A Hennessy-Milner Theorem for ATL with Imperfect Information - - PowerPoint PPT Presentation

A Hennessy-Milner Theorem for ATL with Imperfect Information

Francesco Belardinelli, C˘ at˘ alin Dima, Vadim Malvone, Ferucio T ¸iplea

Imperial College London, LACL – Universit´ e Paris-Est Cr´ eteil, IBISC – Universit´ e d’Evry, University of Ia¸ si

LICS 2020 & Highlights 2020

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ATL with common knowledge semantics

1 ⊧subj ⟪alice, bob⟫X win missing arrows are losing 1 2 3 4 win ∼alice ∼bob ∼alice ∼bob HT HH TT TT Common knowledge semantics for s ⊧ ⟪A⟫ϕ requires:

• The existence of a joint strategy profile built over the whole common knowledge

neigbourhood CA(s).

• Which, when applied in each state of CA(s), produces the objective ϕ.

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ATL with common knowledge semantics

Common knowledge = invariant under ∼C

A= ( ⋃a∈A ∼a ) ∗

. 1 ⊧subj ⟪alice, bob⟫X win 1 / ⊧ck ⟪alice, bob⟫X win missing arrows are losing 1 2 3 4 win ∼alice ∼bob ∼alice ∼bob HT HH? TT? TT Common knowledge semantics for s ⊧ ⟪A⟫ϕ requires:

• The existence of a joint strategy profile built over the whole common knowledge

neigbourhood CA(s).

• Which, when applied in each state of CA(s), produces the objective ϕ.

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Alternating bisimulation – the idea

G1 G2 s s′

a2,b1 a1,b2 a1,b1 a1,b2 a1,b2 a1,b1 a1,b2 a3,b1 a1,b1 a2,b2 a3,b1 a′ 2,b′ 1 a′ 1,b′ 2 a′ 1,b′ 2 a′ 1,b′ 1 a′ 1,b′ 2 a′ 1,b′ 2 a′ 1,b′ 2 a′ 1,b′ 2 a′ 1,b′ 1 a′ 1,b′ 2 a′ 3,b′ 1 a′ 1,b′ 1 a′ 2,b′ 2 a′ 3,b′ 1 a′ 3,b′ 1

⇛alice

∀σalice∃σ′

alice∀ρ′ ∈ outG2(σ′, s′)∃ρ ∈ outG1(σ, s) with ρ′ AP = ρ AP

Guarantees that, for any φ, G1, s ⊧ ⟪alice⟫φ ⇒ G2, s′ ⊧ ⟪alice⟫φ

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Alternating (bi)simulation with imperfect information

Definition

⇛A⊆ Hist(G)× ∈ Hist(G ′) for which, whenever h ⇛A h′, then

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π(h) = π′(h′).

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[Compatibility with indistinguishability] For each a ∈ A, ∀k′ ∼a h′ ∃k ∼a h with h′ ⇛A k′

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[Compatibility with uniform strategies] – ∀σ ∈ PStr(CA(h)) ∃σ ∈ PStr(CA(h′)) with... (see next slide). History-based version of [Belardinelli, Condurache, D., Jamroga, Jones, Knapik, 2017, 2020]

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Alternating bisimulation with imperfect information (A = {alice,bob})

∃ST ∶ PStrA(CA(s1)) → PStrA(C′

A(s′ 1)) s1 s2 s3

∼alice ∼bob

t1 t2 t3 t4 t5

a2, b1,c1 a1, b1,c2 a1, b2,c1 a2, b2,c2 a3, b1,c1

s′ 1 s′ 2 s′ 3 s′ 3

∼bob ∼alice ∼bob

t′ 1 t′ 2 t′ 3 t′ 4 t′ 5

a′

2, b′ 1,c′ 1

a′

2, b′ 1,c′ 3

a′

1, b′ 1,c′ 1

a′

1, b′ 1,c′ 2

a′

1, b′ 2,c′ 2

a′

2, b′ 2,c′ 3

a′

3, b′ 2,c′ 2

a′

2, b′ 1,c′ 3

⇛alice ,bob ⇛alice ,bob ⇛alice ,bob ⇛alice ,bob ⇛alice ,bob ⇛alice ,bob ⇛alice ,bob

∀r ∈ CA(s1), ∀r′ ∈ C′

A(s′ 1), r ⇛A r′ implies ∀σA ∈ PStrA(CA(s1)),

∀r′ ST(σA)(r′)

• → s′, ∃s ∈ CA(s1), r

σA(r)

• → s and s ⇛A s′

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The Hennessy-Milner Theorem

Theorem

Assume ⇚ ⇛A is a history-based A-bisimulation between two game structures G and G′. Let h ∈ Hist(G) and h′ ∈ Hist(G′) with h ⇚ ⇛A h′. Then, for every A-formula φ and x ∈ {subj, obj, ck}, (G, h) ⊧x φ iff (G′, h′) ⊧x φ

Theorem

G(h0) and G′(h′

0) are A-equivalent for the common knowledge semantics ⊧ck if and only if they

are A-bisimilar. 2nd theorem fails for ⊧subj.

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The Hennessy-Milner Theorem

Theorem

Assume ⇚ ⇛A is a history-based A-bisimulation between two game structures G and G′. Let h ∈ Hist(G) and h′ ∈ Hist(G′) with h ⇚ ⇛A h′. Then, for every A-formula φ and x ∈ {subj, obj, ck}, (G, h) ⊧x φ iff (G′, h′) ⊧x φ

Theorem

G(h0) and G′(h′

0) are A-equivalent for the common knowledge semantics ⊧ck if and only if they

are A-bisimilar. 2nd theorem fails for ⊧subj.

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The bisimulation game

h, h′ ρ ⊆ CA(h) × C ′

A(h′)

h, h′, k′ ρ ⊆ CA(h) × C ′

A(h′)

Spoiler wins if π(h) ≠ π′(h′) or π(k) ≠ π′(k′)

h, h′ ρ ⊆ CA(h) × C ′

A(h′)

h, h′, σ ρ ⊆ CA(h) × C ′

A(h′)

h, h′σ, σ′ ρ ⊆ CA(h) × C ′

A(h′)

k, k′, σ, σ′ ρ ⊆ CA(h) × C ′

A(h′)

k, k′, σ, σ′, l′ ρ ⊆ CA(h) × C ′

A(h′)

P-Spoil chooses k′ ∼a h′ P-Dupl chooses k ∼a h ρ ∶= ρ ∪ {(k, k′)} P-Spoil passes when ρ is ”complete” I-Spoil chooses σ ∈ PStr(CA(h)) I-Dupl chooses σ′ ∈ PStr(C ′

A(h′))

P-Spoil chooses (k, k′) ∈ ρ P-Spoil chooses l′ ∈ out(σ′, k′) P-Dupl chooses l ∈ out(σ, k) ρ ∶= ∅

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The bisimulation game

✟ ✟

h, h′l, l′ ρ ⊆ CA(h) × C ′

A(h′)

h, h′, k′ ρ ⊆ CA(h) × C ′

A(h′)

Spoiler wins if π(h) ≠ π′(h′) or π(k) ≠ π′(k′)

h, h′ ρ ⊆ CA(h) × C ′

A(h′)

h, h′, σ ρ ⊆ CA(h) × C ′

A(h′)

h, h′σ, σ′ ρ ⊆ CA(h) × C ′

A(h′)

k, k′, σ, σ′ ρ ⊆ CA(h) × C ′

A(h′)

k, k′, σ, σ′, l′ ρ ⊆ CA(h) × C ′

A(h′)

P-Spoil chooses k′ ∼a h′ P-Dupl chooses k ∼a h ρ ∶= ρ ∪ {(k, k′)} P-Spoil passes when ρ is ”complete” I-Spoil chooses σ ∈ PStr(CA(h)) I-Dupl chooses σ′ ∈ PStr(C ′

A(h′))

P-Spoil chooses (k, k′) ∈ ρ P-Spoil chooses l′ ∈ out(σ′, k′) P-Dupl chooses l ∈ out(σ, k) ρ ∶= {(l, l′)}

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Determinacy for bisimulation games

Gale-Stewart games between 4 players:

• Reachability objective for Spoilers, safety objective for Duplicators.
• Defensive strategy: does not put the game into a winning state for the opponent coalition.
• Defensive strategies against reachability objectives can be transformed into winning

strategies (for the safety objective).

• The game is determined, since when Spoilers do not win, a defensive strategy for

Duplicators exists.

• From a winning strategy for Spoilers one may build an ATL formula containing the

Yesterday modality which is distinguishes the two CGS.

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Concluding remarks

• Bisimulations can be adapted to zero-sum game structures with imperfect information.
• Simple combinations of (perfect information) alternating bisimulations and epistemic

bisimulations don’t work.

• History-based alternating bisimulation is undecidable (see paper).

Further work

• Strategy logic with imperfect information?
• Determinacy for other types of zero-sum multy-player games?

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