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 ν, 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 Var be a countably infinite set of variables. The set of formulas of μ-calculus is defined as follows:
  • each proposition and each variable is a formula;
  • if and are formulas, then is a formula;
  • if is a formula, then is a formula;
  • if is a formula and is an action, then is a formula;
  • if is a formula and a variable, then is a formula, provided that every free occurrence of in occurs positively, i.e. within the scope of an even number of negations.
Given the above definitions, we can enrich the syntax with:
  • meaning
  • meaning
  • means, where means substituting for in all free occurrences of in.
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:
  • is a set of states;
  • maps to each label a binary relation on ;
  • , maps each proposition to the set of states where the proposition is true.
Given a labelled transition system and an interpretation of the variables of the -calculus,, is the function defined by the following rules:
  • ;
  • ;
  • ;
  • ;
  • ;
  • , where maps to while preserving the mappings of everywhere else.
By duality, the interpretation of the other basic formulas is:
  • ;
  • ;
Less formally, this means that, for a given transition system :
  • holds in the set of states ;
  • holds in every state where and both hold;
  • holds in every state where does not hold.
  • holds in a state if every -transition leading out of leads to a state where holds.
  • holds in a state if there exists -transition leading out of that leads to a state where holds.
  • holds in any state in any set such that, when the variable is set to, then holds for all of.
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

  • is interpreted as " is true along every a-path". The idea is that " is true along every a-path" can be defined axiomatically as that sentence which implies and which remains true after processing any a-label.
  • is interpreted as the existence of a path along a-transitions to a state where holds.
  • The property of a state being deadlock-free, meaning no path from that state reaches a dead end, is expressed by the formula

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.
Actually, the complexity of the satisfiability problem of graded modal μ-calculus is also EXPTIME-complete, even if the number in the modalities are written in binary.