Programming Languages in String Diagrams [ two ] Asynchronous Games Paul-André Melliès Oregon Summer School in Programming Languages June 2011
Starting point: game semantics Every proof of formula A initiates a dialogue where Proponent tries to convince Opponent Opponent tries to refute Proponent An interactive approach to logic and programming languages 2
A formal proof of the drinker formula Axiom A ( x 0 ) ⊢ A ( x 0 ) Right Weakening A ( x 0 ) ⊢ A ( x 0 ) , ∀ x . A ( x ) Right ⇒ ⊢ A ( x 0 ) , A ( x 0 ) ⇒ ∀ x . A ( x ) Right ∃ ⊢ A ( x 0 ) , ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } Right ∀ ⊢ ∀ x . A ( x ) , ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } Left Weakening A ( y 0 ) ⊢ ∀ x . A ( x ) , ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } Right ⇒ ⊢ A ( y 0 ) ⇒ ∀ x . A ( x ) , ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } Right ∃ ⊢ ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } , ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } Contraction ⊢ ∃ y . { A ( y ) ⇒ ∀ x . A ( x ) } 3
Duality Proponent Opponent Program Environment plays the game plays the game A ¬ A Negation permutes the rôles of Proponent and Opponent 4
Duality Opponent Proponent Environment Program plays the game plays the game ¬ A A Negation permutes the rôles of Opponent and Proponent 5
Tensor product ⊗ Play the two games in parallel 6
Sum ⊕ Proponent selects one component 7
Exponentials ⊗ ⊗ ⊗ · · · Opponent opens as many copies as necessary to beat Proponent 8
Sequential games An idea dating back to André Joyal 9
Traditional game semantics alternating A proof π A proof π sequences of moves Game semantics: an interleaving semantics of proofs. 10
Sequential games A sequential game ( M , P , λ ) consists of a set of moves , M P ⊆ M ∗ a set of plays , a polarity function on moves λ : M → {− 1 , + 1 } such that every play is alternating and starts by Opponent. Alternatively, a sequential game is an alternating decision tree. 11
� � Sequential games The boolean game B : Player in red � ����������������� � Opponent in blue � � � � � � � � true � false � � � � � � � question 12
Deterministic strategies A strategy is a set of alternating plays of even-length σ s m 1 · · · m 2 k = such that: – σ contains the empty play, – is closed by even-length prefix: σ ∀ s , ∀ m , n ∈ M , s · m · n ∈ σ = ⇒ s ∈ σ – is deterministic: σ s · m · n 1 ∈ σ and s · m · n 2 ∈ σ ∀ s ∈ σ, ∀ m , n 1 , n 2 ∈ M , = ⇒ n 1 = n 2 . 13
� � Three strategies on the boolean game B Player in red � ����������������� Opponent in blue � � � � � � � � � � true false � � � � � � � question 14
Total strategies A strategy σ is total when – for every play s of the strategy σ – for every Opponent move m such that s · m is a play there exists a Proponent move n such that s · m · n is a play of σ . 15
� � Two total strategies on the boolean game B Player in red � ����������������� Opponent in blue � � � � � � � � � � true false � � � � � � � question 16
Tensor product Given two sequential games A and B , define A ⊗ B as the sequential game = M A ⊗ B M A + M B = [ λ A , λ B ] λ A ⊗ B = P A ⊗ B P A ⊗ P B where P A ⊗ P B denotes the set of alternating plays in M ∗ A ⊗ B obtained by interleaving a play s ∈ P A and a play t ∈ P B . 17
Linear implication Given two sequential games A and B , define A ⊸ B as the sequential game = M A ⊸ B M A + M B = [ − λ A , λ B ] λ A ⊸ B = P A ⊸ B P A ⊸ P B where P A ⊸ P B denotes the set of alternating plays in M ∗ A ⊸ B obtained by interleaving a play s ∈ P A and a play t ∈ P B . 18
A category of sequential games The category Games has – the sequential games as objects – the strategies of the sequential game σ ∈ A ⊸ B as maps : A −→ B σ 19
The copycat strategy The identity map id A A A : −→ is the copycat strategy id A : A A ⊸ defined as id A { s ∈ P A ⊸ A | s = m 1 · m 1 · m 2 · m 2 · · · m k · m k } = 20
The copycat strategy id A −→ A m 1 m 1 m 2 m 2 m 3 m 3 . . . . . . m k m k 21
Composition Given two strategies σ τ A −→ B −→ C the composite strategy σ ; τ A −→ C is defined as u ↾ A = s ↾ A � � � s ↾ B = t ↾ B σ ; τ = u ∈ P A ⊸ C ∃ s ∈ σ , ∃ t ∈ τ � � u ↾ C = t ↾ C � The definition of composition is associative. 22
Illustration id true 1 −→ −→ B B question question true true 23
Illustration id false 1 −→ −→ B B question question false false 24
Illustration negation true 1 −→ −→ B B question question true false 25
An important isomorphism The sequential game A B C ( ⊗ ) ⊸ is isomorphic to the sequential game A ( B C ) ⊸ ⊸ for all sequential games A , B , C . Here, isomorphism means tree isomorphism. 26
The category of sequential games Theorem. The category Games is symmetric monoidal closed. As such, it defines a model of linear λ -calculus. 27
Illustration eval ( B B ) ⊗ −→ ⊸ B B question question bool bool f : B ⊸ B , x : B ⊢ f ( x ) : B 28
Illustration eval ( B ⊸ B ) ⊗ B −→ B question question question question bool bool f ( bool ) f ( bool ) f : B ⊸ B , x : B ⊢ f ( x ) : B 29
Illustration id ( B ⊸ B ) −→ B ⊸ B question question question question bool bool f ( bool ) f ( bool ) f : B ⊸ B ⊢ λ x . f ( x ) : B ⊸ B 30
Currification More generally, the transformation of the term , x : A ⊢ f : B Γ into the term Γ ⊢ λ x . f : A ⊸ B does not alter the associated strategy, simply reorganizes it. 31
Cartesian product Given two sequential games A and B , define A B & as the sequential game = M A & B M A M B + = λ A & B λ A + λ B = P A & B P A ⊕ P B where P A ⊕ P B is the coalesced sum of the pointed sets P A and P B . This means that every nonempty play in A & B is either in A or in B . 32
The cartesian product For every sequential game X , there exists an isomorphism X ⊸ A & B ( X ⊸ A ) & ( X ⊸ B ) � This means that the game A & B is the cartesian product of A and B . 33
The cartesian product For every game X , there exists a bijection between the strategies X −→ A & B and the pair of strategies X −→ A X −→ B . 34
� � � � � � � � � The cartesian product The cartesian product A & B of two objects A and B is an object equipped with two maps A & B A A & B B π 1 : −→ π 2 : −→ such that for all diagrams f A π 1 � � � � � � � � � X A & B � � � � � π 2 � � � � B g there exists a unique map h : X −→ A × B making the diagram f A π 1 � � � � � � � � � X A & B h � � � � � � π 2 � � � B g commute. 35
The exponential game Given a sequential game A , define ! A as the game = M ! A × M A N : ( i , m ) �→ λ A ( m ) λ ! A = P ! A ! P A where ! P A is the set of alternating plays in M ! A such that for all i ∈ N the restriction s ↾ i of s to the i -th component is a play of A 1 . for every prefix of s : 2 . u u ↾ i = ε ⇒ u ↾ i + 1 = ε. 36
Two important isomorphisms Define A ⇒ B = ! A ⊸ B Observe the isomorphism ! ( A & B ) ! A ⊗ ! B � and the isomorphism ( A & B ) ⇒ C = ! ( A & B ) ⊸ C ( ! A ⊗ ! B ) ⊸ C � ! A ⊸ ( ! B ⊸ C ) � A ⇒ ( B ⇒ C ) = 37
A model of the simply-typed λ -calculus Fact. Every simply-typed λ -term x 1 : A 1 · · · x k : A k ⊢ M : B , , may be interpreted as a strategy in the game ! A 1 ⊗ · · · ⊗ ! A k −→ B Here, the modality ! enables to backtrack in the sequential games 38
Illustration M ! ( N ⊸ N ) ⊗ N −→ N question question 1 question 1 question 2 question 2 question n n f ( n ) f ( n ) f f ( n ) f f ( n ) f : ! ( N ⊸ N ) , x : N ⊢ f f ( x ) : N 39
Asynchronous games An asynchronous reconstruction of the λ -calculus 40
Geometry of rewriting b ( λy.M ) Q ( λx. ( λy.x )) MQ c a M ( λx. ( λy.x )) MN u ( λy.M ) N v Can we do the same thing for open systems ? 41
� � The traditional interleaving semantics The boolean game B : Player in red � ����������������� � Opponent in blue � � � � � � � � true � false � � � � � � � question 42
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