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| In [[theoretical computer science]], the '''modal μ-calculus''' ('''Lμ''', '''L<sub>μ</sub>''', sometimes just '''μ-calculus''', although this can have a more general meaning) is an extension of [[propositional logic|propositional]] [[modal logic]] (with [[Multimodal logic|many modalities]]) by adding a [[least fixpoint]] operator μ and a [[greatest fixpoint]] operator <math>\nu</math>, thus a [[fixed-point logic]].
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| The (propositional, modal) μ-calculus originates with [[Dana Scott]] and [[Jaco de Bakker]],<ref>Kozen p. 333.</ref> and was further developed by [[Dexter Kozen]] into the version most used nowadays. It is used to describe properties of [[labelled transition system]]s and for [[model checking|verifying]] these properties. Many [[temporal logic]]s can be encoded in the μ-calculus, including [[CTL*]] and its widely used fragments—[[linear temporal logic]] and [[computational tree logic]].<ref>Clarke p.108, Theorem 6; Emerson p. 196</ref>
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| An algebraic view is to see it as an [[universal algebra|algebra]] of [[monotonic function]]s 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]].<ref>Arnold and Niwiński, pp. viii-x and chapter 6</ref> The [[game semantics]] of μ-calculus is related to [[two-player game]]s with [[perfect information]], particularly infinite [[parity game]]s.<ref>Arnold and Niwiński, pp. viii-x and chapter 4</ref>
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| == Syntax ==
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| 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:
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| * each proposition and each variable is a formula;
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| * if <math>\phi</math> and <math>\psi</math> are formulas, then <math>\phi \wedge \psi</math> is a formula.
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| * if <math>\phi</math> is a formula, then <math>\neg \phi</math> is a formula;
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| * if <math>\phi</math> is a formula and <math>a</math> is an action, then <math>[a] \phi</math> is a formula;(pronounced either: <math>a</math> box <math>\phi </math> or after <math>a</math> necessarily <math>\phi</math>)
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| * if <math>\phi</math> is a formula and <math>Z</math> a variable, then <math>\nu Z. \phi</math> is a formula, provided that every free occurrence of <math>Z</math> in <math>\phi</math> occurs positively, i.e. within the scope of an even number of negations.
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| (The notions of free and bound variables are as usual, where <math>\nu</math> is the only binding operator.)
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| Given the above definitions, we can enrich the syntax with:
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| * <math>\phi \lor \psi</math> meaning <math>\neg (\neg \phi \land \neg \psi)</math>
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| * <math>\langle a \rangle \phi</math> (pronounced either: <math>a</math> diamond <math>\phi </math> or after <math>a</math> possibly <math>\phi</math>) meaning <math>\neg [a] \neg \phi</math>
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| * <math>\mu Z. \phi</math> means <math>\neg \nu Z. \neg \phi [Z:=\neg Z]</math>, where <math>\phi [Z:=\neg Z]</math> means substituting <math>\neg Z</math> for ''Z'' in all [[free occurrence]]s of ''Z'' in <math>\phi </math>.
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| The first two formulas are the familiar ones from the classical [[propositional calculus]] and respectively the minimal [[multimodal logic]] '''K'''.
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| The notation <math>\mu Z. \phi</math> (and its dual) are inspired from the [[lambda calculus]]; the intent is to denote the least (and respectively greatest) fixed point of the expression <math>\phi</math> where the "minimization" (and respectively "maximization") are in the variable ''Z'', much like in lambda calculus <math>\lambda Z. \phi</math> is a function with formula <math>\phi</math> in [[Lambda_calculus#Free_and_bound_variables|bound variable]] ''Z'';<ref>Arnold and Niwiński, p. 14</ref> see the denotational semantics below for details.
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| == Denotational semantics ==
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| Models of (propositional) μ-calculus is given are [[labelled transition system]]s <math>(S, R, V)</math> where:
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| * <math>S</math> is a set of states;
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| * <math>R</math> maps to each label <math>a</math> a relation on <math>S</math>;
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| * <math> V : \mbox{Var} \rightarrow 2^S </math>, maps to each proposition <math>p \in \mbox{Prop}</math> the set of states where the proposition is true.
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| Given a labelled transition system <math>(S, R, V)</math> and an interpretation of the fórmulas <math>\phi</math> of <math>\mu</math>-calculus, <math>[\![\underline{~\,}]\!]_i : \phi \rightarrow 2^S </math>, is the function definded by the following rules:
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| * <math>[\![p]\!]_i = V(p)</math>;
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| * <math>[\![\phi \wedge \psi]\!]_i = [\![\phi]\!]_i \cap [\![\psi]\!]_i</math>;
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| * <math>[\![\neg \phi]\!]_i = S \smallsetminus [\![\phi]\!]_i</math>;
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| * <math>[\![[a] \phi]\!]_i = \{s \in S \mid \forall t \in S, (s, t) \in R_a \rightarrow t \in [\![\phi]\!]_i\}</math>;
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| * <math>[\![\nu Z. \phi]\!]_i = \bigcup \{T \subseteq S \mid T \subseteq [\![\phi]\!]_{i[Z := T]}\}</math>, where <math>i[Z := T]</math> maps Z to T while preserving the mappings of <math>i</math> everywhere else.
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| By duality, the interpretation of the other basic formulas is:
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| * <math>[\![\phi \vee \psi]\!]_i = [\![\phi]\!]_i \cup [\![\psi]\!]_i</math>;
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| * <math>[\![\langle a \rangle \phi]\!]_i = \{s \in S \mid \exists t \in S, (s, t) \in R_a \wedge t \in [\![\phi]\!]_i\}</math>;
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| * <math>[\![\mu Z. \phi]\!]_i = \bigcap \{T \subseteq S \mid [\![\phi]\!]_{i[Z := T]} \subseteq T \}</math>
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| Less formally, this means that, for a given transition system <math>(S, R, V)</math>:
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| * <math>p</math> holds in the set of states <math>V(p)</math>;
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| * <math>\phi \wedge \psi</math> holds in every state where <math>\phi</math> and <math>\psi</math> both hold;
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| * <math>\neg \phi</math> holds in every state where <math>\phi</math> does not hold.
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| * <math>[a] \phi</math> holds in a state <math>s</math> if every <math>a</math>-transition leading out of <math>s</math> leads to a state where <math>\phi</math> holds.
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| * <math>\langle a\rangle \phi</math> holds in a state <math>s</math> if there exists <math>a</math>-transition leading out of <math>s</math> that leads to a state where <math>\phi</math> holds.
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| * <math>\nu Z.\phi</math> holds in any state in any set <math>T</math> such that, when the variable <math>Z</math> is set to <math>T</math>, then <math>\phi</math> holds for all of <math>T</math>. (From the [[Knaster–Tarski theorem]] it follows that <math>[\![\nu Z.\phi]\!]_i</math> is the greatest [[fixpoint]] of <math>[\![\phi]\!]_{i[Z := T]}</math>, and <math>[\![\mu Z. \phi]\!]_i</math> its least fixpoint.)
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| The interpretations of <math>[a] \phi</math> and <math>\langle a\rangle \phi</math> are if fact the "classical" ones from [[Dynamic logic (modal logic)|dynamic logic]]. Additionally, the operator μ can be interpreted as [[liveness]] ("something good eventually happens") and ν as [[safety (computer science)|safety]] ("nothing bad ever happens") in [[Leslie Lamport]]'s informal classification.<ref name="Bradfield and Stirling, p. 731">Bradfield and Stirling, p. 731</ref>
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| === Examples ===
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| * <math>\nu Z.\phi \wedge [a]Z</math> is interpreted as "<math>\phi</math> is true along every ''a''-path".<ref name="Bradfield and Stirling, p. 731"/>
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| * <math>\mu Z.\phi \vee \langle a \rangle Z</math> is interpreted as the existence of a path along ''a''-transitions to a state where <math>\phi</math> holds.<ref name="GrädelKolaitis2007"/>
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| * The property of a system of being [[deadlock]]-free, understood as having no states without outgoing transitions and furthermore there does not exists a path to such a state, is expressed by formula<ref name="GrädelKolaitis2007">{{cite book|author1=Erich Grädel|author2=Phokion G. Kolaitis|author3=Leonid Libkin|coauthors=Maarten Marx, Joel Spencer, Moshe Y. Vardi, Yde Venema, Scott Weinstein|title=Finite Model Theory and Its Applications|url=http://books.google.com/books?id=e6ST-AVC27AC&pg=PA159|year=2007|publisher=Springer|isbn=978-3-540-00428-8|page=159}}</ref>
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| *: <math>\nu Z.(\bigvee_{a\in A}\langle a\rangle\top\wedge \bigwedge_{a\in A}[a]Z)</math>
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| ==Satisfiability==
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| [[Satisfiability]] of a modal μ-calculus formula is [[EXPTIME-complete]].<ref name="Schneider2004">{{cite book|author=Klaus Schneider|title=Verification of reactive systems: formal methods and algorithms|url=http://books.google.com/books?id=Z92bL1VrD_sC&pg=PA521|year=2004|publisher=Springer|isbn=978-3-540-00296-3|page=521}}</ref>
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| == See also ==
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| * [[Finite model theory]]
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| * [[Alternation-free modal μ-calculus]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{cite book
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| | last = Clarke, Jr.
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| | first = Edmund M.
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| | coauthors = Orna Grumberg, Doron A. Peled
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| | title = Model Checking
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| | year = 1999
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| | publisher = MIT press
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| | location = Cambridge, Massachusetts, USA
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| | isbn = 0-262-03270-8
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| }}, chapter 7, Model checking for the μ-calculus, pp. 97–108
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| *{{cite book
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| | last = Stirling
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| | first = Colin.
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| | title = Modal and Temporal Properties of Processes
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| | year = 2001
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| | publisher = Springer Verlag
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| | location = New York, Berlin, Heidelberg
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| | isbn = 0-387-98717-7
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| }}, chapter 5, Modal μ-calculus, pp. 103–128
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| * {{cite book|author1=André Arnold|author2=Damian Niwiński|title=Rudiments of μ-Calculus|year=2001|publisher=Elsevier|isbn=978-0-444-50620-7}}, chapter 6, The μ-calculus over powerset algebras, pp. 141–153 is about the modal μ-calculus
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| * Yde Venema (2008) [http://philo.ruc.edu.cn/logic/reading/Yde/mu0612.pdf Lectures on the Modal μ-calculus]; was presented at The 18th European Summer School in Logic, Language and Information
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| *{{cite book
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| | author = Bradfield, Julian and Stirling, Colin
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| | year = 2006
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| | chapter = Modal mu-calculi
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| | title = The Handbook of Modal Logic
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| | editor = P. Blackburn, J. van Benthem and F. Wolter (eds.)
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| | publisher = [[Elsevier]]
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| | pages = 721–756
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| | url = http://homepages.inf.ed.ac.uk/jcb/Research/pubs.html#mlh-chapter
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| }}
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| *{{cite conference
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| | first = E. Allen
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| | last = Emerson
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| | title = Model Checking and the Mu-calculus
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| | booktitle = Descriptive Complexity and Finite Models
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| | year = 1996
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| | pages = 185–214
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| | publisher = [[American Mathematical Society]]
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| | isbn = 0-8218-0517-7
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| }}
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| *{{cite journal
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| | author = Kozen, Dexter
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| | year = 1983
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| | title = Results on the Propositional μ-Calculus
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| | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]]
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| | volume = 27
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| | issue = 3
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| | pages = 333–354
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| | doi = 10.1016/0304-3975(82)90125-6
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| }}
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| == External links ==
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| * Sophie Pinchinat, [http://videolectures.net/ssll09_pinchinat_lag/ Logic, Automata & Games] video recording of a lecture at ANU Logic Summer School '09
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| [[Category:Modal logic]]
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| [[Category:Model checking]]
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