Towards an algebra of negation (work in progress) Paul-Andr Mellis - PowerPoint PPT Presentation

Towards an algebra of negation (work in progress) Paul-André Melliès CNRS, Université Paris Denis Diderot Foundational Methods in Computer Science Halifax 31 May 2008 1

Proof-knots Aim: formulate an algebra of these logical knots 2

A proof of the drinker’s formula Axiom A ( x 0 ) ⊢ A ( x 0 ) Right Weakening A ( x 0 ) ⊢ ∀ x . A ( x ) , A ( x 0 ) Right ⇒ Axiom B ⊢ B Left ⇒ ⊢ A ( x 0 ) ⇒ ∀ x . A ( x ) , A ( x 0 ) ( A ( x 0 ) ⇒ ∀ x . A ( x )) ⇒ B ⊢ A ( x 0 ) , B Left ∀ ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ A ( x 0 ) , B Right ∀ ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ ∀ x . A ( x ) , B Left Weakening ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } , A ( y ) 0 ⊢ ∀ x . A ( x ) , B Right ⇒ Axiom B ⊢ B Left ⇒ ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ A ( y ) 0 ⇒ ∀ x . A ( x ) , B ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } , ( A ( y ) 0 ⇒ ∀ x . A ( x )) ⇒ B ⊢ B , B Left ∀ ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } , ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ B , B Contraction ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ B , B Contraction ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⊢ B Right ⇒ ⊢ ∀ y . { ( A ( y ) ⇒ ∀ x . A ( x )) ⇒ B } ⇒ B 3

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 4

Duality Proponent Opponent Program Environment plays the game plays the game A ¬ A Negation permutes the rôles of Proponent and Opponent 5

Duality Opponent Proponent Environment Program plays the game plays the game ¬ A A Negation permutes the rôles of Opponent and Proponent 6

A brief history of games and linear logic 1977 André Joyal A category of games and strategies 1986 Jean-Yves Girard Linear logic 1992 Andreas Blass A semantics of linear logic Samson Abramsky 1994 Radha Jagadeesan A category of history-free strategies Pasquale Malacaria 1994 Martin Hyland A category of innocent strategies Luke Ong A schism between game semantics and linear logic 7

Sequential game semantics alternating A proof π A proof π sequences of moves Game semantics: an interleaving semantics of proofs. 8

� � An interleaving semantics The boolean game B : Player in red V F Opponent in blue � � � � � � � � � � � � � � � � � � � � true false � � � � � � � � � � q � question ∗ 9

� � � � An interleaving semantics The tensor product of two boolean games B 1 et B 2 : � ���������� � � � � � true 1 false 2 � � � � � � ���������� � � � � q 2 q 1 � � � � � � � ���������� � � � � true 1 � false 2 � � � � � � � � � � � � � � � � � � � q 1 � � q 2 � � � � � � � � � 10

� � � � A step towards true concurrency: bend the branches! � ������������������ � � false 2 true 1 � � � � � � � � � � � � � � � � � � q 2 � � q 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � true 1 � false 2 � � � � � � � � ������������������ � � � � � � � � � � q 1 q 2 � � � � � � � � 11

� � � � � � Asynchronous games: 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 � � � � � � � � ∗ ⊗ ∗ 12

Asynchronous game semantics trajectories in A proof π 1 A proof π 2 asynchronous transition spaces Main result: innocent strategies are positional. 13

� � � � � � Illustration: the strategy (true ⊗ false) V ⊗ F � ������������� false 2 � true 1 � � � � � � � � � � � � � V ⊗ q q ⊗ F ∼ � ����� � ������������� � � q 2 � � q 1 � � � � � � true 1 � false 2 � Strategies seen as � � � ����� � � � � � � � � closure operators q ⊗ q ∼ ∼ V ⊗ ∗ ∗ ⊗ F � ������ � � �������������� � on complete lattices � � � � � � q 2 � q 1 ������ � � � � true 1 � false 2 � � � � � � � � � � � q ⊗ ∗ ∗ ⊗ q ∼ � �������������� � � � � � � � � q 1 � q 2 � � � � � ∗ ⊗ ∗ 14

Part 1 The topological nature of negation At the interface between topology and algebra 15

Cartesian closed categories A cartesian category C is closed when there exists a functor C op × C ⇒ : −→ C and a natural bijection : C ( A × B , C ) C ( A , B ⇒ C ) ϕ A , B , C � � × ⇒ C C A B A B 16

The free cartesian closed category The objects of the category free-ccc ( C ) are the formulas A , B :: = X | A × B | A ⇒ B | 1 where X is an object of the category C . The morphisms are the simply-typed λ -terms, modulo βη -conversion. 17

The simply-typed λ -calculus Variable x : X ⊢ x : X Γ , x : A ⊢ P : B Abstraction Γ ⊢ λ x . P : A ⇒ B Γ ⊢ P : A ⇒ B ∆ ⊢ Q : A Application Γ , ∆ ⊢ PQ : B Γ ⊢ P : B Weakening Γ , x : A ⊢ P : B Γ , x : A , y : A ⊢ P : B Contraction Γ , z : A ⊢ P [ x , y ← z ] : B Γ , x : A , y : B , ∆ ⊢ P : C Permutation Γ , y : B , x : A , ∆ ⊢ P : C 18

� � Proof invariants Every ccc D induces a proof invariant [ − ] modulo execution. [ − ] free-ccc ( C ) � D C Hence the prejudice that proof theory is intrinsically syntactical... 19

However, a striking similarity with knot invariants A tortile category is a monoidal category with A B A A ∗ A A ∗ A B A A braiding twists duality unit duality counit The free tortile category is a category of framed tangles 20

� � Knot invariants Every tortile category D induces a knot invariant [ − ] free-tortile ( C ) � D C A deep connection between algebra and topology first noticed by Joyal and Street 21

Dialogue categories A symmetric monoidal category C equipped with a functor C op ¬ : −→ C and a natural bijection : C ( A ⊗ B , ¬ C ) C ( A , ¬ ( B ⊗ C ) ) ϕ A , B , C � ¬ ¬ � ⊗ ⊗ C C A B A B 22

The free dialogue category The objects of the category free-dialogue ( C ) are dialogue games constructed by the grammar A , B :: = X | A ⊗ B | ¬ A | 1 where X is an object of the category C . The morphisms are total and innocent strategies on dialogue games. As we will see: proofs are 3-dimensional variants of knots... 23

� � A presentation of logic by generators and relations Negation defines a pair of adjoint functors L C op C ⊥ R witnessed by the series of bijection: C op ( ¬ A , B ) C ( A , ¬ B ) C ( B , ¬ A ) � � 24

The 2-dimensional topology of adjunctions The unit and counit of the adjunction L ⊣ R are depicted as η : Id −→ R ◦ L ε : L ◦ R −→ Id R L η ε L R Opponent move = functor R Proponent move = functor L 25

A typical proof R L R R L R L R L L Reveals the algebraic nature of game semantics 26

A purely diagrammatic cut elimination R L 27

The 2-dimensional dynamic of adjunction L R ε ε L R L R = = η η L R Recovers the usual way to compose strategies in game semantics 28

Interlude: a combinatorial observation Fact: there are just as many canonical proofs 2 p 2 q � � �� � � � � �� � � R R ¬ · · · ¬ ¬ · · · ¬ A ⊢ A as there are increasing functions [ p ] −→ [ q ] between the ordinals [ p ] = { 0 < 1 < · · · < p − 1 } and [ q ] . This fragment of logic has the same combinatorics as simplices. 29

The two generators of a monad Every increasing function is composite of faces and degeneracies : : η [0] ⊢ [1] : [2] [1] µ ⊢ Similarly, every proof is composite of the two generators: : A ⊢ ¬¬ A η : µ ¬¬¬¬ A ⊢ ¬¬ A The unit and multiplication of the double negation monad 30