Modal μ-calculus

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In theoretical computer science, the modal μ-calculus (, 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,[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

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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   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; (pronounced either:   box   or after   necessarily  )
  • 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.

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

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

  •   meaning  
  •   (pronounced either:   diamond   or after   possibly  ) 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   (and its dual) are inspired from the lambda calculus; the intent is to denote the least (and respectively greatest) fixed point of the expression   where the "minimization" (and respectively "maximization") are in the variable  , much like in lambda calculus   is a function with formula   in bound variable  ;[6] see the denotational semantics below for details.

Denotational semantics

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Models of (propositional) 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  . (From the Knaster–Tarski theorem it follows that   is the greatest fixed point of  , and   its least fixed point.)

The interpretations of   and   are in fact the "classical" ones from dynamic logic. Additionally, the operator   can be interpreted as liveness ("something good eventually happens") and   as safety ("nothing bad ever happens") in Leslie Lamport's informal classification.[7]

Examples

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  •   is interpreted as "  is true along every a-path".[7] The idea is that "  is true along every a-path" can be defined axiomatically as that (weakest) sentence   which implies   and which remains true after processing any a-label. [8]
  •   is interpreted as the existence of a path along a-transitions to a state where   holds.[9]
  • The property of a state being deadlock-free, meaning no path from that state reaches a dead end, is expressed by the formula[9]  

Decision problems

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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

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Notes

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

References

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  • Clarke, Edmund M. Jr.; 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 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
  • Bradfield, Julian & Stirling, Colin (2006). "Modal mu-calculi". In P. Blackburn; J. van Benthem & F. Wolter (eds.). The Handbook of Modal Logic. Elsevier. pp. 721–756.
  • 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.
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