On Domination and Control in Strategic Ability Micha l Knapik - PowerPoint PPT Presentation

On Domination and Control in Strategic Ability Micha� l Knapik Institute of Computer Science Polish Academy of Sciences (joint work with Wojtek Jamroga and Damian Kurpiewski) TDCS Seminar, 30 May 2019

Outline Enforceability in Concurrent Epistemic Game Structures Comparing Partial Strategies: Strategic Domination Applications: Model Checking Conclusions 2 / 17

Enforceability in CEGS • Models: Concurrent Epistemic Game Structures • Verif. properties: Alternating-time Temporal Logic ( ATL ) - enforceability � � A � � F goal : coalition A has a collective strategy to enforce goal eventually. I.e., ∃ σ A � σ A � F goal . • Difficult: NP -complete and no fixed-point algorithms! • Our contributions: ◦ new method of comparing partial strategies ◦ application: alleviation of brute-force winning strategy synthesis ◦ application: strategy optimisation ◦ a step towards distributed and parallel synthesis 3 / 17

Concurrent Epistemic Game Structures CEGS over A gt = { 1 , . . . , k } and Act : • automaton with states St • protocol d i ( q ) ⊆ Act for agent i q init start • transition function o ( q , α 1 , . . . , α k ) agent i selects action α i in q. . . ( A , U ) ( A , V ) • labeling V ( p ) ⊆ St of states with ( B , ⋆ ) propositions 1 • ∼ i indistinguishability relation for i q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 4 / 17

Concurrent Epistemic Game Structures - Strategies Uniform Strategy for agent i : • σ i : St → Act s.t. q ∼ i q ′ = ⇒ σ i ( q ) = σ i ( q ′ ) q init start ( A , U ) ( A , V ) ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 5 / 17

Concurrent Epistemic Game Structures - Strategies Uniform Strategy for agent i : • σ i : St → Act s.t. q ∼ i q ′ = ⇒ σ i ( q ) = σ i ( q ′ ) q init start Uniform Strategy for coalition A : • set σ A of uniform strat. for all i ∈ A ( A , U ) ( A , V ) ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 5 / 17

Concurrent Epistemic Game Structures - Strategies Uniform Strategy for agent i : • σ i : St → Act s.t. q ∼ i q ′ = ⇒ σ i ( q ) = σ i ( q ′ ) q init start Uniform Strategy for coalition A : • set σ A of uniform strat. for all i ∈ A ( A , U ) ( A , V ) Outcome out ( q , σ A ) of σ A from q : ( B , ⋆ ) • all paths resulting from A following 1 σ A and A gt \ A unrestricted q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 5 / 17

Concurrent Epistemic Game Structures - Strategies Uniform Strategy for agent i : • σ i : St → Act s.t. q ∼ i q ′ = ⇒ σ i ( q ) = σ i ( q ′ ) q init start Uniform Strategy for coalition A : • set σ A of uniform strat. for all i ∈ A ( A , U ) ( A , V ) Outcome out ( q , σ A ) of σ A from q : ( B , ⋆ ) • all paths resulting from A following 1 σ A and A gt \ A unrestricted q 2 q 1 Example: ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) • let σ 1 ( q init ) = σ 1 ( q 1 ) = σ 1 ( q 2 ) = A • ∀ π ∈ out ( q init , σ 1 ) ∃ i π i is red q 3 q 4 q 5 so: q init | = � σ 1 � F red 5 / 17

Some Problems and Previous Results • Checking � � A � � F goal under no memory and imperfect knowledge is NP -complete (and ∆ P 2 -complete for whole ATL ir ). • Standard fixed-point ATL equivalences don’t work . . . and ATL ir cannot be embedded in AE µ C = ⇒ no fixed-point procedures? [1,2] • (Symbolic) algorithms and methods mostly based on brute-force. [3,4,5] • Exception: in [6] an approximate fixed-point verification put forward (but sometimes inconclusive). 1. Bulling, Jamroga: Alternating Epistemic Mu-Calculus, 2011. 2. Dima, Maubert, Pinchinat: Relating Paths in Transition Systems. . . , 2015. 3. Lomuscio, Raimondi: Automatic Verification of Multi-agent Systems by Model Checking. . . , 2007. 4. Busard et. al: Reasoning About Memoryless Strategies. . . , 2015. 5. Pilecki et. al: SMC: Synthesis of Uniform Strategies. . . , 2017. 6. Jamroga et. al: Approximate Verification of Strategic Abilities. . . , 2018. 6 / 17

Partial Strategies • Partial Strategy for agent i : partial function that can be extended to strategy for i . (Accord. σ A for coalition A .) 7 / 17

Partial Strategies • Partial Strategy for agent i : partial function that can be extended to strategy for i . (Accord. σ A for coalition A .) • Partial strategies σ A and σ ′ A are conflictless iff dom ( σ A ) ∩ dom ( σ ′ A ) = ∅ and σ A ∪ σ ′ A is a partial strategy. 7 / 17

Partial Strategies • Partial Strategy for agent i : partial function that can be extended to strategy for i . (Accord. σ A for coalition A .) • Partial strategies σ A and σ ′ A are conflictless iff dom ( σ A ) ∩ dom ( σ ′ A ) = ∅ and σ A ∪ σ ′ A is a partial strategy. • Fusion of conflictless partial strategies: σ A ∪ · σ ′ A = σ A ∪ σ ′ A ∪ { assignments induced for all agents from A by σ A ∪ σ ′ A } . 7 / 17

Partial Strategies • Partial Strategy for agent i : partial function that can be extended to strategy for i . (Accord. σ A for coalition A .) • Partial strategies σ A and σ ′ A are conflictless iff dom ( σ A ) ∩ dom ( σ ′ A ) = ∅ and σ A ∪ σ ′ A is a partial strategy. • Fusion of conflictless partial strategies: σ A ∪ · σ ′ A = σ A ∪ σ ′ A ∪ { assignments induced for all agents from A by σ A ∪ σ ′ A } . • Notions of outcome, etc. adapted to partial strategies. 7 / 17

Partial Strategies, ct’d Example (part. strat. for 1 ): q init start • dom ( σ 1 ) = { q init } with σ 1 ( q init ) = A • dom ( σ ′ 1 ) = { q 2 } with σ ′ 1 ( q 2 ) = A ( A , U ) ( A , V ) • σ 1 ∪ · σ ′ 1 with dom ( σ 1 ∪ · σ ′ 1 ) = { q init , q 1 , q 2 } ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 8 / 17

Partial Strategies, ct’d Example (part. strat. for 1 ): q init start ⋆ dom ( σ 1 ) = { q init } with σ 1 ( q init ) = A • dom ( σ ′ 1 ) = { q 2 } with σ ′ 1 ( q 2 ) = A ( A , U ) ( A , V ) • σ 1 ∪ · σ ′ 1 with dom ( σ 1 ∪ · σ ′ 1 ) = { q init , q 1 , q 2 } ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 8 / 17

Partial Strategies, ct’d Example (part. strat. for 1 ): q init start • dom ( σ 1 ) = { q init } with σ 1 ( q init ) = A ⋆ dom ( σ ′ 1 ) = { q 2 } with σ ′ 1 ( q 2 ) = A ( A , U ) ( A , V ) • σ 1 ∪ · σ ′ 1 with dom ( σ 1 ∪ · σ ′ 1 ) = { q init , q 1 , q 2 } ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 8 / 17

Partial Strategies, ct’d Example (part. strat. for 1 ): q init start • dom ( σ 1 ) = { q init } with σ 1 ( q init ) = A • dom ( σ ′ 1 ) = { q 2 } with σ ′ 1 ( q 2 ) = A ( A , U ) ( A , V ) ⋆ σ 1 ∪ · σ ′ 1 with dom ( σ 1 ∪ · σ ′ 1 ) = { q init , q 1 , q 2 } ( B , ⋆ ) 1 q 2 q 1 ( B , ⋆ ) ( B , ⋆ ) ( A , V ) ( A , V ) ( A , U ) ( A , U ) q 3 q 4 q 5 8 / 17

Comparing Strategies: Strategic Domination • Let σ C A and σ A be conflictless. Call σ C A context. 9 / 17

Comparing Strategies: Strategic Domination • Let σ C A and σ A be conflictless. Call σ C A context. • Call all states in dom ( σ A ∪ · σ A ) found along � A ) out ( q , σ C A ) q ∈ dom ( σ C inputs of σ C A into σ A : I ( σ C A , σ A ). (For technical reasons also q init is an input if q init ∈ dom ( σ A ) ). 9 / 17

Comparing Strategies: Strategic Domination • Let σ C A and σ A be conflictless. Call σ C A context. • Call all states in dom ( σ A ∪ · σ A ) found along � A ) out ( q , σ C A ) q ∈ dom ( σ C inputs of σ C A into σ A : I ( σ C A , σ A ). (For technical reasons also q init is an input if q init ∈ dom ( σ A ) ). • Let q ∈ I ( σ C A , σ A ). Outputs of σ A (w.r.t. σ C A ) in q are states along · σ A ) but not in dom ( σ A ): IO ( σ C out ( q , σ A ∪ A , σ A )( q ). 9 / 17

Comparing Strategies: Strategic Domination • Let σ C A and σ A be conflictless. Call σ C A context. • Call all states in dom ( σ A ∪ · σ A ) found along � A ) out ( q , σ C A ) q ∈ dom ( σ C inputs of σ C A into σ A : I ( σ C A , σ A ). (For technical reasons also q init is an input if q init ∈ dom ( σ A ) ). • Let q ∈ I ( σ C A , σ A ). Outputs of σ A (w.r.t. σ C A ) in q are states along · σ A ) but not in dom ( σ A ): IO ( σ C out ( q , σ A ∪ A , σ A )( q ). Strategic domination Let σ C A , σ A and σ C A , σ ′ A be conflictless. If I ( σ C A , σ A ) = I ( σ C A , σ ′ A ) and for each q ∈ I ( σ C A , σ A ) we have IO ( σ C A , σ ′ A )( q ) ⊆ IO ( σ C A , σ A )( q ) then σ ′ A dominates σ A w.r.t. σ C A : A σ ′ σ A � σ C A . 9 / 17