches to gaining an intuitive understanding of formulae are described, and the main theorem about the semantics is established. The modal mu-calculus has the tree model property and relates to some other temporal logics, to automata and to games.