Modal μ-calculus


In theoretical computer science, the modal μ-calculus is an extension of propositional modal logic by adding the least fixed point operator μ and the greatest fixed point operator, thus a fixed-point logic.
The 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 and A be two finite sets of symbols, and let V be a countably infinite set of variables. The set of formulas of μ-calculus is defined as follows:
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 are inspired from the lambda calculus; the intent is to denote the least fixed point of the expression where the "minimization" are in the variable, much like in lambda calculus is a function with formula in bound variable ; see the denotational semantics below for details.

Denotational semantics

Models of μ-calculus are given as labelled transition systems where:
Given a labelled transition system and an interpretation of the variables of the -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 :
The interpretations of and are in fact the "classical" ones from dynamic logic. Additionally, the operator can be interpreted as liveness and as safety in Leslie Lamport's informal classification.

Examples

of a modal μ-calculus formula is EXPTIME-complete. As for Linear Temporal Logic, model checking, satisfiability and validity problems of linear modal μ-calculus are PSPACE-complete.