Specification of Dynamic Strategy Switching Soumya Paul Joint work - PowerPoint PPT Presentation

Strategies • A strategy for player i is a partial function: σ : T G ⇀ A i from the nodes of the tree unfolding T G of the arena G (histories) to her action set. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

Strategies • A strategy for player i is a partial function: σ : T G ⇀ A i from the nodes of the tree unfolding T G of the arena G (histories) to her action set. • If σ is not defined for some history, she may play any action there. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

Strategies • A strategy for player i is a partial function: σ : T G ⇀ A i from the nodes of the tree unfolding T G of the arena G (histories) to her action set. • If σ is not defined for some history, she may play any action there. • Σ i denotes the set of all strategies of player i . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

Strategies • A strategy for player i is a partial function: σ : T G ⇀ A i from the nodes of the tree unfolding T G of the arena G (histories) to her action set. • If σ is not defined for some history, she may play any action there. • Σ i denotes the set of all strategies of player i . • A strategy σ may be viewed as a subtree T σ G of T G . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

Example Let σ be the strategy of player 1 which is undefined at the empty history but prescribes her to play the action a for all subsequent histories. Then T σ G looks like: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 10 / 64

� � � � � � � � Example Let σ be the strategy of player 1 which is undefined at the empty history but prescribes her to play the action a for all subsequent histories. Then T σ G looks like: ǫ � � ������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � ( a , c ) ( a , d ) ( b , c ) ( b , d ) � � � � � � � � � � � � � � � � � � ( a , c ) ( a , d ) ( a , c ) ( a , d ) ( a , c ) ( a , d ) ( a , c ) ( a , d ) . . . . . . . . . . . . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 10 / 64

Strategy Specifications • Players change/compose/form strategies based on certain observable properties of the game. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

Strategy Specifications • Players change/compose/form strategies based on certain observable properties of the game. • P is a countable set of propositions that talk about the observables in the game. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

Strategy Specifications • Players change/compose/form strategies based on certain observable properties of the game. • P is a countable set of propositions that talk about the observables in the game. • V : W → 2 P is a valuation function. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

Strategy Specifications • Players change/compose/form strategies based on certain observable properties of the game. • P is a countable set of propositions that talk about the observables in the game. • V : W → 2 P is a valuation function. • V may be lifted to T G , V : T G → 2 P in the usual way. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: • T G , t | = p iff p ∈ V ( t ). Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: • T G , t | = p iff p ∈ V ( t ). • T G , t | = ψ 1 ∨ ψ 2 iff T G | = ψ 1 or T G | = ψ 2 . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: • T G , t | = p iff p ∈ V ( t ). • T G , t | = ψ 1 ∨ ψ 2 iff T G | = ψ 1 or T G | = ψ 2 . • T G , t | = ¬ ψ iff T G , t �| = ψ . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: • T G , t | = p iff p ∈ V ( t ). • T G , t | = ψ 1 ∨ ψ 2 iff T G | = ψ 1 or T G | = ψ 2 . • T G , t | = ¬ ψ iff T G , t �| = ψ . • T G , t | = ⊖ ψ iff k > 0 and T G , t k − 1 | = ψ . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[2] An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ 1 ∨ ψ 2 | ¬ ψ | ⊖ ψ | ψ 1 S ψ 2 The truth of a formula at a node t = ¯ a 1 . . . ¯ a k of the game tree is defined as: • T G , t | = p iff p ∈ V ( t ). • T G , t | = ψ 1 ∨ ψ 2 iff T G | = ψ 1 or T G | = ψ 2 . • T G , t | = ¬ ψ iff T G , t �| = ψ . • T G , t | = ⊖ ψ iff k > 0 and T G , t k − 1 | = ψ . • T G , t | = ψ 1 S ψ 2 iff ∃ l : 1 ≤ l < k such that T G , t l | = ψ 2 and ∀ m : l < m ≤ k , T G , t m | = ψ 1 . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 | ( π 1 + π 2 ) Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 | ( π 1 + π 2 ) | ψ ? π where ψ ∈ Ψ. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 | ( π 1 + π 2 ) | ψ ? π where ψ ∈ Ψ.Intuitively: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 | ( π 1 + π 2 ) | ψ ? π where ψ ∈ Ψ.Intuitively: • π 1 ∪ π 2 means that the player plays according to the strategy π 1 or the strategy π 2 . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[3] A strategy of player i can be of the form: Π i ::= σ ∈ Σ i | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 � π 2 | ( π 1 + π 2 ) | ψ ? π where ψ ∈ Ψ.Intuitively: • π 1 ∪ π 2 means that the player plays according to the strategy π 1 or the strategy π 2 . • π 1 ∩ π 2 means that if at a history t ∈ T G , π 1 is defined then the player plays according to π 1 ; else if π 2 is defined at t then the player plays according to π 2 . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

Strategy Specifications[4] • π 1 � π 2 means that the player plays according to the strategy π 1 and then after some history, switches to playing according to π 2 . The position at which she makes the switch is not fixed in advance. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

Strategy Specifications[4] • π 1 � π 2 means that the player plays according to the strategy π 1 and then after some history, switches to playing according to π 2 . The position at which she makes the switch is not fixed in advance. • ( π 1 + π 2 ) says that at every point, the player can choose to follow either π 1 or π 2 . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

Strategy Specifications[4] • π 1 � π 2 means that the player plays according to the strategy π 1 and then after some history, switches to playing according to π 2 . The position at which she makes the switch is not fixed in advance. • ( π 1 + π 2 ) says that at every point, the player can choose to follow either π 1 or π 2 . • ψ ? π says at every history, the player tests if the property ψ holds of that history. If it does then she plays according to π . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

Strategy Specifications[4] • π 1 � π 2 means that the player plays according to the strategy π 1 and then after some history, switches to playing according to π 2 . The position at which she makes the switch is not fixed in advance. • ( π 1 + π 2 ) says that at every point, the player can choose to follow either π 1 or π 2 . • ψ ? π says at every history, the player tests if the property ψ holds of that history. If it does then she plays according to π . For a specification π for player i , let [ [ π ] ] G ⊆ Σ i be the strategies (partial functions), it denotes. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

Examples • Σ bowler = { σ short , σ good , σ outside − off , σ legs } P = { p ( short , single ) , p ( short , boundary ) , . . . , p ( legs , sixer ) , . . . } - ( p ( good , sixer ) ∧ p ( legs , sixer ) )?( σ short + σ good + σ outside − off + σ legs ) ∪ ¬ ✸ - ( p ( good , sixer ) ∧ p ( legs , sixer ) )?( σ short + σ outside − off ) ✸ Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 15 / 64

Examples • Σ bowler = { σ short , σ good , σ outside − off , σ legs } P = { p ( short , single ) , p ( short , boundary ) , . . . , p ( legs , sixer ) , . . . } - ( p ( good , sixer ) ∧ p ( legs , sixer ) )?( σ short + σ good + σ outside − off + σ legs ) ∪ ¬ ✸ - ( p ( good , sixer ) ∧ p ( legs , sixer ) )?( σ short + σ outside − off ) ✸ • Σ bowler = { σ 5 , σ 2 , . . . } σ 5 � σ 2 Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 15 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: • A finite set M , the memory of the strategy. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: • A finite set M , the memory of the strategy. • m I ∈ M , the initial memory. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: • A finite set M , the memory of the strategy. • m I ∈ M , the initial memory. • A function δ : A × M → M , the memory update. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: • A finite set M , the memory of the strategy. • m I ∈ M , the initial memory. • A function δ : A × M → M , the memory update. • A function g : A × M → A , the action update. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Bounded Memory Strategies A strategy σ is said to be bounded memory if there exists: • A finite set M , the memory of the strategy. • m I ∈ M , the initial memory. • A function δ : A × M → M , the memory update. • A function g : A × M → A , the action update. such that when ¯ a 1 . . . ¯ a k − 1 is a play and the sequence m 0 , m 1 , . . . , m k is determined by m 0 = m I and m i +1 = δ (¯ a i − 1 , m i ) then σ (¯ a 1 . . . ¯ a k − 1 ) = g (¯ a k − 1 , m k ). Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

Finite State Transducer A finite state transducer FST over input alphabet A and output alphabet A i is a tuple A = ( Q , → , I , f ) such that Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

Finite State Transducer A finite state transducer FST over input alphabet A and output alphabet A i is a tuple A = ( Q , → , I , f ) such that • Q is a finite set (of states). Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

Finite State Transducer A finite state transducer FST over input alphabet A and output alphabet A i is a tuple A = ( Q , → , I , f ) such that • Q is a finite set (of states). • → : Q × A → 2 Q is the transition function. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

Finite State Transducer A finite state transducer FST over input alphabet A and output alphabet A i is a tuple A = ( Q , → , I , f ) such that • Q is a finite set (of states). • → : Q × A → 2 Q is the transition function. • I ⊆ Q is the set of initial states. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

Finite State Transducer A finite state transducer FST over input alphabet A and output alphabet A i is a tuple A = ( Q , → , I , f ) such that • Q is a finite set (of states). • → : Q × A → 2 Q is the transition function. • I ⊆ Q is the set of initial states. • f : Q → A i is the output function. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

FST for Bounded Memory Strategy Given a bounded memory strategy σ for player i we can construct an FST A σ = ( Q , → , I , f ) over A and A i such that the output of the transducer correctly reflects whatever the strategy σ prescribes. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

FST for Bounded Memory Strategy Given a bounded memory strategy σ for player i we can construct an FST A σ = ( Q , → , I , f ) over A and A i such that the output of the transducer correctly reflects whatever the strategy σ prescribes. • Q = M × A ∪ { ǫ } × A i × W . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

FST for Bounded Memory Strategy Given a bounded memory strategy σ for player i we can construct an FST A σ = ( Q , → , I , f ) over A and A i such that the output of the transducer correctly reflects whatever the strategy σ prescribes. • Q = M × A ∪ { ǫ } × A i × W . • → : Q × A → 2 Q such that if for any ( m , ¯ a , a , w ) ∈ Q , δ (¯ a , m ) = a , a , w ) ¯ a ′ m ′ and g (¯ a , m ′ ) = a ′ then ( m , ¯ → ( m ′ , ¯ a ′ , a ′ , w ′ ) such that a ′ a ′ ( i ) = a and w ¯ → w ′ . If g is not defined at (¯ a , m ′ ) then ¯ a , a , w ) ¯ a ′ a ′ , a ′ , w ′ ) for all a ′ ∈ A i and w ¯ a ′ → ( m ′ , ¯ → w ′ . ( m , ¯ Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

FST for Bounded Memory Strategy Given a bounded memory strategy σ for player i we can construct an FST A σ = ( Q , → , I , f ) over A and A i such that the output of the transducer correctly reflects whatever the strategy σ prescribes. • Q = M × A ∪ { ǫ } × A i × W . • → : Q × A → 2 Q such that if for any ( m , ¯ a , a , w ) ∈ Q , δ (¯ a , m ) = a , a , w ) ¯ a ′ m ′ and g (¯ a , m ′ ) = a ′ then ( m , ¯ → ( m ′ , ¯ a ′ , a ′ , w ′ ) such that a ′ a ′ ( i ) = a and w ¯ → w ′ . If g is not defined at (¯ a , m ′ ) then ¯ a , a , w ) ¯ a ′ a ′ , a ′ , w ′ ) for all a ′ ∈ A i and w ¯ a ′ → ( m ′ , ¯ → w ′ . ( m , ¯ • I = { ( m I , ǫ, a , w 0 ) | a = g ( ǫ, m I ) if defined, else a ∈ A i } . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

FST for Bounded Memory Strategy Given a bounded memory strategy σ for player i we can construct an FST A σ = ( Q , → , I , f ) over A and A i such that the output of the transducer correctly reflects whatever the strategy σ prescribes. • Q = M × A ∪ { ǫ } × A i × W . • → : Q × A → 2 Q such that if for any ( m , ¯ a , a , w ) ∈ Q , δ (¯ a , m ) = a , a , w ) ¯ a ′ m ′ and g (¯ a , m ′ ) = a ′ then ( m , ¯ → ( m ′ , ¯ a ′ , a ′ , w ′ ) such that a ′ a ′ ( i ) = a and w ¯ → w ′ . If g is not defined at (¯ a , m ′ ) then ¯ a , a , w ) ¯ a ′ a ′ , a ′ , w ′ ) for all a ′ ∈ A i and w ¯ a ′ → ( m ′ , ¯ → w ′ . ( m , ¯ • I = { ( m I , ǫ, a , w 0 ) | a = g ( ǫ, m I ) if defined, else a ∈ A i } . • f : Q → A i such that f (( m , ¯ a , a , w )) = a . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

Language of an FST • A run χ of an fst A = ( Q , → , I , f , λ ) on a (total) strategy µ is a labelling of the nodes of strategy tree T µ G with the states of Q such that the transitions of A are respected. That is, if there is an edge a k +1 in T µ from node from ¯ a 1 . . . ¯ a k to ¯ a 1 . . . ¯ G then χ (¯ a 1 . . . ¯ a k ) ∈→ ( χ (¯ a 1 . . . ¯ a k ) , ¯ a k +1 ). Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

Language of an FST • A run χ of an fst A = ( Q , → , I , f , λ ) on a (total) strategy µ is a labelling of the nodes of strategy tree T µ G with the states of Q such that the transitions of A are respected. That is, if there is an edge a k +1 in T µ from node from ¯ a 1 . . . ¯ a k to ¯ a 1 . . . ¯ G then χ (¯ a 1 . . . ¯ a k ) ∈→ ( χ (¯ a 1 . . . ¯ a k ) , ¯ a k +1 ). • A strategy µ is said to be accepted by A if there exists a run χ of A a k ∈ T µ on µ such that ∀ t = ¯ a 1 . . . ¯ G , ¯ a ( i ) = f ( χ ( t )). Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

Language of an FST • A run χ of an fst A = ( Q , → , I , f , λ ) on a (total) strategy µ is a labelling of the nodes of strategy tree T µ G with the states of Q such that the transitions of A are respected. That is, if there is an edge a k +1 in T µ from node from ¯ a 1 . . . ¯ a k to ¯ a 1 . . . ¯ G then χ (¯ a 1 . . . ¯ a k ) ∈→ ( χ (¯ a 1 . . . ¯ a k ) , ¯ a k +1 ). • A strategy µ is said to be accepted by A if there exists a run χ of A a k ∈ T µ on µ such that ∀ t = ¯ a 1 . . . ¯ G , ¯ a ( i ) = f ( χ ( t )). • The language of A , L ( A ) is defined to be the set of all strategies that are accepted by it. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

Transducer Lemma Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Π i , there is a transducer A π such that for all µ ∈ Ω i , µ ∈ [ [ π ] ] G iff µ ∈ L ( A π ) . Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

Transducer Lemma Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Π i , there is a transducer A π such that for all µ ∈ Ω i , µ ∈ [ [ π ] ] G iff µ ∈ L ( A π ) . Proof Sketch: By induction on π Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

Transducer Lemma Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Π i , there is a transducer A π such that for all µ ∈ Ω i , µ ∈ [ [ π ] ] G iff µ ∈ L ( A π ) . Proof Sketch: By induction on π • For σ ∈ Σ i we have an FST A σ from the construction above. Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 21 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 22 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 23 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 24 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 25 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 26 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 27 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 28 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 29 / 64

Proof Sketch • π 1 ∪ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 30 / 64

Proof Sketch[2] • π 1 ∩ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 31 / 64

Proof Sketch[2] • π 1 ∩ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 32 / 64

Proof Sketch[2] • π 1 ∩ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 33 / 64

Proof Sketch[2] • π 1 ∩ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 34 / 64

Proof Sketch[2] • π 1 ∩ π 2 : A π 1 · · · A π 2 · · · Soumya Paul Joint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 35 / 64