An algebraic presentation of innocence Paul-Andr Mellis CNRS, - PowerPoint PPT Presentation

An algebraic presentation of innocence Paul-André Melliès CNRS, Université Paris Denis Diderot Peripatetic Seminar on Sheaves and Logic in honor of Martin Hyland and Peter Johnstone Cambridge Sunday 4 April 2009 1

Le poisson soluble Is proof theory soluble in algebra ? 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

Guided by innocence Martin Luke A purely interactive description of formal proofs 4

An algebra of duality Proponent Opponent Program Environment plays the game plays the game A ¬ A Negation permutes the rôles of Proponent and Opponent 5

An algebra of duality Opponent Proponent Environment Program plays the game plays the game ¬ A A Negation permutes the rôles of Opponent and Proponent 6

Proof-knots Revealing the topological nature of proofs 7

An algebraic presentation of innocence Part 1 : Negation 8

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 9

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. 10

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 11

� � 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... 12

A striking similarity with representation theory A ribbon category is a monoidal category with A B A A ∗ A A ∗ A B A A braiding twists duality unit duality counit The free ribbon category is a category of framed tangles 13

� � Knot invariants Every ribbon category D induces a knot invariant [ − ] free-ribbon ( C ) � D C A deep connection between algebra and topology 14

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 15

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... 16

� � 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 ) � � 17

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 18

A typical proof R L R R L R L R L L Reveals the algebraic nature of game semantics 19

A purely diagrammatic cut elimination R L 20

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 21

Interesting fact There are just as many canonical proofs 2 p 2 q � � �� � � � � �� � � R R ¬ · · · ¬ ¬ · · · ¬ A ⊢ A as there are maps [ p ] −→ [ q ] between the ordinals [ p ] = { 0 < 1 < · · · < p − 1 } and [ q ] . This fragment of logic has the same combinatorics as simplices. 22

The two generators of a monad Every increasing function is composite of faces and degeneracies : : η [0] ⊢ [1] : µ [2] ⊢ [1] Similarly, every proof is composite of the two generators: : A ⊢ ¬¬ A η : ¬¬¬¬ A ⊢ ¬¬ A µ The unit and multiplication of the double negation monad 23

The two generators in sequent calculus A ⊢ A 6 A , ¬ A ⊢ 5 A ⊢ A ¬ A ⊢ ¬ A 2 4 A , ¬ A ⊢ ¬ A , ¬¬ A ⊢ 1 3 A ⊢ ¬¬ A ¬ A ⊢ ¬¬¬ A 2 ¬¬¬¬ A , ¬ A ⊢ 1 ¬¬¬¬ A ⊢ ¬¬ A 24

The two generators in string diagrams The unit and multiplication of the monad R ◦ L are depicted as η : Id −→ R ◦ L µ : R ◦ L ◦ R ◦ L −→ R ◦ L R R R L µ η L R L L 25

An algebraic presentation of innocence Part 2 : Tensor and negation 26

Tensor vs. negation A well-known fact: the continuation monad is strong ( ¬¬ A ) ⊗ B −→ ¬¬ ( A ⊗ B ) The starting point of the algebraic theory of side effects 27

Tensor vs. negation Proofs are generated by a parametric strength : ¬ ( X ⊗ ¬ A ) ⊗ B −→ ¬ ( X ⊗ ¬ ( A ⊗ B )) κ X which generalizes the usual notion of strong monad : κ : ¬¬ A ⊗ B −→ ¬¬ ( A ⊗ B ) 28

Proofs as 3-dimensional string diagrams The left-to-right proof of the sequent ¬¬ A ⊗ ¬¬ B ⊢ ¬¬ ( A ⊗ B ) is depicted as κ + κ + R ε L R R B L L B A A 29

Tensor vs. negation : conjunctive strength � � � ����� � � R � A 2 R � κ � � −→ � � � � � � � � � � � � � � B L � � � � � � � � B L � � ���� � � � A 1 A 2 A 1 Linear distributivity in a continuation framework 30

Tensor vs. negation : disjunctive strength � � � � � � L � � � � � � B 2 L � κ � � −→ � � � � � � � � � � � � A R � � � � � � � A R � � � � � � � � � B 1 B 2 B 1 Linear distributivity in a continuation framework 31

A factorization theorem The four proofs η, ǫ, κ � and κ � generate every proof of the logic. Moreover, every such proof −→ κ � η −→ κ � η η −→ κ � η η ǫ −→ ǫ −→ ǫ −→ ǫ −→ ǫ X −→ −→ −→ −→ −→ −→ Z factors uniquely as X κ � −→ κ � η ǫ −→ −→ −→ −→ −→ −→ −→ Z Corollary: two proofs are equal iff they are equal as strategies 32

Categorical combinatorics (Russ Harmer, Martin Hyland, PAM) Define a distributivity law λ ! ? −→ ? ! between a monad ? and a comonad ! on a category of games. The category of dialogue games and innocent strategies recovered by a bi-Kleisli construction ! A −→ ? B Big step instead of small step 33

Multi-threaded strategies (Samuel Mimram, PAM) κ L κ R R L R L Additional hypothesis that negation defines a monoidal functor 34

Part 3 Game cobordism Logical interaction as a material event 35

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 36

Frobenius objects A Frobenius object F is a monoid and a comonoid satisfying d m m = = m d d A deep relationship with ∗ -autonomous categories discovered by Day and Street. 37

Game cobordism C ( x, y ) C ( x, ¬ y ) S = C op C op C op C op C op C op Idea: replace the elementary particles by the game boards 38

Game cobordism C ( x, ¬ y ) C ( x, ¬ y ) = C op C op C op C op C op C op Idea: replace the elementary particles by the game boards 39

Game cobordism C ( x, ¬ y ) C ( x, y ) = C op C op C op C op C op C op Idea: replace the elementary particles by the game boards 40

Conclusion Logic = Data Structure + Duality This point of view is accessible thanks to 2-dimensional algebra 41

Abstract is concrete ! My intellectual debt to Martin 42