In theoretical computer science, the modal μ-calculus (Lμ, Lμ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic. The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker, and was further developed by Dexter Kozen into the version most used nowadays. It is used to describe properties of labelled transition systems and for verifying these properties. Many temporal logics can be encoded in the μ-calculus, including CTL* and its widely used fragments—linear temporal logic and computational tree logic. An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators consisting of functional composition plus the least and greatest fixed point operators; from this viewpoint, the modal μ-calculus is over the lattice of a power set algebra. The game semantics of μ-calculus is related to two-player games with perfect information, particularly infinite parity games.

Syntax

Let P (propositions) and A (actions) be two finite sets of symbols, and let Var be a countably infinite set of variables. The set of formulas of (propositional, modal) μ-calculus is defined as follows: (The notions of free and bound variables are as usual, where \nu is the only binding operator.) Given the above definitions, we can enrich the syntax with: The first two formulas are the familiar ones from the classical propositional calculus and respectively the minimal multimodal logic K. The notation \mu Z. \phi (and its dual) are inspired from the lambda calculus; the intent is to denote the least (and respectively greatest) fixed point of the expression \phi where the "minimization" (and respectively "maximization") are in the variable Z, much like in lambda calculus is a function with formula \phi in bound variable Z; see the denotational semantics below for details.

Denotational semantics

Models of (propositional) μ-calculus are given as labelled transition systems (S, R, V) where: Given a labelled transition system (S, R, V) and an interpretation i of the variables Z of the \mu-calculus,, is the function defined by the following rules: By duality, the interpretation of the other basic formulas is: Less formally, this means that, for a given transition system (S, R, V): The interpretations of [a] \phi and are in fact the "classical" ones from dynamic logic. Additionally, the operator \mu can be interpreted as liveness ("something good eventually happens") and \nu as safety ("nothing bad ever happens") in Leslie Lamport's informal classification.

Examples

Decision problems

Satisfiability of a modal μ-calculus formula is EXPTIME-complete. Like for linear temporal logic, the model checking, satisfiability and validity problems of linear modal μ-calculus are PSPACE-complete.