Asynchronous Games 4 A fully complete model of propositional linear logic Paul-Andr´ e Melli` es CNRS, Universit´ e Paris 7 Logic in Computer Science Chicago, Wednesday 29th June 2005 1
An epic in 50 slides Twenty-four seconds each 2
A crash course on Mazurkiewicz traces The 2-dimensional geometry of concurrency 3
Trace semantics Interpret any process = a | b π by the sequences of actions it performs in the course of time : ab ba 4
� � The synchronization tree of a process a | b interpreted as · · � ��������� � � � � � a � b � � � · · � ��������� � � � � a � � b � � � · 5
� � The synchronization tree of a process + ab ba interpreted as · · � ��������� � � � � � a � b � � � · · � ��������� � � � � a � � b � � � · 6
This is a problem Trace semantics cannot see the difference between a | b no interference and + ab ba interference 7
� � Idea : replace synchronization trees... a | b interpreted as · · � ��������� � � � � � a � b � � � · · � ��������� � � � � a � � b � � � · 8
� � ... by Mazurkiewicz traces a | b interpreted as · � � � � � � � � b � � a � � � � � � � � � � � � � � � � � � · � ∼ · � � � � � � � � � � � � � � � � a � � � � b � � � � � � � � · � 9
� � True concurrency = homotopy · � � � � � � � � b � a � � � � � � � � � � � � � � � � � � � ∼ · � · � � � � � � � � � � � � � � � � a � � � � b � � � � � � � � · � Think of this permutation as a 2-dimensional tile. 10
� � Interference = holes + ab ba interpreted as · � � � � � � � � b � a � � � � � � � � � � � � � � � � � � � ∼ · � · � � � � � � � � � � � � � � � � a � � � � b � � � � � � � � · � 11
Asynchronous games Games played on Mazurkiewicz traces 12
� � Game semantics Player in red V F Opponent in blue � � � � � � � � � � � � � � � � � � � � true false � � � � � � � � � � � q q ∗ The boolean game B 13
� � Traditional game semantics Player in red � ����������������� � Opponent in blue � � � � � � � � � true false � � � � � � � q The boolean game B 14
� � � � Traditional game semantics : an interleaving semantics � ���������� � � � � � false 2 true 1 � � � � � � ���������� � � � � q 2 q 1 � � � � � � � ���������� � � � � � true 1 false 2 � � � � � � � � � � � � � � � � � � � q 1 � � q 2 � � � � � � � � � The tensor product of two boolean games B 1 and B 2 15
� � � � Bend the branches ! � �������������������� � � � false 2 true 1 � � � � � � � � � � � � � � � � � � � � � q 2 q 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � true 1 � � false 2 � � � � � � � � �������������������� � � � � � � � � � � � q 1 q 2 � � � � � � � � � 16
� � � � � � Tile the diagram ! � ������������������ � � � � false 2 � true 1 � � � � � � � � � � � � ∼ � � ������� � ������������������ � � � � � � � � q 2 � q 1 � � � � � � true 1 false 2 � � � ������� � � � � � � � � � � � � ∼ ∼ � � � ������������������ � � � � � � � � � � � � � � � � � � � � � � � � q 2 q 1 � � � true 1 � � false 2 � � � � � � � � � � � � � � � � � � � � ∼ � � � ������������������ � � � � � � � � � q 1 � q 2 � � � � � � � � 17
� � � � � � Tag the positions ! 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 � � � � � � � ∗ ⊗ ∗ 18
� � � � � � A 2-dimensional space of interaction 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 � � � � � � � ∗ ⊗ ∗ 19
� � Asynchronous games A 2-dimensional graph equipped with tiles of the shape · � ���������� � n � m � � � � � � � ∼ · � · � ���������� � � � � � m n � � � � · � in which : ◦ every edge is polarized Player or Opponent ◦ an initial position ∗ is distinguished 20
Sequential play A sequential play is defined as an alternated path ∗ m 1 m 2 m 3 m k − → x 1 − → x 2 − → · · · x k − 1 − → x k starting by an Opponent move. 21
Strategies A strategy is a set of sequential 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. 22
Innocence 1994 Martin Hyland, Luke Ong, Hanno Nickau An interactive characterization of λ -terms 23
� � � � � � � � Innocence : strategies with partial information · p · p · p · p m · n · n · n · m · n The Player view � s � : what the Player can remember of the play s . An innocent strategy plays according to the current Player view. 24
Innocence 2004 From amnesia to positionality 25
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