Modal μ-calculus explained

In theoretical computer science, the modal μ-calculus (Lμ, L_μ, 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,^[1] and was further developed by Dexter Kozen^[2] 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.^[3]

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.^[4] The game semantics of μ-calculus is related to two-player games with perfect information, particularly infinite parity games.^[5]

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:

• each proposition and each variable is a formula;
• if

\phi and \psi are formulas, then \phi \wedge \psi is a formula;

• if

\phi is a formula, then \neg \phi is a formula;

• if

\phi is a formula and a is an action, then [a] \phi is a formula; (pronounced either: a box \phi or after a necessarily \phi )

• if

\phi is a formula and Z a variable, then \nu Z. \phi is a formula, provided that every free occurrence of Z in \phi occurs positively, i.e. within the scope of an even number of negations.

(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:

\phi \lor \psi meaning \neg (\neg \phi \land \neg \psi) \langle a \rangle \phi (pronounced either: a diamond \phi or after a possibly \phi ) meaning \neg [a] \neg \phi \mu Z. \phi means \neg \nu Z. \neg \phi [Z: = \neg Z] , where \phi [Z: = \neg Z] means substituting \neg Z for Z in all free occurrences of Z in \phi .

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 λ Z. \phi is a function with formula \phi in bound variable Z ;^[6] see the denotational semantics below for details.

Denotational semantics

Models of (propositional) μ-calculus are given as labelled transition systems

(S, R, V) where:S is a set of states;R maps to each label a a binary relation on S ;V : P \to 2 ^{S} , maps each proposition p \in P to the set of states where the proposition is true.

Given a labelled transition system

(S, R, V) and an interpretation i of the variables Z of the \mu -calculus, [[ ⋅ ]] _{i:\phi} \to 2 ^{S} , is the function defined by the following rules:[[p]] _{i} = V(p) ;[[Z]] _{i} = i(Z) ;[[\phi \wedge \psi]] _{i} = [[\phi]] _{i} \cap [[\psi]] _{i} ;[[\neg \phi]] _{i} = S \smallsetminus [[\phi]] _{i} ;[[[a] \phi]] _{i} = \{s \in S \mid \forall t \in S, (s, t) \in R _{a} → t \in [[\phi]] _{i\}} ;[[\nu Z. \phi]] _{i} = c up \{T \subseteq S \mid T \subseteq [[\phi]] _{i[Z} \} , where i[Z : = T] maps Z to T while preserving the mappings of i everywhere else.

By duality, the interpretation of the other basic formulas is:

[[\phi \vee \psi]] _{i} = [[\phi]] _{i} \cup [[\psi]] _{i} ;[[\langle a \rangle \phi]] _{i} = \{s \in S \mid \exists t \in S, (s, t) \in R _{a} \wedge t \in [[\phi]] _{i\}} ;[[\mu Z. \phi]] _{i} = c ap \{T \subseteq S \mid [[\phi]] _{i[Z} \subseteq 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 \phi and \psi both hold;\neg \phi holds in every state where \phi does not hold.[a] \phi holds in a state s if every a -transition leading out of s leads to a state where \phi holds.\langle a\rangle \phi holds in a state s if there exists a -transition leading out of s that leads to a state where \phi holds.\nu Z.\phi holds in any state in any set T such that, when the variable Z is set to T , then \phi holds for all of T . (From the Knaster–Tarski theorem it follows that [[\nu Z.\phi]] _{i} is the greatest fixed point of T\mapsto[[\phi]] _{i[Z} , and [[\mu Z. \phi]] _{i} its least fixed point.)

The interpretations of

[a] \phi and \langle a\rangle \phi 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.^[7]

Examples

\nu Z.\phi \wedge [a]Z is interpreted as "\phi is true along every a-path".^[7] The idea is that "\phi is true along every a-path" can be defined axiomatically as that (weakest) sentence Z which implies \phi and which remains true after processing any a-label. ^[8] \mu Z.\phi \vee \langle a \rangle Z is interpreted as the existence of a path along a-transitions to a state where \phi holds.

• The property of a state being deadlock-free, meaning no path from that state reaches a dead end, is expressed by the formula^[9] $$\backslash nu\; Z.\backslash left\; (\backslash bigvee\_\backslash langle\; a\backslash rangle\backslash top\backslash wedge\; \backslash bigwedge\_[a]Z\; \backslash right)$$

Decision problems

Satisfiability of a modal μ-calculus formula is EXPTIME-complete.^[10] Like for linear temporal logic,^[11] the model checking, satisfiability and validity problems of linear modal μ-calculus are PSPACE-complete.^[12]

See also

• Finite model theory
• Alternation-free modal μ-calculus

Notes

1. Scott . Dana . Dana Scott. Bakker . Jacobus . 1969 . A theory of programs . Unpublished Manuscript.
2. Kozen. Dexter. Results on the propositional μ-calculus. ICALP. 1982. Automata, Languages and Programming. 140. 348–359. 978-3-540-11576-2. 10.1007/BFb0012782.
3. Clarke p.108, Theorem 6; Emerson p. 196
4. Arnold and Niwiński, pp. viii-x and chapter 6
5. Arnold and Niwiński, pp. viii-x and chapter 4
6. Arnold and Niwiński, p. 14
7. Bradfield and Stirling, p. 731
8. Bradfield and Stirling, p. 6
9. Book: Erich Grädel . Phokion G. Kolaitis . Leonid Libkin. Leonid Libkin . Maarten Marx . . . Yde Venema . Scott Weinstein . Finite Model Theory and Its Applications . 2007 . Springer . 978-3-540-00428-8 . 159.
10. Book: Klaus Schneider. Verification of reactive systems: formal methods and algorithms. 2004. Springer. 978-3-540-00296-3. 521.
11. Sistla. A. P.. Clarke. E. M.. 1985-07-01. The Complexity of Propositional Linear Temporal Logics. J. ACM. 32. 3. 733–749. 10.1145/3828.3837. 0004-5411. free.
12. Book: Vardi, M. Y.. Proceedings of the 15th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '88 . A temporal fixpoint calculus . 1988-01-01. New York, NY, USA. ACM. 250–259. 10.1145/73560.73582. 0897912527. free.

References

• Book: Clarke , Edmund M. Jr. . Orna Grumberg . Doron A. Peled . Model Checking. 1999. MIT press. Cambridge, Massachusetts, USA. 0-262-03270-8., chapter 7, Model checking for the μ-calculus, pp. 97–108
• Book: Stirling , Colin. . Modal and Temporal Properties of Processes. 2001. Springer Verlag. New York, Berlin, Heidelberg. 0-387-98717-7., chapter 5, Modal μ-calculus, pp. 103–128
• Book: André Arnold. Damian Niwiński. Rudiments of μ-Calculus. 2001. Elsevier. 978-0-444-50620-7., 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; was presented at The 18th European Summer School in Logic, Language and Information
• Book: Bradfield, Julian . Stirling, Colin . amp . 2006 . Modal mu-calculi . The Handbook of Modal Logic . P. Blackburn . J. van Benthem . F. Wolter . . 721–756 . http://homepages.inf.ed.ac.uk/jcb/Research/pubs.html#mlh-chapter .
• E. Allen . Emerson. E. Allen Emerson . Model Checking and the Mu-calculus . Descriptive Complexity and Finite Models . 1996 . 185–214 . . 0-8218-0517-7.
• Kozen, Dexter . 1983 . Results on the Propositional μ-Calculus . . 27 . 3 . 333–354 . 10.1016/0304-3975(82)90125-6 .

External links

• Sophie Pinchinat, Logic, Automata & Games video recording of a lecture at ANU Logic Summer School '09