Reasoning about Knowledge and Strategies: Epistemic Strategy Logic Francesco Belardinelli Laboratoire IBISC, Universit´ e d’Evry Strategic Reasoning – 5 April 2014 1
Overview Motivation and Background 1 ◮ logics for reasoning about strategies and knowledge Epistemic Strategy Logic 2 ◮ semantics ◮ syntax Main Contribution 3 ◮ model checking ESL is no harder than SL Imperfect Information 4 ◮ benefits of combining epistemic and strategy modalities Conclusions and Future Work 5 2
Motivation and Background Logics of strategic abilities • Logics for strategic reasoning are a thriving area of research in AI and MAS. 3
Motivation and Background Logics of strategic abilities • Logics for strategic reasoning are a thriving area of research in AI and MAS. • Two lines of research: 1 multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups: ⋆ A lternating-time T emporal L ogic [AHK02] ⋆ C oalition L ogic [Pau02] ⋆ S trategy L ogic [CHP10, MMV10] 2 extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution: ⋆ combinations of CTL and LTL with multi-modal epistemic logic S5 n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN + 08, LQR09] 3
Motivation and Background Logics of strategic abilities • Logics for strategic reasoning are a thriving area of research in AI and MAS. • Two lines of research: 1 multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups: ⋆ A lternating-time T emporal L ogic [AHK02] ⋆ C oalition L ogic [Pau02] ⋆ S trategy L ogic [CHP10, MMV10] 2 extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution: ⋆ combinations of CTL and LTL with multi-modal epistemic logic S5 n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN + 08, LQR09] • Along these lines, [vdHW03] introduced ATEL. ◮ spawned a wealth of contributions: ⋆ imperfect information/uniform strategies [Sch04, JvdH04] ⋆ constructive knowledge [J˚ A07] ⋆ irrevocable/feasible strategies [AGJ07, Jon03] 3
Motivation and Background Logics of strategic abilities • Logics for strategic reasoning are a thriving area of research in AI and MAS. • Two lines of research: 1 multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups: ⋆ A lternating-time T emporal L ogic [AHK02] ⋆ C oalition L ogic [Pau02] ⋆ S trategy L ogic [CHP10, MMV10] 2 extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution: ⋆ combinations of CTL and LTL with multi-modal epistemic logic S5 n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN + 08, LQR09] • Along these lines, [vdHW03] introduced ATEL. ◮ spawned a wealth of contributions: ⋆ imperfect information/uniform strategies [Sch04, JvdH04] ⋆ constructive knowledge [J˚ A07] ⋆ irrevocable/feasible strategies [AGJ07, Jon03] E pistemic S trategy L ogic = strategies + knowledge ◮ topic of interest [HvdM14b, HvdM14a] 3
The Prisoner’s Dilemma Games in Normal Form • A nne and B ob can either C ooperate or D efect • payoff ordering: a > b > c > d B ob C ooperate D efect C ooperate b , b d , a A nne D efect a , d c , c • can A nne achieve payoff a ? • does A nne know whether she can achieve payoff a ? • does B ob know whether A nne has a strategy to achieve payoff a ? • do A nne and B ob know ( de dicto ) whether they can reach a Nash equilibrium? • are there strategies such that A nne and B ob know ( de re ) that they can reach a Nash equilibrium? 4
Epistemic Concurrent Game Models Agents We adopt an agent-oriented perspective. Definition (Agent) An agent i is • situated in some local state l i ∈ L i and . . . • performs the actions in Act i • . . . according to her protocol function Pr i : L i �→ 2 Act i The setting is reminiscent of the interpreted systems semantics for MAS [FHMV95]. Example (Prisoner’s Dilemma) Agent A nne = � L A , Act A , Pr A � is defined as • L A = { ǫ A , a , b , c , d } • Act A = { C , D , ∗} , where ∗ is the skip action • Pr A ( ǫ A ) = { C , D } and Pr A ( a ) = Pr A ( b ) = Pr A ( c ) = Pr A ( d ) = {∗} The definition of agent B ob is symmetric. 5
Epistemic Concurrent Game Models ECGM The interactions amongst agents generate ECGM. • related to CGS [AHK02, MMV10] and AETS [vdHW03] • global states are not primitive: s = � l 0 , . . . , l ℓ � ∈ G = Π i ∈ Ag L i • joint actions are tuples σ = � σ 0 , . . . , σ ℓ � ∈ Act = Π i ∈ Ag Act i Definition (ECGM) Given ◮ a set Ag = { i 0 , . . . , i ℓ } of agents ◮ a set AP of atomic propositions an ECGM P includes ◮ a finite set I ⊆ G of initial global states ◮ a transition function τ : G × Act → G ◮ an interpretation π : AP → 2 G of atomic propositions • the epistemic indistinguishability relation is not primitive: s ∼ i s ′ iff l i = l ′ i 6
The Prisoner’s Dilemma as an ECGM Let AP = { a i , b i , c i , d i } for i ∈ { A , B } . s 0 ǫ A , ǫ B ( D , D ) ( C , C ) ( C , D ) ( D , C ) c , c b , b d , a a , d ( ∗ , ∗ ) ( ∗ , ∗ ) ( ∗ , ∗ ) ( ∗ , ∗ ) Example (ECGM P pd ) For the set Ag = { A , B } of agents, the prisoner’s dilemma ECGM P pd includes • the set I = { s 0 } of initial states, with s 0 = ( ǫ A , ǫ B ) • the transition function τ , given as ◮ τ ( s 0 , ( C , C )) = ( b , b ) ◮ τ ( s 0 , ( C , D )) = ( d , a ) ◮ τ ( s 0 , ( D , C )) = ( a , d ) ◮ τ ( s 0 , ( D , D )) = ( c , c ) ◮ τ ( s , ( ∗ , ∗ )) = s , for every state s different from s 0 • the interpretation π s.t. a state ( l A , l B ) belongs to π ( p i ) iff l i = p . 7
Epistemic Strategy Logic ESL ESL extends SL with epistemic operators K i for individual knowledge. • we introduce a set Var i of strategy variables for each agent i ∈ Ag Definition (ESL) ESL formulas are defined in BNF as follows: φ ::= p | ¬ φ | φ → φ | X φ | φ U φ | ∃ x i φ | K i φ • we consider a multi-agent setting ( � = [CHP10]) • the language does not include the binding operator ( α, x ) ( � = [MMV10]) The questions above can be recast as model checking problems: ? P pd | = ∃ x A Fa A ? P pd | = K A ( ∃ x A Fa A ∨ ¬∃ x A Fa A ) ? P pd | = K B ( ∃ x A Fa A ∨ ¬∃ x A Fa A ) 8
Epistemic Concurrent Game Models Strategies Definition (Strategy) An A-strategy is a mapping f A : G + �→ Act A from finite sequences of states to enabled A-actions. • a run λ is a sequence s 0 → s 1 → . . . of global states • a run λ belongs to outcome out ( s , f A ) iff λ ( i + 1) ∈ ˆ τ ( λ ( i ) , f A ( λ [ . . . , i ])) ⇒ a group strategy is really the composition of its members’ strategies • an assignment χ maps each agent i ∈ Ag to an i -strategy f i ◮ f χ is the Ag -strategy χ ( i 0 ) × . . . × χ ( i ℓ ) Definition (Satisfaction) An ECGM P satisfies an ESL formula ϕ in a state s for an assignment χ , iff ( P , s , χ ) | = p iff s ∈ π ( p ) for λ = out ( s , f χ ), ( P , λ (1) , χ ) | ( P , s , χ ) | = X ψ iff = ψ for λ = out ( s , f χ ) there is k ≥ 0 s.t. ( P , λ ( k ) , χ ) | ( P , s , χ ) | = ψ U ψ ′ iff = ψ ′ and 0 ≤ j < k implies ( P , λ ( j ) , χ ) | = ψ there exists an i -strategy f i s.t. ( P , s , χ i ( P , s , χ ) | = ∃ x i ψ iff f i ) | = ψ for every s ∈ S , s ∼ i s ′ implies ( P , s ′ , χ ) | ( P , s , χ ) | = K i ψ iff = ψ 9
Expressiveness Knowledge of Nash Equilibria • given an n -player game in normal form with payoff ordering a 1 > . . . > a k , define n k i − 1 � � � ∧ ∃ y i Xa i → Xa i ψ NE ::= ¬∃ y j Xa j i =1 i =1 j =1 Proposition ( P pd , s 0 , χ ) | = ψ NE iff ( χ (1)( s 0 ) , . . . , χ ( n )( s 0 )) is a Nash equilibrium • for the prisoner’s dilemma, ( P pd , s 0 , χ ) | = ψ NE iff ( χ (1)( s 0 ) , χ (2)( s 0 )) is a Nash equilibrium iff χ (1)( s 0 ) = χ (2)( s 0 ) = D The questions above can be recast as model checking problems: ? P pd | = K A ∃ x A , x B ψ NE ∧ K B ∃ x A , x B ψ NE ? P pd | = ∃ x A , x B ( K A ψ NE ∧ K B ψ NE ) 10
Expressiveness Knowledge de re v. Knowledge de dicto • knowledge de re ⇒ knowledge de dicto : | = ∃ x i K j φ → K j ∃ x i φ also, knowledge de dicto ⇒ knowledge de re : | = K j ∃ x i φ → ∃ x i K j φ indeed, agents have perfect information of the game • individual strategies depend on global states [JvdH04] 11
Model Checking ESL Theorem (Hardness) The model checking problem for ESL is Non-ElementarySpace -hard. • reduction to satisfiability for quantified propositional temporal logic (QPTL) • differently from [MMV10] the syntax does not include the binding operator Theorem (Completeness) The model checking problem for ESL is PTime -complete w.r.t. the size of the model and Non-Elementary w.r.t. the size of the formula. • reduction to non-emptyness for alternating tree automata [MMV10] ⇒ The model checking problem is no harder for ESL than for SL. 12
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