Department of Computer Science Institute for Theoretical Computer Science Modal µ -Calculus and Alternating Tree Automata Seminar “Automata, Logics, and Infinite Games” Patrick Bahr s0404888@inf.tu-dresden.de Dresden, February 1, 2008
Outline Modal µ -Calculus Introduction Syntax Semantics Auxiliary Notions Translation into Alternating Tree Automata [Alternating Tree Automata] Construction of Equivalent Alternating Tree Automata Model-Checking and Satisfiability Conclusion TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 2 of 28
Outline Modal µ -Calculus Introduction Syntax Semantics Auxiliary Notions Translation into Alternating Tree Automata [Alternating Tree Automata] Construction of Equivalent Alternating Tree Automata Model-Checking and Satisfiability Conclusion TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 3 of 28
01 Introduction to µ -Calculus • introduced in the context of systems verification • basic modal language (i.e. Boolean connectives plus modal connectives) • interpret Boolean and modal connectives as set-theoretic operations • additional connectives to define fixed points • great expressive power • includes many temporal logics e.g. CTL* and PDL • computationally “well-behaved” TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 4 of 28
01 Syntax of the µ -Calculus L µ • In the following we fix a set of atomic propositions P . Definition ( L µ formulae) The set of formulae of the modal µ -calculus, denoted by L µ is defined by the following grammar: ⊥ | ⊤ | p | ¬ ϕ | ϕ ∧ ψ | ϕ ∨ ψ | � ϕ | ♦ ϕ | µ p .ϕ p | ν p .ϕ p ϕ, ψ ::= TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 5 of 28
01 Syntax of the µ -Calculus L µ • In the following we fix a set of atomic propositions P . Definition ( L µ formulae) The set of formulae of the modal µ -calculus, denoted by L µ is defined by the following grammar: ⊥ | ⊤ | p | ¬ ϕ | ϕ ∧ ψ | ϕ ∨ ψ | � ϕ | ♦ ϕ | µ p .ϕ p | ν p .ϕ p ϕ, ψ ::= where p ∈ P and ϕ p is restricted to only contain p in the scope of an even number of negation connectives “ ¬ ”. TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 5 of 28
01 Example L µ -Formulae Example ϕ 1 = µ p . ¬ ( µ p ′ . ( ¬ q ∨ � p ′ ) ∨ ¬ � p ) is well-formed ϕ 1 : µp ¬ µp ′ ∨ ∨ ¬ � � ¬ p ′ q p TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 6 of 28
01 Example L µ -Formulae Example Example ϕ 1 = µ p . ¬ ( µ p ′ . ( ¬ q ∨ � p ′ ) ∨ ¬ � p ) is ϕ 2 = µ p . ¬ ( µ p ′ . ( ¬ q ∨ � p ′ ) ∨ ( ¬ � p ∧ p )) is well-formed not well-formed ϕ 2 : µp ϕ 1 : µp ¬ ¬ µp ′ µp ′ ∨ ∨ ∨ ∧ ∨ ¬ � ¬ ¬ p � � ¬ p ′ � q p ′ q p p TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 6 of 28
01 Example L µ -Formulae Example Example ϕ 1 = µ p . ¬ ( µ p ′ . ( ¬ q ∨ � p ′ ) ∨ ¬ � p ) is ϕ 2 = µ p . ¬ ( µ p ′ . ( ¬ q ∨ � p ′ ) ∨ ( ¬ � p ∧ p )) is well-formed not well-formed ϕ 2 : µp ϕ 1 : µp ¬ ¬ µp ′ µp ′ ∨ ∨ ∨ ∧ ∨ ¬ � ¬ ¬ p � � ¬ p ′ � q p ′ q p p TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 6 of 28
01 Fixed Point Connectives bind Atomic Propositions Definition (Free Occurrences of Atomic Propositions) The set free ( ϕ ) of atomic propositions occurring freely in ϕ : free ( ⊤ ) = free ( ⊥ ) = ∅ , free ( p ) = { p } , free ( ϕ ∨ ψ ) = free ( ϕ ∧ ψ ) = free ( ϕ ) ∪ free ( ψ ) , free ( ¬ ϕ ) = free ( � ϕ ) = free ( ♦ ϕ ) = free ( ϕ ) , free ( µ p .ϕ ) = free ( ν p .ϕ ) = free ( ϕ ) \ { p } . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 7 of 28
01 Kripke Models Definition (Kripke Models) A Kripke model is a triple K = ( W , R , λ ) where • W , the universe of K , is a set of worlds, • R ⊆ W × W is an accessibility relation, and • λ : P → 2 W is a valuation of the atomic propositions. For w ∈ W we will call ( K , w ) a pointed Kripke model. Definition (Update of Kripke Models) Let K = ( W , R , λ ) be a Kripke model. � W ′ if p = p ′ • λ [ p ′ �→ W ′ ]( p ) = for all p ∈ P . λ ( p ) otherwise • K [ p ′ �→ W ′ ] = ( W , R , λ [ p ′ �→ W ′ ]) TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 8 of 28
01 Semantics of the µ -Calculus Definition (Semantics of µ -Calculus) Let K = ( W , R , λ ) be a Kripke model. For every L µ formula ϕ the set � ϕ � K ⊆ W is defined as follows: �⊥� K = ∅ , �⊤� K = W , � p � K = λ ( p ) for all p ∈ P , �¬ ϕ � K = W \ � ϕ � K , � ϕ ∨ ψ � K = � ϕ � K ∪ � ψ � K , � ϕ ∧ ψ � K = � ϕ � K ∩ � ψ � K , � � ϕ � K = { w ∈ W | wR ⊆ � ϕ � K } , � ♦ ϕ � K = { w ∈ W | wR ∩ � ϕ � K � = ∅} , { W ′ ⊆ W | � ϕ � K [ p �→ W ′ ] ⊆ W ′ } , � µ p .ϕ � K = � { W ′ ⊆ W | � ϕ � K [ p �→ W ′ ] ⊇ W ′ } . � ν p .ϕ � K = � • ( K , w ) | = ϕ if w ∈ � ϕ � K . • ϕ ≡ ψ if for all pointed Kripke models ( K , w ) we have that ( K , w ) | = ϕ iff ( K , w ) | = ψ . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 9 of 28
01 Semantics of Fixed Point Connectives in Detail { W ′ ⊆ W | � ϕ � K [ p �→ W ′ ] ⊆ W ′ } . • Reconsider � µ p .ϕ � K = � • This implicitly refers to the function: g : 2 W → 2 W , W ′ �→ � ϕ � K [ p �→ W ′ ] TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 10 of 28
01 Semantics of Fixed Point Connectives in Detail { W ′ ⊆ W | � ϕ � K [ p �→ W ′ ] ⊆ W ′ } . • Reconsider � µ p .ϕ � K = � • This implicitly refers to the function: g : 2 W → 2 W , W ′ �→ � ϕ � K [ p �→ W ′ ] { W ′ ⊆ W | g ( W ′ ) ⊆ W ′ } • Now � µ p .ϕ � K = � • Due to syntactic restriction w.r.t. bound propositions: g is monotone! � � µ p .ϕ � K = least fixed point of g . [Knaster-Tarski-Theorem] TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 10 of 28
01 Semantics of Fixed Point Connectives in Detail { W ′ ⊆ W | � ϕ � K [ p �→ W ′ ] ⊆ W ′ } . • Reconsider � µ p .ϕ � K = � • This implicitly refers to the function: g : 2 W → 2 W , W ′ �→ � ϕ � K [ p �→ W ′ ] { W ′ ⊆ W | g ( W ′ ) ⊆ W ′ } • Now � µ p .ϕ � K = � • Due to syntactic restriction w.r.t. bound propositions: g is monotone! � � µ p .ϕ � K = least fixed point of g . [Knaster-Tarski-Theorem] • Dually � ν p .ϕ � K = greatest fixed point of g . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 10 of 28
01 Equivalences Lemma (Equivalences) Let ϕ, ψ ∈ L µ and p , q ∈ P , s.t. q does not occur in ϕ . Then we have the following equivalences: ( i ) µ p .ϕ ≡ µ q .ϕ [ p / q ] ν p .ϕ ≡ ν q .ϕ [ p / q ] , ( ii ) ( iii ) ¬¬ ϕ ≡ ϕ, ( iv ) ϕ ∨ ψ ≡ ¬ ( ¬ ϕ ∧ ¬ ψ ) , ( v ) ♦ ϕ ≡ ¬ � ¬ ϕ, ( vi ) µ p .ϕ ≡ ¬ ν p . ¬ ϕ [ p / ¬ p ] . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 11 of 28
01 Some Examples Example Consider ϕ 1 = µ p . � p . ( K , w ) | = ϕ 1 iff all paths in K starting in w are finite. TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 12 of 28
01 Some Examples Example Consider ϕ 1 = µ p . � p . ( K , w ) | = ϕ 1 iff all paths in K starting in w are finite. Example Consider ϕ 2 = µ p . ( ψ ∨ ♦ p ) . = ϕ 2 iff there is a world w ′ in K in which ψ holds and which is reachable ( K , w ) | from w . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 12 of 28
01 Some Examples Example Consider ϕ 1 = µ p . � p . ( K , w ) | = ϕ 1 iff all paths in K starting in w are finite. Example Consider ϕ 2 = µ p . ( ψ ∨ ♦ p ) . = ϕ 2 iff there is a world w ′ in K in which ψ holds and which is reachable ( K , w ) | from w . Example Consider ϕ 3 = ν p .µ q . (( ψ ∧ p ) ∨ ♦ q ) ( K , w ) | = ϕ 2 iff ψ holds infinitely often on a path starting in w . TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 12 of 28
01 Some Examples Example Consider ϕ 1 = µ p . � p . ( K , w ) | = ϕ 1 iff all paths in K starting in w are finite. Example Consider ϕ 2 = µ p . ( ψ ∨ ♦ p ) . = ϕ 2 iff there is a world w ′ in K in which ψ holds and which is reachable ( K , w ) | from w . Example Consider ϕ 3 = ν p .µ q . (( ψ ∧ p ) ∨ ♦ q ) ( K , w ) | = ϕ 2 iff ψ holds infinitely often on a path starting in w . Intuitive Semantics of Fixed Point Connectives • µ specifies properties of finite paths • ν specifies properties of infinite paths TU Dresden, February 1, 2008 µ -Calculus & Alternating Tree Automata slide 12 of 28
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