Logic of Joint Action Natasha Alechina (joint work with Thomas - PowerPoint PPT Presentation

Logic of Joint Action Natasha Alechina (joint work with Thomas ˚ Agotnes) November 2011, St Andrews Workshop in Honour of Roy Dyckhoff ˚ Agotnes & Alechina Logic of Joint Action November 2011 1 / 24

Motivation Translate Coalition Logic (which has complicated semantics) into multimodal K with intersection of modalities Benefits: for example, a more standard tableaux procedure ˚ Agotnes & Alechina Logic of Joint Action November 2011 2 / 24

Outline Outline Syntax and Semantics of Coalition Logic 1 Syntax and Semantics of K ∩ 2 n Idea of the Embedding 3 Logic of Joint Actions 4 ˚ Agotnes & Alechina Logic of Joint Action November 2011 3 / 24

Syntax and Semantics of Coalition Logic Syntax Syntax of CL is defined relative to a set of primitive propositions Θ and a set of agents N (assume | N | = g ) φ ::= p | ¬ φ | φ ∧ φ | [ C ] φ where p ∈ Θ and C ⊆ N [ C ] φ means: coalition C can enforce the outcome φ ˚ Agotnes & Alechina Logic of Joint Action November 2011 4 / 24

Syntax and Semantics of Coalition Logic CL Models (Also called Concurrent Game Structures, CGS) M = � S , V , Act , d , δ � where S is a set of states ; V is a valuation function , assigning a set V ( s ) ⊆ Θ to each state s ∈ S ; Act is a set of actions ; For each i ∈ N and s ∈ S , d i ( s ) ⊆ Act is a non-empty set of actions available to agent i in s . D ( s ) = d 1 ( s ) × · · · × d g ( s ) is the set of full joint actions in s . δ is a transition function , mapping each state s ∈ S and full joint action a ∈ D ( s ) to a state δ ( s , a ) ∈ S . ˚ Agotnes & Alechina Logic of Joint Action November 2011 5 / 24

Syntax and Semantics of Coalition Logic Truth in CL Models M , s | = p ⇔ p ∈ V ( s ) M , s | = ¬ φ ⇔ M , s �| = φ M , s | = ( φ 1 ∧ φ 2 ) ⇔ ( M , s | = φ 1 and M , s | = φ 2 ) M , s | = [ C ] ψ ⇔ ∃ a C ∈ D C ( s ) ∀ a C ∈ D C ( s ) , M , δ ( s , ( a C , a C )) | = ψ ˚ Agotnes & Alechina Logic of Joint Action November 2011 6 / 24

Syntax and Semantics of K ∩ n K n with intersection of modalities Assume a set of primitive propositions Θ and actions A : φ ::= p ∈ Θ | ¬ φ | φ ∧ φ | [ π ] φ π ::= a | π ∩ π where a ∈ A . As usual, � π � φ is defined as ¬ [ π ] ¬ φ ˚ Agotnes & Alechina Logic of Joint Action November 2011 7 / 24

Syntax and Semantics of K ∩ n K ∩ n models M = � S , V , { R π : π ∈ Π }� where S is a set of states ; V : S → 2 Θ is a valuation function ; For each π ∈ Π , R π ⊆ S × S R π 1 ∩ π 2 = R π 1 ∩ R π 2 (INT) The modality truth definition clause: = [ π ] φ iff ∀ ( s , s ′ ) ∈ R π , M , s ′ | M , s | = φ ˚ Agotnes & Alechina Logic of Joint Action November 2011 8 / 24

Idea of the Embedding Idea of embedding A CGS like this: s t <a,b> ˚ Agotnes & Alechina Logic of Joint Action November 2011 9 / 24

Idea of the Embedding Idea of embedding 2 can be represented by a K ∩ n model like this: (1,a) ∩ (2,b) s t (1,a) (2,b) ˚ Agotnes & Alechina Logic of Joint Action November 2011 10 / 24

Idea of the Embedding Idea of embedding and CL formulas translated as follows: ([ { i 1 , . . . , i k } ] φ ) ′ ≡ � � � ( i j , a j ) �⊤ ∧ [( i 1 , a 1 ) ∩ . . . ∩ ( i k , a k )] φ ′ a 1 ,..., a k ∈ Act φ 1 ≤ j ≤ k where Act φ is a finite set of actions that is “read off” the formula ˚ Agotnes & Alechina Logic of Joint Action November 2011 11 / 24

Idea of the Embedding Problem t1 s t2 <a1, b1> <a1, b2> <a2, b2> <a2, b1> ˚ Agotnes & Alechina Logic of Joint Action November 2011 12 / 24

Idea of the Embedding Problem (1,a1), (2,b1), (1,a1), (2,b1), t1 s t2 (1,a2), (2,b2) (1,a2), (2,b2) (1,a1)  (2,b1), (1,a1)  (2,b1), (1,a2)  (2,b1), (1,a2)  (2,b1), (1,a1)  (2,b2), (1,a1)  (2,b2), (1,a2)  (2,b2) (1,a2)  (2,b2) ˚ Agotnes & Alechina Logic of Joint Action November 2011 13 / 24

Idea of the Embedding Injective CGS CGS without two or more different full joint actions between the same two states are injective . Injective CGSs do not suffer this problem. Theorem (Goranko 2007) For every CGS M = � S , V , Act , d , δ � there is an injective CGS M ′ with states S ′ such that S ⊆ S ′ and for all CL formulae φ and states s ∈ S , = φ iff M ′ , s | = φ . Moreover, if M is finite, then | S ′ | ≤ | S | + | δ | . M , s | ˚ Agotnes & Alechina Logic of Joint Action November 2011 14 / 24

Logic of Joint Actions Logic of Joint Actions Let Act be a finite set of actions and N a set of g agents. Define a set of atomic modalities as follows: A = N × Act an atomic modality in A is an individual action a composite modality π = π 1 ∩ π 2 is a joint action . joint actions of the form ( 1 , a 1 ) ∩ . . . ∩ ( g , a n ) with one individual action for every agent in N will be called complete (joint) actions . ˚ Agotnes & Alechina Logic of Joint Action November 2011 15 / 24

Logic of Joint Actions Models for Logic of JA A K ∩ n model over A (where Act is finite) is a joint action model if it satisfies: Seriality (SER) For any state s and agent i , at least one action is enabled in s for i . Independent Choice (IC) For any state s , agents C = { i 1 , . . . , i k } and actions a 1 , . . . , a k ∈ Act , if for every j a j is enabled for i j in s , then there is a state s ′ such that ( s , s ′ ) ∈ R ( i 1 , a 1 ) ∩···∩ ( i k , a k ) . Deterministic Joint Actions (DJA) For any complete joint action α and states s , s 1 , s 2 , ( s , s 1 ) , ( s , s 2 ) ∈ R α implies that s 1 = s 2 . Unique Joint Actions (UJA) For any complete joint actions α and β and states s , t , if ( s , t ) ∈ R α ∩ R β then α = β . ˚ Agotnes & Alechina Logic of Joint Action November 2011 16 / 24

Logic of Joint Actions Translation Given an injective CGS M = � S , V , Act , d , δ � where Act is finite, the corresponding joint action model ˆ M = � S , V , { R π : π ∈ Π }� over Θ and A is defined as follows: R ( i , a ) = { ( s , s ′ ) : ∃ a ∈ D ( s ) s.t. a i = a and s ′ = δ ( s , a ) } , when ( i , a ) ∈ A R π 1 ∩ π 2 = R π 1 ∩ R π 2 ˚ Agotnes & Alechina Logic of Joint Action November 2011 17 / 24

Logic of Joint Actions Translation p ′ ≡ p ( ¬ φ ) ′ ≡ ¬ φ ′ ( φ 1 ∧ φ 2 ) ′ ≡ φ ′ 1 ∧ φ ′ 2 ([ { i 1 , . . . , i k } ] φ ) ′ ≡ � � 1 ≤ j ≤ k � ( i j , a j ) �⊤ a 1 ,..., a k ∈ Act ∧ [( i 1 , a 1 ) ∩ . . . ∩ ( i k , a k )] φ ′ ˚ Agotnes & Alechina Logic of Joint Action November 2011 18 / 24

Logic of Joint Actions Axiomatisation K [ π ]( φ → ψ ) → ([ π ] φ → [ π ] ψ ) A1 � a ∈ Act � ( i , a ) �⊤ A2 � π � φ → � a ∈ Act � π ∩ ( i , a ) � φ A3 � i ∈ N � ( i , a i ) �⊤ → � ( 1 , a 1 ) ∩ . . . ∩ ( g , a g ) �⊤ A4 � ( 1 , a 1 ) ∩ · · · ∩ ( g , a g ) � φ → [( 1 , a 1 ) ∩ . . . ∩ ( g , a g )] φ A5 [ π ] φ → [ π ∩ π ′ ] φ A6 [( i , a ) ∩ ( i , b )] ⊥ when a � = b MP From φ → ψ and φ infer ψ G From φ infer [ π ] φ ˚ Agotnes & Alechina Logic of Joint Action November 2011 19 / 24

Logic of Joint Actions Tableaux procedure 1 Based on Lutz and Sattler 2001 for K ∩ n . in what follows, assume that formulas do not contain diamond modalities and disjunctions notation: for two modalities π 1 and π 2 we say π 1 ≤ π 2 if the set of individual actions of π 1 is a subset of that of π 2 . For example, ( 2 , b ) ≤ ( 1 , a ) ∩ ( 2 , b ) ∩ ( 3 , c ) ˚ Agotnes & Alechina Logic of Joint Action November 2011 20 / 24

Logic of Joint Actions Tableaux procedure 2 Given a set of formulas X , we use Cl ( X ) to denote the smallest set containing all subformulas of formulas in X such that: (a) for each agent i and action a , [( i , a )] ⊥ ∈ Cl ( X ) (b) for every complete joint action α ∈ JA , [ α ] ⊥ ∈ Cl ( X ) (c) if ¬ [( 1 , a 1 ) , . . . , ( g , a g )] ψ ∈ Cl ( X ) , then [( 1 , a 1 ) , . . . , ( g , a g )] ∼ ψ ∈ Cl ( X ) , where ∼ ψ = ¬ ψ if ψ is not of the form ¬ χ , and ∼ ψ = χ otherwise (d) for each i and a � = b , [( i , a ) ∩ ( i , b )] ⊥ ∈ Cl ( X ) (e) if ψ ∈ Cl ( X ) , then ∼ ψ ∈ Cl ( X ) ˚ Agotnes & Alechina Logic of Joint Action November 2011 21 / 24

Logic of Joint Actions Tableaux procedure 3 For sets of formulas ∆ and S where S is closed as above, Tab (∆ , S ) returns true iff (A) ∆ is a maximally propositionally consistent subset of S , that is, for each ¬ ψ ∈ S , ψ ∈ ∆ iff ¬ ψ �∈ ∆ and for each ψ 1 ∧ ψ 2 ∈ S , ψ 1 ∧ ψ 2 ∈ ∆ iff ψ 1 ∈ ∆ and ψ 2 ∈ ∆ . (B) There is a partition of the set {¬ [ π ] ψ : ¬ [ π ] ψ ∈ ∆ } into sets W α (at most one for each α ∈ JA ) such that if ¬ [ π ] ψ ∈ W α then π ≤ α and (i) ¬ ψ ∈ ∆ α (ii) for each π ′ and ψ ′ , if [ π ′ ] ψ ′ ∈ ∆ and π ′ ≤ α , then ψ ′ ∈ ∆ α (iii) Tab (∆ α , S ′ ) returns true, where S ′ = Cl ( { ψ ′ : [ π ′ ] ψ ′ ∈ ∆ and π ′ ≤ α } ∪ {¬ ψ : ¬ [ π ] ψ ∈ W α } ) ˚ Agotnes & Alechina Logic of Joint Action November 2011 22 / 24