Modal μ-calculus

In theoretical computer science, the modal μ-calculus (also μ-calculus, but this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding a least fixpoint operator μ and a greatest fixpoint operator ν.

The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker,^[1] 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.^[2]

An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators functional composition, and least and greatest fixed point operators; from this viewpoint, the modal μ-calculus is over the lattice of a powerset algebra.^[3] The semantics of μ-calculus in general is related to two-player games with perfect information, particularly infinite parity games.^[4]

Syntax

Let P (propositions) and A (actions) be two finite sets of symbols, and let V be a countably infinite set of variables. The set of formulas of (propositional, modal) μ-calculus is defined as follows:

• each proposition and each variable is a formula;
• if ϕ and ψ are formulas, then $\phi \wedge \psi$ is a formula.
• if ϕ is a formula, then $\neg \phi$ is a formula;
• if ϕ is a formula and a is an action, then [a]ϕ is a formula;
• if ϕ is a formula and Z a variable, then νZ.ϕ is a formula, provided that every free occurrence of Z in ϕ occurs positively, i.e. within the scope of an even number of negations.

(The notions of free and bound variables are as usual, where ν is the only binding operator.)

Given the above definitions, we can enrich the syntax with:

• $\phi \lor \psi$ meaning $\neg (\neg \phi \land \neg \psi)$
• $\langle a \rangle \phi$ meaning $\neg [a] \neg \phi$
• μZ.ϕ means $\neg \nu Z. \neg \phi (\neg Z)$, where $\phi (\neg Z)$ means substituting $\neg Z$ for Z in all free occurrences of Z in φ.

The first two formulas are the familiar ones from the classical propositional calculus and respectively the minimal multimodal logic K.

Semantics

Models of (propositional) μ-calculus is given are labelled transition systems (S,R,V) where:

• S is a set of states;
• R maps to each label a a relation on S;
• V maps to each proposition $p \in PROP$ the set of states where the proposition is true.

Given a labelled transition system (S,R,V) and an interpretation $i : VAR \rightarrow 2^S$, we interpret a formula:

• | | p | | _i = V(p);
• $||\phi \wedge \psi||_i = ||\phi||_i \cap ||\psi||_i$;
• $||\neg \phi||_i = S \backslash ||\phi||_i$;
• $||[a] \phi||_i = \{s \in S \mid \forall t \in S, (s, t) \in R_a \rightarrow t \in ||\phi||_i\}$;
• $||\nu Z. \phi||_i = \bigcup \{T \subseteq S \mid T \subseteq ||\phi||_{i[Z := T]}\}$.

Less formally, this means that, for a given transition system (S,R,V):

• p holds in the set of states V(p);
• $\phi \wedge \psi$ holds in every state where ϕ and ψ both hold;
• $\neg \phi$ holds in every state where ϕ does not hold.
• [a]ϕ holds in a state s if every a-transition leading out of s leads to a state where ϕ holds.
• $\langle a\rangle \phi$ holds in a state s if any a-transition leading out of s leads to a state where ϕ holds.
• $\nu Z.\phi$ holds in any state in any set T such that, when the variable Z is set to T, then ϕ holds for all of T

Satisfiability

Satisfiability of a modal μ-calculus formula is EXPTIME-complete.^[5]

See also

• Alternation-free modal μ-calculus
• Knaster–Tarski theorem

Notes

1. ^ Kozen p. 333.
2. ^ Clarke p.108, Theorem 6; Emerson p. 196
3. ^ Arnold and Niwiński, pp. viii-x and chapter 6
4. ^ Arnold and Niwiński, pp. viii-x and chapter 4
5. ^ Klaus Schneider (2004). Verification of reactive systems: formal methods and algorithms. Springer. p. 521. ISBN 9783540002963. http://books.google.com/books?id=Z92bL1VrD_sC&pg=PA521. 

References

• Clarke, Jr., Edmund M.; Orna Grumberg, Doron A. Peled (1999). Model Checking. Cambridge, Massachusetts, USA: MIT press. ISBN 0-262-03270-8. , chapter 7, Model checking for the μ-calculus, pp. 97-108
• Stirling, Colin. (2001). Modal and Temporal Properties of Processes. New York, Berlin, Heidelberg: Springer Verlag. ISBN 0-387-98717-7. , chapter 5, Modal μ-calculus, pp. 103-128
• André Arnold; Damian Niwiński (2001). Rudiments of μ-Calculus. Elsevier. ISBN 9780444506207. , chapter 6, The μ-calculus over powerset algebras, pp. 141-153 is about the modal μ-calculus
• Yde Venema (2008) Lectures on the Modal μ-calculus; the 2006 version was presented at The 18th European Summer School in Logic, Language and Information
• Bradfield, Julian and Stirling, Colin (2006). "Modal mu-calculi". In P. Blackburn, J. van Benthem and F. Wolter (eds.). The Handbook of Modal Logic. Elsevier. pp. 721–756. http://homepages.inf.ed.ac.uk/jcb/Research/pubs.html#mlh-chapter. 
• Emerson, E. Allen (1996). "Model Checking and the Mu-calculus". Descriptive Complexity and Finite Models. American Mathematical Society. pp. 185–214. ISBN 0-8218-0517-7. 
• Kozen, Dexter (1983). "Results on the Propositional μ-Calculus". Theoretical Computer Science 27 (3): 333–354. doi:10.1016/0304-3975(82)90125-6. 
• Videolectures.net - ANU Logic Summer School '09

P ≟ NP

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