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

5

An algebra of duality

Opponent Environment plays the game ¬ A Proponent Program plays the game 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 ϕA,B,C :

C (A × B , C)

• C (A , B ⇒ C)

C B A ×

• C

B A ⇒

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 Abstraction Γ, x :A ⊢ P :B Γ ⊢ λx.P :A ⇒ B Application Γ ⊢ P :A ⇒ B ∆ ⊢ Q :A Γ, ∆ ⊢ PQ :B Weakening Γ ⊢ P :B Γ, x :A ⊢ P :B Contraction Γ, x :A, y :A ⊢ P :B Γ, z :A ⊢ P[x, y ← z] :B Permutation Γ, x :A, y :B, ∆ ⊢ P :C 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

B B A A

A A

A A∗ 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 ϕA,B,C :

C (A ⊗ B , ¬ C)

• C (A , ¬ ( B ⊗ C ) )

¬ C B A ⊗

• ¬

C B A ⊗ 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

C

L

• ⊥

C op

R

• witnessed by the series of bijection:

C (A, ¬ B)

• C (B, ¬ A)
• C op(¬ A, B)

17

The 2-dimensional topology of adjunctions

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

18

A typical proof

L L L L L R R R R R

Reveals the algebraic nature of game semantics

19

A purely diagrammatic cut elimination

R L

20

The 2-dimensional dynamic of adjunction

ε η L L

=

L L η ε R R

=

R R

Recovers the usual way to compose strategies in game semantics

21

Interesting fact

There are just as many canonical proofs

2p

• R

¬ · · · ¬ A ⊢

2q

• R

¬ · · · ¬ 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: η : A ⊢ ¬¬A µ : ¬¬¬¬A ⊢ ¬¬A The unit and multiplication of the double negation monad

23

The two generators in sequent calculus A ⊢ A

2

A , ¬A ⊢

1

A ⊢ ¬¬A A ⊢ A

6

A , ¬A ⊢

5

¬A ⊢ ¬A

4

¬A , ¬¬A ⊢

3

¬A ⊢ ¬¬¬A

2

¬¬¬¬A , ¬A ⊢

1

¬¬¬¬A ⊢ ¬¬A

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The two generators in string diagrams

The unit and multiplication of the monad R ◦ L are depicted as η : Id −→ R ◦ L

L R η

µ : R ◦ L ◦ R ◦ L −→ R ◦ L

L L L R R R µ

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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 κX : ¬ (X ⊗ ¬ A) ⊗ B −→ ¬ (X ⊗ ¬ (A ⊗ B)) 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+ ε B A R A B R R L L L 29

Tensor vs. negation : conjunctive strength

• R

A2

• B

L A1

κ

−→ R

• B

L

• A1

A2 Linear distributivity in a continuation framework

30

Tensor vs. negation : disjunctive strength

L

• A

R

• B1

B2

κ

−→

• L

B2

• A

R B1 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 X

ǫ

−→ κ −→ ǫ −→ ǫ −→

η

−→

η

−→ κ −→ ǫ −→

η

−→ ǫ −→ κ −→

η

−→

η

−→ Z factors uniquely as X κ −→ −→

ǫ

−→ −→

η

−→ −→ κ −→ −→ 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)

R R κL κR L 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 ϕA,B,C :

C (A ⊗ B , ¬ C)

• C (A , ¬ ( B ⊗ C ) )

¬ C B A ⊗

• ¬

C B A ⊗ 36

Frobenius objects

A Frobenius object F is a monoid and a comonoid satisfying

m d

=

m d

=

m d

A deep relationship with ∗-autonomous categories discovered by Day and Street.

37

Game cobordism

=

C op C op C op C op C op C op S C (x, y) C (x, ¬ y)

Idea: replace the elementary particles by the game boards

38

Game cobordism

=

C op C op C op C op C op C op C (x, ¬ y) C (x, ¬ y)

Idea: replace the elementary particles by the game boards

39

Game cobordism

=

C op C op C op C op C op C op C (x, y) C (x, ¬ y)

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