From asynchronous games to coherence spaces Paul-André Melliès CNRS, Université Paris Denis Diderot Workshop on Geometry of Interaction, Traced Monoidal Categories, Implicit Complexity Kyoto, Tuesday 25 August 2009 1
contraction contraction contraction contraction An anomaly of the Geometry of Interaction $! A$ $? (A^{\bot})$ 1 1 ?( A ⊥ ) ! A Very much studied in the field of game semantics 2
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 3
Four basic operations on logical games the negation ¬ A A ⊕ B the sum the tensor A ⊗ B the exponential ! A Algebraic structure similar to linear algebra ! 4
Negation Proponent Opponent Program Environment plays the game plays the game ¬ A A Negation permutes the rôles of Proponent and Opponent 5
Negation Opponent Proponent Environment Program plays the game plays the game ¬ A A Negation permutes the rôles of Opponent and Proponent 6
Sum ⊕ Proponent selects one component 7
Tensor product ⊗ Opponent plays the two games in parallel 8
Exponentials ⊗ ⊗ ⊗ · · · Opponent opens as many copies as necessary to beat Proponent 9
Policy of the talk In order to clarify game semantics, compare it to relational semantics... Key idea: the strategy σ associated to a proof π should contain its clique. 10
Part I Additives in sequential games Sequential strategies at the leaves 11
Sequential games alternating A proof π A proof π sequences of moves 12
Sequential games A sequential game ( M, P, λ ) consists of a set of moves , M P ⊆ M ∗ a set of plays , λ : M → {− 1 , +1 } a polarity function on moves such that every play is alternating and starts by Opponent. Alternatively, a sequential game is an alternating decision tree. 13
� � Sequential games The boolean game B : Player in red V F Opponent in blue � � � � � � � � � � � � � � � � � � � � true false � � � � � � � � � � � q question ∗ 14
Strategies A strategy σ is a set of alternating plays of even-length = m 1 · · · m 2 k s such that: — σ contains the empty play, — σ is closed by even-length prefix: ∀ s, ∀ m, n ∈ M, s · m · n ∈ σ ⇒ s ∈ σ — σ is deterministic: ∀ s ∈ σ, ∀ m, n 1 , n 2 ∈ M, s · m · n 1 ∈ σ and s · m · n 2 ∈ σ ⇒ n 1 = n 2 . 15
� � Three strategies on the boolean game B Player in red V F Opponent in blue � � � � � � � � � � � � � � � � � � � � true false � � � � � � � � � � � q question ∗ 16
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 . 17
� � Two total strategies on the boolean game B Player in red V F Opponent in blue � � � � � � � � � � � � � � � � � � � � true false � � � � � � � � � � � q question ∗ 18
� � � From sequential games to coherence spaces The diagram commutes strategy � � � � � � � � � � � � � � � � � � � � � proof leaves � � � � � � � � � � � � � � � � � � � � � clique for every proof of a purely additive formula. 19
From sequential games to coherence spaces Let G denote the category — with families of sequential games as objects, — with families of sequential strategies as morphisms. Proposition. The category G is the free category with sums, equipped with a contravariant functor G op : ¬ G − → and a bijection ∼ : G ( x, ¬ y ) = G ( y, ¬ x ) ϕ x,y natural in x and y . 20
� A theorem for free There exists a functor : leaves G Coh which preserves the sum, and transports the non-involutive negation of the category G into the involutive negation of the category Coh . This functor collapses the dynamic semantics into a static one 21
Part II (a) Multiplicatives in asynchronous games From trajectories to positions 22
� � � � Sequential games: an interleaving semantics The tensor product of two boolean games B 1 et B 2 : � ���������� � � � � � false 2 true 1 � � � � � � ���������� � � � � q 2 q 1 � � � � � � � ���������� � � � � � true 1 false 2 � � � � � � � � � � � � � � � � � � � q 1 � � q 2 � � � � � � � � � 23
� � � � A step towards true concurrency: bend the branches! � ������������������ � � false 2 true 1 � � � � � � � � � � � � � � � � � � q 2 � � q 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � true 1 false 2 � � � � � � � � ������������������ � � � � � � � � � � q 1 q 2 � � � � � � � � 24
� � � � � � True concurrency: tile the diagram! V ⊗ F � ���������������� � � false 2 true 1 � � � � � � � � � � � � � � V ⊗ q q ⊗ F ∼ � ������ � ���������������� � � � � q 2 � q 1 � � � � � � � true 1 false 2 � � � � � ������ � � � � � � � � � � � q ⊗ q V ⊗ ∗ ∼ ∼ ∗ ⊗ F � ���������������� � ������ � � � � � � � � � � � q 2 q 1 ������ � � � � � true 1 false 2 � � � � � � � � � � � � q ⊗ ∗ ∗ ⊗ q ∼ � ����������������� � � � � � � � � � q 1 � q 2 � � � � � � � ∗ ⊗ ∗ 25
Asynchronous game semantics trajectories in A proof π A proof π asynchronous transition spaces The phenomenon refined: a truly concurrent semantics of proofs. 26
Asynchronous games An asynchronous game is an event structure equipped with a polarity function : → {− 1 , +1 } λ M − indicating whether a move is Player ( +1 ) or Opponent ( − 1 ). 27
Legal plays A legal play is a path ∗ m 1 m 2 m 3 m k − → x 1 − → x 2 − → · · · x k − 1 − → x k starting from the empty position ∗ of the transition space, and satisfying: λ ( m i ) = ( − 1) i . ∀ i ∈ [1 , ..., k ] , So, a legal play is alternated and starts by an Opponent move. 28
Strategies A strategy is a set of legal plays of even length , such that: — σ contains the empty play, — σ is closed under even-length prefix s · m · n ∈ σ ⇒ s ∈ σ, — σ is deterministic s · m · n 1 ∈ σ and s · m · n 2 ∈ σ ⇒ n 1 = n 2 . A strategy plays according to the current play. 29
Innocence: strategies with partial information Full abstraction result [Martin Hyland, Luke Ong, Hanno Nickau, 1994] Innocence characterizes the interactive behaviour of λ -terms. An innocent strategy plays according to the current view. 30
� � � � � � � � Where are the pointers in asynchronous games? · p · p · p · p m · n · n · n · m · n Play = sequence of moves with pointers 31
� � � � � � � � � � � � � � Event structure = generalized arena E E R 2 � � L 2 � � ������� � � � � � � � � � � � � � � � � � � ∼ � � R 1 � � � � L 1 L 2 R 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∼ ∼ � � � � � � � � � � � � � � � � � � � � � � L 1 R 1 � � � � L 2 � � � � R 2 � � � � ∼ � � ������� � � � � � � � � � � � � � � � � � � L 1 � � R 1 � � � � B B B · L 1 · L 2 · R 1 · R 2 · E 32
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