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The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems

Alessandro Facchini1 Luca Alberucci2

1 University of Lausanne and LaBRI, Bordeaux 2 IAM, University of Berne

Logic Colloquium Berne, July 4th 2008

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Goal of our works

Understand the expressive power of modal µ-calculus over different classes of models.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

What is the modal µ-calculus

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

What is the modal µ-calculus

the propositional modal µ-calculus = propositional modal logic + least and greatest fixpoint operators

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Expressive power

Eventually ”p”: µx.p ∨ ✸x Allways ”p”: νx.p ∧ x Allways eventually ”p”: νx.(µy.p ∨ ✸y) ∧ x There is a branch such that infinitely often ”p”: νx.µy.(p ∧ ✸x) ∨ ✸y

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

The fixpoint alternation depth

The fixpoint alternation of a formula is the number of non-trivial nestings of alternating least and greatest fixpoints.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Example

ϕ1 := p ∨ ♦q

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Example

ϕ2 := µx.p ∨ ♦x

µx ∨

❄

x

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . p

✛

♦

❄ ✲

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Example

ϕ3 := νx.µy.(p ∧ ♦x) ∨ ♦y

νx x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✲

µy

❄

♦

✲

∨

❄

y

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . p ✛ ∧

✛ ✛

♦

❄ ✲

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Example

ϕ4 := µx(νy(p ∧ ♦y) ∨ x)

µx ∨

❄

x

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . νy

✛

• ✻

✲

y . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✲

∧

❄

♦

✻ ✛

p

✲

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

∆µ

3

Σµ

2

• Πµ

2

• ∆µ

2

• Σµ

1

• Πµ

1

• ∆µ

1

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Is the semantical hierarchy strict?

∆Tµ

3

ΣTµ

2

• ΠTµ

2

• ∆Tµ

2

• ΣTµ

1

• ΠTµ

1

• ∆Tµ

1

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Semantical complexity

Bradfield (1996): Stictness of semantical modal µ-calculus hierarchy

The semantical modal µ-calculus hierarchy is strict on the class of all transition systems. ⇒ For each n there is a formula ϕ with ad(ϕ) = n such that for all formulae ψ with ad(ψ) < n we do not have for all transition systems T : (T | = ϕ ⇔ T | = ψ).

Proof.

Game formulae are complete for their corresponding level.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Question

What happens for restricted classes of transition systems? refl sym tr sym & tr tr & wf strict ? ? ? ? ? collapse ? ? ? ? ?

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems Introduction

Overview

The modal µ-calculus The Hierarchy on Transitive and Symmetric Transition Systems The Hierarchy on Transitive Transition Systems A final picture

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Syntax of µ-calculus

ϕ ::≡ p | ∼ p | ⊤ | ⊥ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | ✸ϕ | ϕ . . . . . . | µx.ϕ | νx.ϕ where p, x ∈ Prop and x occurs only positively.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Semantics

A transition system T is a triple (S, →T , λT ) consisting of

◮ a set S of states, ◮ a binary relation →T ⊆ S × S called transition relation, ◮ the valuation λ : P → ℘(S) assigning to each propositional

variable p a subset λ(p) of S.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Denotation and validity of a formula

Given a transition system T = (S, →T , λT ). We define ϕT inductively such that

◮ pT = λT (p) and ∼ pT = S \ λT (p) ◮ α ∧ βT = αT ∩ βT and α ∨ βT = αT ∪ βT , ◮ αT = {s ∈ S | ∀s′(s →T s′ =

⇒ s′ ∈ αT )},

◮ ✸αT = {s ∈ S | ∃s′(s →T s′ and s′ ∈ αT )}.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Denotation and validity of a formula

◮ . . . ◮ νx.ϕ(x)T =

{S′ ⊆ S | S′ ⊆ ϕ(x)T [x→S′]} GFP(ϕ(x)T )

◮ µx.ϕ(x)T =

{S′ ⊆ S | ϕ(x)T [x→S′] ⊆ S′} LFP(ϕ(x)T )

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

An equivalence

A formula is well-named if bound and free variables are pairwise distinct and if all bound variable occur only once and are guarded.

Lemma

Every formula ϕ is equivalent to a well-named formula nf(ϕ).

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation games

◮ The player are V (verifier) and F (falsifier). Consider w0 ∈ S,

V tries to show that w0 ϕ, while F tries to show that w0 ϕ.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation games

◮ The player are V (verifier) and F (falsifier). Consider w0 ∈ S,

V tries to show that w0 ϕ, while F tries to show that w0 ϕ.

◮ the play starts at ϕ, w0 ◮ the admissible moves are choices of subformulas and points of

S which respect the following rules:

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation game for modal logic

position player next position pi, w F iff pi ∈ λ(w)

• ¬pi, w

V iff pi ∈ λ(w)

• ψ ∨ φ, w

V chooses between ψ, w and φ, w V choice ψ ∧ φ, w F chooses between ψ, w and φ, w F choice ♦ψ, w V chooses a point w ′ s.t. wRw ′ ψ, w ′ ψ, w F chooses a point w ′ s.t. wRw ′ ψ, w ′

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation game for modal logic

◮ V wins iff F cannot move.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Example

(♦♦p ∨ p) ∧ r

✄∧ ∨

✛

r

✲

♦

✛

• ❄

♦

❄

p❄ p❄ {p, q, r} {p}

✛

∅

❄

∅

✛

∅

❄

∅

❄

{p}

✲

. . . . . . . . . . . .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation game for µ-calculus

position player next position pi, w F iff pi ∈ λ(w)

• ¬pi, w

V iff pi ∈ λ(w)

• ψ ∨ φ, w

V chooses between ψ, w and φ, w V choice ψ ∧ φ, w F chooses between ψ, w and φ, w F choice ♦ψ, w V chooses a point w ′ s.t. wRw ′ ψ, w ′ ψ, w F chooses a point w ′ s.t. wRw ′ ψ, w ′ µx.ψ, w

• ψ, w

νx.ψ, w

• ψ, w

x, w

• ηx.ψ(x), w

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Evaluation game for µ-calculus

◮ . . . the game behind the µ-calculus is a parity game ◮ . . . “µ means finite looping” ◮ . . . “ν means infinite looping”

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Ω : V → ω is priority function defined on ηx.δ, s ∈ V , where η ∈ {µ, ν}. In this case we have that: Ω(ηx.δ, s) =        ad(ηx.δ) if η = µ and ad is odd, or η = ν and ad is even; ad(ηx.δ) − 1 if η = µ and ad is even, or η = ν and ad is odd.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Winning Conditions

Given a play π of E(ϕ, (T , wI ))

• 1. if π is finite, V wins iff F can’t move anymore;
• 2. if π is infinite iff V wins iff the highest priority appearing

infinitely often is even.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Example

µx(p ∨ ♦x)

✄µ ✛ ♦ ∨

❄ ✲

p❄ ∅ ∅

✛

∅

❄

∅

✛

∅

❄

∅

❄

∅

✲

. . . . . . . . . . . . . . . . . . . . . . . .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Example

µx(p ∨ ♦x)

✄µ ✛ ♦ ∨

❄ ✲

p❄ ∅ ∅

✛

∅

❄

∅

✛

∅

❄

∅

❄

∅

✲

. . . . . . . . . . . . . . . . . . . . . {p}

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The modal µ-calculus

Game-theoretical version of the “fundamental theorem”

Theorem [Streett Emerson 89]

s ∈ ||ϕ||T iff V has a winning strategy in E(ϕ, (T , wI)).

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive and Symmetric Transition Systems

The Hierarchy on Transitive and Symmetric Transition Systems

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive and Symmetric Transition Systems

Theorem (AF (08a))

Let T be a transitive and symmetric transition system. We have that νx.ϕ(x)T = ϕ(ϕ(⊤))T .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive and Symmetric Transition Systems

The syntactical translation ()t : Lµ → LM is defined as:

◮ . . . ◮ (µx.ϕ)t = (ϕ(ϕ(⊥)))t ◮ (νx.ϕ)t = (ϕ(ϕ(⊤)))t

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive and Symmetric Transition Systems

A first collapse

Corollary (AF (08a))

On transitive and symmetric (and reflexive) transition systems we have that ϕT = ϕtT .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

The Hierarchy on Transitive Transition Systems

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

What was already known

Theorem (Lenzi (96), Lenzi-d’Agostino (08))

For every class A of transitive transition systems, A is definable by a µ-formula iff it is recognizable by B¨ uchi automata.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Conjecture

Over transitive transition systems, the µ-calculus collapses to its alternation free fragment.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Lemma

Let T be a transitive transition system and let s, s′ be two states such that s →T s′. For all µ-formulae ϕ we have that s ∈ ϕT = ⇒ s′ ∈ ϕT and s′ ∈ ✸ϕT = ⇒ s ∈ ✸ϕT .

Theorem (AF (08a))

Let T be a transitive transition system and let νx.ϕ(x) be a formula such that x is in the scope of a modality. We have that νx.ϕ(x)T = ϕ(ϕ(⊤))T .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

τ : Lµ → Lµ is defined as:

◮ . . . ◮ τ(µx.ϕ) = τ(ϕ(ϕ(⊥))), x is in the scope of a ✸ in ϕ ◮ τ(µx.ϕ) = µx.τ(ϕ), x is not in the scope of a ✸ in ϕ ◮ τ(νx.ϕ) = τ(ϕ(ϕ(⊤))), x is in the scope of a in ϕ ◮ τ(νx.ϕ) = νx.τ(ϕ), x is not in the scope of a in ϕ

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

A first step towards the collapse

Corollary

On transitive transition systems we have that ϕT = τ(ϕ)T .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Finite model theorem

Theorem (AF (08a))

For all modal µ-formulae ϕ for which there is a transitive transition system T and a state s in T such that s ∈ ϕT there is a finite transitive transition system T F and a state sF such that sF ∈ ϕT F .

Proof.

For every ϕ, s ∈ (ϕ)trT iff s ∈ ϕ(T )tr . Then, use the fmp of the µ-calculus.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

Let T be a finite transitive transition system and ϕ a µ-formula. Consider an arbitrary (memoryless) strategy σ for Player 0, not necessarily winning. We define the restriction of E(ϕ, (T , s0)) on σ, denoted by E|σ(ϕ, (T , s0)), the parity game starting from ϕ, s0 induced by σ.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

Given a winning strategy σ for player 0, we define a measure d on E|σ(ϕ, (T , s0)):

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

• 1. scc(ψ, s) = ∅ :

d(ψ, s) = ( if E|σ(ψ, s) = ∅ max{d(φ, s′) : φ, s′ ∈ E|σ(ψ, s)} + 1 else

• 2. scc(ψ, s) = ∅ :

d(ψ, s) = 0 if [ {E|σ(α, s) : α, s ∈ scc(ψ, s)} \ scc(ψ, s) = ∅, else d(ψ, s) = max{d(φ, s′) : φ, s′ / ∈ scc(ψ, s) and exists ξ, s′′ ∈ scc(ψ, s) with φ, s′ ∈ E|σ(ξ, s′′)} + 1.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

Lemma

Let T be a finite transitive transition system and ϕ ∈ Σµ

• 2. Suppose

there is a winning strategy σ for Player 0 in the parity game E(ϕ, (T , s0)). If y ∈ bound(ϕ) is a µ-variable, then for every position y, s ∈ V |σ, we have that scc(y, s) = ∅.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

Given a winning strategy σ for Player 0 on E(ϕ, (T , s0)), the normalized (winning) strategy σN for Player 0 is given by adapting σ such that for all vertexes of the form ✸β, s′ Player 0 moves to a vertex β, s′′ whose measure is the minimal measure of all positions of the type β,¯ s reachable (in one step) from ✸β, s′ in E|σ(ϕ, (T , s0)).

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Normalizing winning strategies

Theorem

Given a finite transitive transition system T , a formula ϕ ∈ Σµ

2

such that all ν-variables are weakly existential and a normalized winning strategy, σN, of Player 0 in E(ϕ, (T , s)). If in a play π consistent with σN there is a regeneration of a ν-variable x then

◮ either there is no more regeneration of a µ-variable after the

first regeneration of x

◮ or, if there is such a regeneration of a µ-variable, then after

this position there is no more regeneration of x.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Another collapse

The collapse of the µ-calculus over transitive models is then given by first “coding” the previous Theorem via a translation ρ ◦ τ : Σµ

2 → ∆µ 2

which preserves logical equivalence over finite transitive models.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Another collapse

Lemma

For all ϕ ∈ Σµ

2 and all finite transitive transition systems T we

have that ϕT = ρ(τ(ϕ))T .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Another collapse

R : Lµ → ∆µ

2 is defined as ◮ R(p) = p and R(¬p) = ¬p ◮ R(⊥) = ⊥ and R(⊤) = ⊤ ◮ R(α ◦ β) = R(α) ◦ R(β), where ◦ ∈ {∧, ∨} ◮ R(△ β) =△ R(β), where △∈ {, ✸} ◮ R(µx.ϕ) = nf

• ρ(τ(nf(µx.(R(ϕ)))))
• ◮ R(νx.ϕ) = ¬(R(µx.¬ϕ[x/¬x]))

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems The Hierarchy on Transitive Transition Systems

Another collapse

Theorem (AF (08a))

For all ϕ ∈ Lµ and all transitive transition systems T we have that ϕT = R(ϕ)T .

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems A final picture

The modal µ-calculus over restricted classes of models: a final picture

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems A final picture

A final picture

refl sym tr sym & tr tr & wf strict [AF(08a)] ? collapse ? [AF(08a)]: ∆µ

2

[AF(08a)]: LM [Vis(05)], [vBe(06)], [AF(08b)]: LM

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems A final picture

The symmetric case

Conjecture

The modal µ-calculus hierarchy is strict over symmetric models.

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems A final picture

The symmetric case

Conjecture

The modal µ-calculus hierarchy is strict over symmetric models.

Theorem (F(08))

The fixpoint alternation hierarchy of the modal µ-calculus with backward modalities is strict (over all transition systems).

The modal µ-calculus Hierarchy on Restricted Classes of Transition Systems A final picture

Thank you!