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 1

ATL with common knowledge semantics 1 ⊧ subj ⟪ alice , bob ⟫ X win ∼ alice ∼ bob 2 ∼ bob ∼ alice 1 3 4 HH missing arrows are losing HT TT TT win Common knowledge semantics for s ⊧ ⟪ A ⟫ ϕ requires: ● The existence of a joint strategy profile built over the whole common knowledge neigbourhood C A ( s ) . ● Which, when applied in each state of C A ( s ) , produces the objective ϕ . 2

ATL with common knowledge semantics ∗ Common knowledge = invariant under ∼ C A = ( ⋃ a ∈ A ∼ a ) . 1 ⊧ subj ⟪ alice , bob ⟫ X win ∼ alice ∼ bob 2 1 / ⊧ ck ⟪ alice , bob ⟫ X win ∼ bob ∼ alice 1 3 4 HH ? missing arrows are losing TT ? HT TT win Common knowledge semantics for s ⊧ ⟪ A ⟫ ϕ requires: ● The existence of a joint strategy profile built over the whole common knowledge neigbourhood C A ( s ) . ● Which, when applied in each state of C A ( s ) , produces the objective ϕ . 2

Alternating bisimulation – the idea G 1 G 2 ⇛ alice s ′ s a ′ 2, b ′ a ′ 1, b ′ a 2, b 1 a 1, b 2 1 2 a ′ 1, b ′ 2 a ′ 1, b ′ a ′ 1, b ′ a 1, b 1 1 2 a ′ 1, b ′ a ′ 1, b ′ a ′ 1, b ′ a 1, b 2 a 1, b 2 2 2 2 a ′ 1, b ′ a ′ 3, b ′ a 1, b 1 a 1, b 1 a 3, b 1 1 1 a ′ 1, b ′ a ′ 3, b ′ a ′ 1, b ′ a ′ 2, b ′ a ′ 3, b ′ a 1, b 2 a 3, b 1 a 2, b 2 2 1 1 2 1 alice ∀ ρ ′ ∈ out G 2 ( σ ′ , s ′ )∃ ρ ∈ out G 1 ( σ, s ) with ρ ′ ∀ σ alice ∃ σ ′ AP = ρ AP Guarantees that, for any φ , G 2 , s ′ ⊧ ⟪ alice ⟫ φ G 1 , s ⊧ ⟪ alice ⟫ φ ⇒ 3

Alternating (bi)simulation with imperfect information Definition ⇛ A ⊆ Hist ( G )× ∈ Hist ( G ′ ) for which, whenever h ⇛ A h ′ , then π ( h ) = π ′ ( h ′ ) . 1 [Compatibility with indistinguishability] For each a ∈ A , 2 ∀ k ′ ∼ a h ′ ∃ k ∼ a h with h ′ ⇛ A k ′ [Compatibility with uniform strategies] – ∀ σ ∈ PStr ( C A ( h )) ∃ σ ∈ PStr ( C A ( h ′ )) with... (see 3 next slide). History-based version of [Belardinelli, Condurache, D., Jamroga, Jones, Knapik, 2017, 2020] 4

Alternating bisimulation with imperfect information ( A = { alice , bob } ) ∃ ST ∶ PStr A ( C A ( s 1 )) → PStr A ( C ′ A ( s ′ 1 )) ⇛ alice , bob ⇛ alice , bob ⇛ alice , bob s ′ s ′ s ′ s ′ s 1 ∼ alice s 2 ∼ bob s 3 ∼ bob ∼ alice ∼ bob 1 2 3 3 a ′ 2 , b ′ 1 , c ′ a ′ 1 , b ′ 1 , c ′ a ′ 1 , b ′ 2 , c ′ a ′ 3 , b ′ 2 , c ′ a 1 , b 1 , c 2 a 1 , b 2 , c 1 a 2 , b 2 , c 2 3 1 2 2 a ′ 2 , b ′ 1 , c ′ a ′ 1 , b ′ 1 , c ′ a ′ 2 , b ′ 2 , c ′ a ′ 2 , b ′ 1 , c ′ a 2 , b 1 , c 1 a 3 , b 1 , c 1 1 2 3 3 t ′ t ′ t ′ t ′ t ′ t 1 t 2 t 3 t 4 t 5 1 2 3 4 5 ⇛ alice , bob ⇛ alice , bob ⇛ alice , bob ⇛ alice , bob 1 ) , r ⇛ A r ′ implies ∀ σ A ∈ PStr A ( C A ( s 1 )) , ∀ r ∈ C A ( s 1 ) , ∀ r ′ ∈ C ′ A ( s ′ ∀ r ′ ST ( σ A )( r ′) σ A ( r ) → s ′ , ∃ s ∈ C A ( s 1 ) , r → s and s ⇛ A s ′ � � � � � � � � � 5

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 ( h 0 ) 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 . 6

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 ( h 0 ) 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 . 6

The bisimulation game P-Spoil chooses k ′ ∼ a h ′ h , h ′ h , h ′ , k ′ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) P-Dupl chooses k ∼ a h ρ ∶ = ρ ∪ {( k , k ′ )} P-Spoil passes when ρ is ”complete” h , h ′ I-Spoil chooses h , h ′ , σ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) σ ∈ PStr ( C A ( h )) P-Dupl chooses l ∈ out ( σ, k ) I-Dupl chooses ρ ∶ = ∅ σ ′ ∈ PStr ( C ′ A ( h ′ )) k , k ′ , σ, σ ′ P-Spoil chooses h , h ′ σ, σ ′ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ( k , k ′ ) ∈ ρ P-Spoil chooses l ′ ∈ out ( σ ′ , k ′ ) k , k ′ , σ, σ ′ , l ′ Spoiler wins if π ( h ) ≠ π ′ ( h ′ ) or π ( k ) ≠ π ′ ( k ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) 7

The bisimulation game P-Spoil chooses k ′ ∼ a h ′ h , h ′ , k ′ h , h ′ l , l ′ ✟ ✟ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) P-Dupl chooses k ∼ a h ρ ∶ = ρ ∪ {( k , k ′ )} P-Spoil passes when ρ is ”complete” h , h ′ I-Spoil chooses h , h ′ , σ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) σ ∈ PStr ( C A ( h )) P-Dupl chooses l ∈ out ( σ, k ) I-Dupl chooses ρ ∶ = {( l , l ′ )} σ ′ ∈ PStr ( C ′ A ( h ′ )) k , k ′ , σ, σ ′ P-Spoil chooses h , h ′ σ, σ ′ ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) ( k , k ′ ) ∈ ρ P-Spoil chooses l ′ ∈ out ( σ ′ , k ′ ) k , k ′ , σ, σ ′ , l ′ Spoiler wins if π ( h ) ≠ π ′ ( h ′ ) or π ( k ) ≠ π ′ ( k ′ ) ρ ⊆ C A ( h ) × C ′ A ( h ′ ) 7

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. 8

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? 9