Modal -Calculus and Alternating Tree Automata Seminar Automata, - - PowerPoint PPT Presentation

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

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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”

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01 Syntax of the µ-Calculus Lµ

• In the following we fix a set of atomic propositions P.

Definition (Lµ formulae)

The set of formulae of the modal µ-calculus, denoted by Lµ is defined by the following grammar: ϕ, ψ ::= ⊥ | ⊤ | p | ¬ϕ | ϕ ∧ ψ | ϕ ∨ ψ | ϕ | ♦ϕ | µp.ϕp | νp.ϕp

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01 Syntax of the µ-Calculus Lµ

• In the following we fix a set of atomic propositions P.

Definition (Lµ formulae)

The set of formulae of the modal µ-calculus, denoted by Lµ is defined by the following grammar: ϕ, ψ ::= ⊥ | ⊤ | p | ¬ϕ | ϕ ∧ ψ | ϕ ∨ ψ | ϕ | ♦ϕ | µp.ϕp | νp.ϕp where p ∈ P and ϕp is restricted to only contain p in the scope of an even number

• f negation connectives “¬”.

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01 Example Lµ-Formulae

Example

ϕ1 = µp.¬(µp′.(¬q ∨ p′) ∨ ¬p) is well-formed

ϕ1 : µp ¬ µp′ ∨ ∨ ¬ ¬

• q

p′ p

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01 Example Lµ-Formulae

Example

ϕ1 = µp.¬(µp′.(¬q ∨ p′) ∨ ¬p) is well-formed

ϕ1 : µp ¬ µp′ ∨ ∨ ¬ ¬

• q

p′ p

Example

ϕ2 = µp.¬(µp′.(¬q ∨ p′) ∨ (¬p ∧ p)) is not well-formed ϕ2 : µp ¬ µp′ ∨ ∨ ∧ ¬

• ¬

p q p′

• p

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01 Example Lµ-Formulae

Example

ϕ1 = µp.¬(µp′.(¬q ∨ p′) ∨ ¬p) is well-formed

ϕ1 : µp ¬ µp′ ∨ ∨ ¬ ¬

• q

p′ p

Example

ϕ2 = µp.¬(µp′.(¬q ∨ p′) ∨ (¬p ∧ p)) is not well-formed ϕ2 : µp ¬ µp′ ∨ ∨ ∧ ¬

• ¬

p q p′

• p

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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(µp.ϕ) = free(νp.ϕ) = free(ϕ) \ {p}.

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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 → 2W 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.

• λ[p′ → W ′](p) =
• W ′

if p = p′ λ(p)

• therwise

for all p ∈ P.

• K[p′ → W ′] = (W, R, λ[p′ → W ′])

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01 Semantics of the µ-Calculus

Definition (Semantics of µ-Calculus)

Let K = (W, R, λ) be a Kripke model. For every Lµ formula ϕ the set ϕK ⊆ W is defined as follows: ⊥K = ∅, ⊤K = W, pK = λ(p) for all p ∈ P, ¬ϕK = W \ ϕK, ϕ ∨ ψK = ϕK ∪ ψK, ϕ ∧ ψK = ϕK ∩ ψK, ϕK = {w ∈ W | wR ⊆ ϕK}, ♦ϕK = {w ∈ W | wR ∩ ϕK = ∅}, µp.ϕK = {W ′ ⊆ W | ϕK[p→W ′] ⊆ W ′}, νp.ϕK = {W ′ ⊆ W | ϕK[p→W ′] ⊇ W ′}.

• (K, w) |

= ϕ if w ∈ ϕK.

• ϕ ≡ ψ if for all pointed Kripke models (K, w) we have that (K, w) |

= ϕ iff (K, w) | = ψ.

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01 Semantics of Fixed Point Connectives in Detail

• Reconsider µp.ϕK =

{W ′ ⊆ W | ϕK[p→W ′] ⊆ W ′}.

• This implicitly refers to the function:

g : 2W → 2W , W ′ → ϕK[p→W ′]

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01 Semantics of Fixed Point Connectives in Detail

• Reconsider µp.ϕK =

{W ′ ⊆ W | ϕK[p→W ′] ⊆ W ′}.

• This implicitly refers to the function:

g : 2W → 2W , W ′ → ϕK[p→W ′]

• Now µp.ϕK =

{W ′ ⊆ W | g(W ′) ⊆ W ′}

• Due to syntactic restriction w.r.t. bound propositions: g is monotone!
• µp.ϕK = least fixed point of g. [Knaster-Tarski-Theorem]

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01 Semantics of Fixed Point Connectives in Detail

• Reconsider µp.ϕK =

{W ′ ⊆ W | ϕK[p→W ′] ⊆ W ′}.

• This implicitly refers to the function:

g : 2W → 2W , W ′ → ϕK[p→W ′]

• Now µp.ϕK =

{W ′ ⊆ W | g(W ′) ⊆ W ′}

• Due to syntactic restriction w.r.t. bound propositions: g is monotone!
• µp.ϕK = least fixed point of g. [Knaster-Tarski-Theorem]
• Dually νp.ϕK = greatest fixed point of g.

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01 Equivalences

Lemma (Equivalences)

Let ϕ, ψ ∈ Lµ and p, q ∈ P, s.t. q does not occur in ϕ. Then we have the following equivalences: (i) µp.ϕ ≡ µq.ϕ[p/q] (ii) νp.ϕ ≡ νq.ϕ[p/q], (iii) ¬¬ϕ ≡ ϕ, (iv) ϕ ∨ ψ ≡ ¬(¬ϕ ∧ ¬ψ), (v) ♦ϕ ≡ ¬¬ϕ, (vi) µp.ϕ ≡ ¬νp.¬ϕ[p/¬p].

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01 Some Examples

Example

Consider ϕ1 = µp.p. (K, w) | = ϕ1 iff all paths in K starting in w are finite.

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01 Some Examples

Example

Consider ϕ1 = µp.p. (K, w) | = ϕ1 iff all paths in K starting in w are finite.

Example

Consider ϕ2 = µp. (ψ ∨ ♦p). (K, w) | = ϕ2 iff there is a world w′ in K in which ψ holds and which is reachable from w.

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01 Some Examples

Example

Consider ϕ1 = µp.p. (K, w) | = ϕ1 iff all paths in K starting in w are finite.

Example

Consider ϕ2 = µp. (ψ ∨ ♦p). (K, w) | = ϕ2 iff there is a world w′ in K in which ψ holds and which is reachable from w.

Example

Consider ϕ3 = νp.µq. ((ψ ∧ p) ∨ ♦q) (K, w) | = ϕ2 iff ψ holds infinitely often on a path starting in w.

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01 Some Examples

Example

Consider ϕ1 = µp.p. (K, w) | = ϕ1 iff all paths in K starting in w are finite.

Example

Consider ϕ2 = µp. (ψ ∨ ♦p). (K, w) | = ϕ2 iff there is a world w′ in K in which ψ holds and which is reachable from w.

Example

Consider ϕ3 = νp.µq. ((ψ ∧ p) ∨ ♦q) (K, w) | = ϕ2 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

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01 Normal Forms

Definitions

To make the translation into an alternating tree automaton easier, normal forms of Lµ formulae are considered.

Definition (Normal Forms)

Let ϕ ∈ Lµ. (i) ϕ is standardised apart if every atomic proposition in ϕ occurs either only free

• r is bound exactly once.

The subformula ηp.ψ uniquely determined by p will be denoted by ϕp. (ii) ϕ is in negation normal form if ¬ψ ≤ ϕ implies ψ ∈ P for every ψ ∈ Lµ. That is, the negation connective only occurs in front of atomic propositions. (iii) ϕ is in normal form if it is standardised apart and in negation normal form.

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01 Normal Forms

Notes

• For every formula ϕ there is an equivalent formula ϕ′ that is in normal form.
• |ϕ′| ≤ 2 · |ϕ|.
• Transformation to normal form takes linear time.

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01 Alternation Depth

Definition (Alternation Depth)

Let ϕ be an Lµ formula being standardised apart. The alternation depth of ϕ, denoted α(ϕ), is recursively defined as follows: α(⊥) = α(⊤) = α(p) = 0, α(ψ1 ∨ ψ2) = α(ψ1 ∧ ψ2) = max({α(ψ1), α(ψ2)}), α(¬ψ) = α(ψ) = α(♦ψ) = α(ψ), α(µp.ψ) = max({1, α(ψ)} ∪ {α(νq.ψ′) + 1 | νq.ψ′ ≤ ψ, p ∈ free(νq.ψ′)}), α(νp.ψ) = max({1, α(ψ)} ∪ {α(µq.ψ′) + 1 | µq.ψ′ ≤ ψ, p ∈ free(µq.ψ′)}).

Note

The alternation depth of all subformulae of an Lµ formula ϕ can be computed in O |ϕ|2 time.

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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 16 of 28

02 Alternating Tree Automata

The Definition

Definition (Alternating Tree Automata)

An alternating tree automaton is a tuple A = (Q, qI, δ, Ω) where

• Q is a finite set of states of the automaton,
• qI ∈ Q is a state called initial state,
• δ : Q → TCQ is called transition function, and
• Ω : Q → ω is called priority function.

Skip alternating tree automata recap TU Dresden, February 1, 2008 µ-Calculus & Alternating Tree Automata slide 17 of 28

02 Alternating Tree Automata

The Definition

Definition (Alternating Tree Automata)

An alternating tree automaton is a tuple A = (Q, qI, δ, Ω) where

• Q is a finite set of states of the automaton,
• qI ∈ Q is a state called initial state,
• δ : Q → TCQ is called transition function, and
• Ω : Q → ω is called priority function.

Definition (Transition Conditions)

Let Q be some set of states. The transition conditions TCQ over Q are defined as follows:

• 0, 1 ∈ TCQ,
• p, ¬p ∈ TCQ

for all p ∈ P,

• q, q, ♦q ∈ TCQ

for all q ∈ Q, and

• q1 ∨ q2, q1 ∧ q2 ∈ TCQ

for all q1, q2 ∈ Q.

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02 Alternating Tree Automata

Runs

Definition (Runs of Alternating Tree Automata)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton and (K, wI) a pointed Kripke model where K = (W, R, λ). A run of A on (K, w) is a (W × Q)-vertex-labelled tree R = (V, E, l) such that the root is labelled with (wI, qI) and for every vertex v with label (w, q) the following conditions are satisfied:

• δ(q) = 0.
• δ(q) = p

⇒ w ∈ λ(p)

• δ(q) = ¬p

⇒ w ∈ λ(p).

• δ(q) = q′

⇒ ∃v′ ∈ vE s.t l(v′) = (w, q′).

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02 Alternating Tree Automata

Runs

Definition (Runs of Alternating Tree Automata)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton and (K, wI) a pointed Kripke model where K = (W, R, λ). A run of A on (K, w) is a (W × Q)-vertex-labelled tree R = (V, E, l) such that the root is labelled with (wI, qI) and for every vertex v with label (w, q) the following conditions are satisfied:

• δ(q) = 0.
• δ(q) = p

⇒ w ∈ λ(p)

• δ(q) = ¬p

⇒ w ∈ λ(p).

• δ(q) = q′

⇒ ∃v′ ∈ vE s.t l(v′) = (w, q′).

• δ(q) = ♦q′

⇒ ∃w′ ∈ wR ∃v′ ∈ vE having l(v′) = (q′, w′)

• δ(q) = q′

⇒ ∀w′ ∈ wR ∃v′ ∈ vE having l(v′) = (q′, w′)

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02 Alternating Tree Automata

Runs

Definition (Runs of Alternating Tree Automata)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton and (K, wI) a pointed Kripke model where K = (W, R, λ). A run of A on (K, w) is a (W × Q)-vertex-labelled tree R = (V, E, l) such that the root is labelled with (wI, qI) and for every vertex v with label (w, q) the following conditions are satisfied:

• δ(q) = 0.
• δ(q) = p

⇒ w ∈ λ(p)

• δ(q) = ¬p

⇒ w ∈ λ(p).

• δ(q) = q′

⇒ ∃v′ ∈ vE s.t l(v′) = (w, q′).

• δ(q) = ♦q′

⇒ ∃w′ ∈ wR ∃v′ ∈ vE having l(v′) = (q′, w′)

• δ(q) = q′

⇒ ∀w′ ∈ wR ∃v′ ∈ vE having l(v′) = (q′, w′)

• δ(q) = q1 ∨ q2 ⇒ ∃v′ ∈ vE s.t. l(v′) = (w, q1) or l(v′) = (w, q2).
• δ(q) = q1 ∧ q2 ⇒ ∃v1, v2 ∈ vE s.t. l(v1) = (w, q1) and l(v2) = (w, q2).

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02 Alternating Tree Automata

Acceptance

Definition (Acceptance)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton and R = (V, E, l) a run of it.

• Let π = v0v1 . . . be an infinite branch of R and l(π) = (w0, q0)(w1, q1) . . .

its labelling trace. π is accepting if max{Ω(q)|q occurs infinitely often in q0q1 . . . } is even.

• R is accepting if every infinite branch of R is accepting.

Definition (Language of Alternating Tree Automata)

Let A be an alternating tree automaton and (K, w) a pointed Kripke model. (K, w) is accepted by A if there is an accepting run of A on (K, w). The set of all pointed Kripke models of an alternating tree automaton A — denoted L(A) — is called the language of A.

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02 Alternating Tree Automata

Transition Graph & Index

Definition (Transition Graph)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton. The transition graph of A, denoted G(A), is the directed graph (V, E) where:

• V = Q
• E = {(q, q′)|q′ appears in the transition condition δ(q)}

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02 Alternating Tree Automata

Transition Graph & Index

Definition (Transition Graph)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton. The transition graph of A, denoted G(A), is the directed graph (V, E) where:

• V = Q
• E = {(q, q′)|q′ appears in the transition condition δ(q)}

Definition (Index)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton and CA the set of all strongly connected components of G(A).

• For every C ∈ CA, mA

C denotes the number of different priorities of states

• ccurring in C, i.e.,

mA

C = |{Ω(q)|q ∈ C}|

• The index ind(A) of the automaton A is the maximum of all mA

C , i.e.,

ind(A) = max({mA

C |C ∈ CA} ∪ {0}) TU Dresden, February 1, 2008 µ-Calculus & Alternating Tree Automata slide 20 of 28

02 Alternating Tree Automata

Complexity Results

Theorem (Word Problem Complexity)

Let A = (Q, qI, δ, Ω) be an alternating tree automaton with d different non-zero priorities and let (K, w) be a finite pointed Kripke model.

• There is an algorithm which computes in time

O

• d|Q|(|R| + 1)

|Q||W|

⌈d/2⌉

⌈d/2⌉

, and in space O (d|Q||W| log (|Q||W|)) . whether A accepts (K, w).

• The problem whether A accepts (K, w) is in UP ∩ co-UP.

Theorem (Emptiness Problem Complexity)

The emptiness problem of alternating tree automata is in Exptime.

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02 Translation into Alternating Tree Automaton

The Idea

• W.l.o.g. the input Lµ formula ϕ is in normal form.
• For every subformula ψ of ϕ there is a state ψ in the automaton.
• ϕ is the initial state.

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02 Translation into Alternating Tree Automaton

The Idea

• W.l.o.g. the input Lµ formula ϕ is in normal form.
• For every subformula ψ of ϕ there is a state ψ in the automaton.
• ϕ is the initial state.
• Boolean and modal connectives of Lµ are directly translated into the

respective connectives of transition conditions. (e.g. δ(ψ1 ∨ ψ2) = ψ1 ∨ ψ2)

• Unbound (“real”) atomic propositions and their negation are translated into

respective terminating transition conditions. (e.g. δ(¬p) = ¬p)

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02 Translation into Alternating Tree Automaton

The Idea

• W.l.o.g. the input Lµ formula ϕ is in normal form.
• For every subformula ψ of ϕ there is a state ψ in the automaton.
• ϕ is the initial state.
• Boolean and modal connectives of Lµ are directly translated into the

respective connectives of transition conditions. (e.g. δ(ψ1 ∨ ψ2) = ψ1 ∨ ψ2)

• Unbound (“real”) atomic propositions and their negation are translated into

respective terminating transition conditions. (e.g. δ(¬p) = ¬p)

• For fixed point connectives loops are constructed by having a transition from

states p to ηp.ψ.

• Least and greatest fixed points are discriminated by appropriately choosing

the priority function.

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02 Translation into Alternating Tree Automaton

Formal Definition

Definition

Let ϕ be an Lµ formula in normal form. The alternating tree automaton corresponding to ϕ, denoted by A(ϕ) = (Q, q0, δ, Ω), is defined as follows:

• Q = {ψ | ψ ≤ ϕ},
• q0 = ϕ,
• δ : Q → TC Q is defined by:

δ(⊥) = 0, δ(⊤) = 1, δ(p) =

• p

if p ∈ free(ϕ), ϕp

• therwise,

δ(¬p) = ¬p, δ(ψ1 ∨ ψ2) = ψ1 ∨ ψ2, δ(ψ1 ∧ ψ2) = ψ1 ∧ ψ2, δ(ψ) = ψ, δ(♦ψ) = ♦ψ, δ(µp.ψ) = ψ, δ(νp.ψ) = ψ,

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02 Translation into Alternating Tree Automaton

Formal Definition (cont.)

Definition (cont.)

• Ω : Q → ω is defined by:

Ω(ψ) =

  

smallest odd number ≥ α(ψ) − 1 if ψ = µp.ψ′, smallest even number ≥ α(ψ) − 1 if ψ = νp.ψ′,

• therwise.

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02 Translation into Alternating Tree Automaton

Formal Definition (cont.)

Definition (cont.)

• Ω : Q → ω is defined by:

Ω(ψ) =

  

smallest odd number ≥ α(ψ) − 1 if ψ = µp.ψ′, smallest even number ≥ α(ψ) − 1 if ψ = νp.ψ′,

• therwise.

Note

This construction takes O |ϕ|2 time and O (|ϕ|) space.

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02 Translation into Alternating Tree Automaton

Formal Definition (cont.)

Definition (cont.)

• Ω : Q → ω is defined by:

Ω(ψ) =

  

smallest odd number ≥ α(ψ) − 1 if ψ = µp.ψ′, smallest even number ≥ α(ψ) − 1 if ψ = νp.ψ′,

• therwise.

Note

This construction takes O |ϕ|2 time and O (|ϕ|) space.

Observation

(i) α(ϕ) = ind(A(ϕ)), (ii) ind(A(ϕ)) = “number of different non-zero priorities”, (iii) |QA(ϕ)| = |ϕ|S.

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02 Equivalence of ϕ and A(ϕ)

Theorem

Let ϕ be am Lµ formula in normal form. Then for every pointed Kripke model (K, w) the following holds: (K, w) | = ϕ iff (K, w) ∈ L(A(ϕ)).

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02 Equivalence of ϕ and A(ϕ)

Theorem

Let ϕ be am Lµ formula in normal form. Then for every pointed Kripke model (K, w) the following holds: (K, w) | = ϕ iff (K, w) ∈ L(A(ϕ)).

Consequences

• The model-checking problem of the modal µ-calculus can be reduced to the

word problem of alternating tree automata.

• The satisfiability problem of the modal µ-calculus can be reduced to the

(non-)emptiness problem of alternating tree automata.

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02 Model-Checking and Satisfiability

Complexity Results

Corollary (Model-Checking Problem Complexity)

Let ϕ be an Lµ formula with α(ϕ) ≥ 2 and (K, w) a pointed Kripke model.

• Then solving the corresponding model-checking problem is bound in time by

O

• α(ϕ)|ϕ|(|R| + 1)

|ϕ||W|

⌈α(ϕ)/2⌉

⌈α(ϕ)/2⌉

, and in space by O (α(ϕ)|ϕ||W| log (|ϕ||W|)) .

• The model-checking problem for the modal µ-calculus is in UP ∩ co-UP.

Corollary (Satisfiability Problem Complexity)

The satisfiability problem for the modal µ-calculus is in Exptime.

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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 27 of 28

03 Conclusion

• The modal µ-calculus is a powerful logic to describe properties of transition

systems (Kripke models).

• There is an effective construction of an equivalent alternating tree automaton

corresponding to an Lµ formula.

• Hence, model-checking and satisfiability of modal µ-calculus can be reduced

to word problem and emptiness problem of alternating tree automata.

• Significant quantities of the formula (“size”, alternation depth) are conveyed

to the equivalent automaton (state space, number of non-zero priorities).

• Hence, complexity results of alternating tree automaton are easily translated

to the modal µ-calculus.

• Considering a fixed Lµ formula the model-checking problem is polynomial in

the size of the state space of the transition system.

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