Braided notions of dialogue categories Paul-Andr Mellis CNRS & - - PowerPoint PPT Presentation

Braided notions of dialogue categories

Paul-André Melliès

CNRS & Université Paris Denis Diderot Groupe de travail Sémantique Laboratoire PPS 24 janvier 2012

Where is the flow of logic?

Looking for a connection between proofs and knots

Revisiting proof-nets in linear logic

Claim: the traditional distinction between proof nets ↔ proof structures deserves to be understood from this topological point of view.

3

Sequent calculus

In linear logic, the two equivalent proofs

π1 · · · ⊢ A π2 · · · ⊢ B, C ⊢ A ⊗ B, C π3 · · · ⊢ D ⊢ A ⊗ B, C ⊗ D π1 · · · ⊢ A π2 · · · ⊢ B, C π3 · · · ⊢ D ⊢ B, C ⊗ D ⊢ A ⊗ B, C ⊗ D

4

Proof nets

are interpreted as the same proof net:

&

D C B A

π3 π2 π1 5

Sequentialization by deformation

&

D C B A

π3 π2 π1

π1 · · · ⊢ A π2 · · · ⊢ B, C π3 · · · ⊢ D ⊢ B, C ⊗ D ⊢ A ⊗ B, C ⊗ D

6

Sequentialization by deformation

&

D C B A

π3 π2 π1

π1 · · · ⊢ A π2 · · · ⊢ B, C ⊢ A ⊗ B, C π3 · · · ⊢ D ⊢ A ⊗ B, C ⊗ D

7

Multiplicative proof nets

axiom & & A A* A* A

&

axiom

&

axiom B* A

&

B A* A B A* B* & A B B A

Unfortunately, proof nets are not exactly string diagrams...

8

Tensorial logic

tensorial logic = a logic of tensor and negation = linear logic without A ¬¬A = the syntax of linear continuations = the syntax of dialogue games Provides a synthesis of linear logic and game semantics Research program: recast there the various aspects of linear logic

9

Tensorial logic

axiom A ⊢ A Γ ⊢ A A , ∆ ⊢ B cut Γ , ∆ ⊢ B Γ ⊢ A left ¬ Γ , ¬ A ⊢ ⊥ Γ , A ⊢ ⊥ right ¬ Γ ⊢ ¬ A Γ , A , B ⊢ C left ⊗ Γ , A ⊗ B ⊢ C Γ ⊢ A ∆ ⊢ B right ⊗ Γ , ∆ ⊢ A ⊗ B Γ ⊢ A left true Γ , true ⊢ A right true ⊢ true

10

Dialogue categories

A monoidal category with a left duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps B −→ A ⊸ ⊥ A familiar situation in tensorial algebra

11

Dialogue categories

A monoidal category with a right duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps A −→ ⊥ B A familiar situation in tensorial algebra

12

Dialogue categories

Definition. A dialogue category is a monoidal category C equipped with ⊲ an object ⊥ ⊲ two natural bijections ϕA,B :

C (A ⊗ B, ⊥)

−→

C (B, A ⊸ ⊥)

ψA,B :

C (A ⊗ B, ⊥)

−→

C (A, ⊥ B)

13

Cyclic dialogue categories

A dialogue category equipped with a family of bijections wheel A,B :

C (A ⊗ B, ⊥)

−→

C (B ⊗ A, ⊥)

natural in A and B making the diagram

C ((B ⊗ C) ⊗ A, ⊥)

associativity

C (A ⊗ (C ⊗ B), ⊥)

wheel B,C⊗A

• C (A ⊗ (B ⊗ C))

wheel A,B⊗C

• associativity
• C ((C ⊗ A) ⊗ B, ⊥)

C ((A ⊗ B) ⊗ C, ⊥)

wheel A⊗B,C

C (C ⊗ (A ⊗ B), ⊥)

associativity

• commutes.

14

Cyclic dialogue categories

The wheel should be understood diagrammatically as: wheel x,y :

x y f

→

x y f

15

The coherence diagram

x z f y x z f y x z f y wheel x y wheel x wheel , y z y , z x ,z

16

An equivalent formulation

A dialogue category equipped with a natural isomorphism turn A : A ⊸ ⊥ −→ ⊥ B making the diagram below commute: ⊥ (⊥ A) ⊗ A

eval

• B ⊗ (B ⊸ ⊥)

eval

• (A ⊸ ⊥) ⊗ A

turn A

• B ⊗ (⊥ B)

turn−1

B

• B ⊗ ((A ⊗ B) ⊸ ⊥) ⊗ A

eval

• turn A⊗B

B ⊗ (⊥ (A ⊗ B)) ⊗ A

eval

• 17

Balanced dialogue categories

A braiding γA,B : A ⊗ B −→ B ⊗ A

B B A A

A twist θA : A −→ A

A A 18

Main theorem

Every category C of atomic formulas induces a functor [−] such that free-dialogue(C )

[−]

free-ribbon(C ∗)

C

• where C ∗ is the category C extended with an object ∗.
• Theorem. The functor [−] is faithful.

Equality of proofs reduces to equality of knots modulo deformation

19

String Diagrams

A notation by Roger Penrose

20

Monoidal Categories

A monoidal category is a category C equipped with a functor: ⊗ : C × C −→ C an object: I and three natural transformations: (A ⊗ B) ⊗ C

α

−→ A ⊗ (B ⊗ C) I ⊗ A

λ

−→ A A ⊗ I

ρ

−→ A satisfying a series of coherence properties.

21

String Diagrams

A morphism f : A ⊗ B ⊗ C −→ D ⊗ E is depicted as:

f A B C D E 22

Composition

The morphism A

f

−→ B

g

−→ C is depicted as

A A C g ◦ f

=

g f A C B

Vertical composition

23

Tensor product

The morphism (A

f

−→ B) ⊗ (C

g

−→ D) is depicted as

A ⊗ C B ⊗ D f ⊗ g

=

g f A B C D

Horizontal tensor product

24

Example

f A B D D

f ⊗ idD

25

Example

g f A B C D

(f ⊗ idD) ◦ (idA ⊗ g)

26

Example

g f A B C D

(idB ⊗ g) ◦ (f ⊗ idC)

27

Meaning preserved by deformation

g f A B C D

=

g f A B C D

(f ⊗ idD) ◦ (idA ⊗ g) = (idB ⊗ g) ◦ (f ⊗ idC)

28

The functorial approach to knot invariants

Ribbon categories

29

Braided categories

A monoidal category C equipped with a family of isomorphisms γA,B : A ⊗ B −→ B ⊗ A natural in A and B, represented pictorially as the positive braiding

B B A A 30

Braided categories

As expected, the inverse map γ−1

A,B

: B ⊗ A −→ A ⊗ B is represented pictorially as the negative braiding

A A B B 31

Coherence diagram for braids [1]

A ⊗ (B ⊗ C)

γ

(B ⊗ C) ⊗ A

α

• (A ⊗ B) ⊗ C

α

• γ⊗C
• B ⊗ (C ⊗ A)

(B ⊗ A) ⊗ C

α

B ⊗ (A ⊗ C)

B⊗γ

• 32

Same coherence diagram in string diagrams

x y z x y z = x y z x y z

33

Coherence diagram for braids [2]

(A ⊗ B) ⊗ C

γ

C ⊗ (A ⊗ B)

α−1

• A ⊗ (B ⊗ C)

α−1

• A⊗γ
• (C ⊗ A) ⊗ B

A ⊗ (C ⊗ B)

α−1

(A ⊗ C) ⊗ B

γ⊗B

• 34

Same coherence diagram in string diagrams

x y z x y z = x y z x y z

35

Balanced categories

A braided monoidal category C equipped with a twist θA : A −→ A defined as a natural family of isomorphisms, and depicted as

A A 36

Coherence for twists

The twist θ is required to satisfy the equality θ I = id I and to make the diagram A ⊗ B

γA,B

• θA⊗B
• B ⊗ A

θB⊗θA

• A ⊗ B

B ⊗ A

γB,A

• commute for all objects A and B.

37

Coherence for twists

θx⊗y =

x y x y

38

Duality

A dual pair A ⊣ B is defined as a pair of morphisms η : I −→ A ⊗ B ε : B ⊗ A −→ I which are depicted as

A B

B A

39

Coherence for duality

The two morphisms η and ε should satisfy the “zig-zag” equalities:

A A

=

A A B B

=

B B

In that case, A is called a right dual of B.

40

Ribbon categories

Definition. A ribbon category is a balanced category C where ⊲ every object A has a right dual A∗ ⊲ the diagram A∗ ⊗ A

A∗⊗ θA

• θA∗⊗A
• A∗ ⊗ A

ε

• A∗ ⊗ A

ε

I

commutes for all objects A.

41

Ribbon categories

• Remark. – In a ribbon category, the object A∗ is also a left dual of A.

= x * x

η’

x * x

η

x = * x x * x

ε ε’ 42

Ribbon categories

Hence – the equations below are satisfied in every ribbon category

η ε’

=

x * x x x x

ε η’

* x x x

=

43

The free ribbon category

Theorem [Shum 1994] The free ribbon category free-ribbon(C ) generated by a category C has ⊲ as objects: signed sequences (Aε1

1 , . . . , Aεk k ) of objects of C ,

⊲ as morphisms: framed tangles with links labelled by maps in C .

44

The free ribbon category

So, a typical morphism in the category free-ribbon(C ) (A+) −→ (B+, C−, D+) looks like this:

g f D+ C− B+ A+

where f : A → B and g : C → D are morphisms in the category C .

45

Knot invariants

Every ribbon category D induces a knot invariant free-ribbon(C )

[−]

D

C

• The free ribbon category is a category of framed tangles

46

Jones polynomial invariant

• 2

x2 + 1 x4 + y2 x2

2x2 − x4 + x2y2

47

A construction of ribbon categories

Categories of modules over Hopf algebras

48

Bialgebras

A bialgebra in a braided category is an object H equipped with four maps µ : H ⊗ H → H η : I → H δ : H → H ⊗ H ε : H → I depicted as

H H H H η ε H H H H µ δ

defining a monoid and a comonoid, and satisfying the four equalities...

49

Bialgebras

H H H H H H H H H H H H

= = =

H H H H id

=

50

Antipode

A Hopf algebra is a bialgebra equipped with a morphism S : H −→ H satisfying the equality:

=

H H H

S

H H H

S

=

51

Hopf algebras

Every Hopf algebra H induces a monoidal closed category Mod(H)

• f left modules, where the action

H ⊗ (W V) −→ W V

• n the function space W V is defined as

h ⊲ f : v →

• h(1) ⊲ f ( S h(2) ⊲ v )

52

Action on the right negation V ⊸ ⊥

H V V

reval

=

reval

H V V

S

53

Hopf algebras

When the antipode S is reversible, the category is also closed on the left, with action H ⊗ (V ⊸ W) −→ V ⊸ W defined as h ⊲ f : v →

• h(2) ⊲ f ( S−1 h(1) ⊲ v )

54

Action on the left negation V ⊸ ⊥

H V

leval

V

leval

=

H V V

S -1

55

Braided Hopf algebras

A braiding on H is defined as an invertible element R ∈ H ⊗ H such that R · ∆(h) = ∆op(h) · R (∆ ⊗ idH)(R) = R13 · R23 (idH ⊗ ∆)(R) = R13 · R12

56

Braided Hopf algebras

H R H H H R H H

=

R H H H R H H H R R H H H

=

R H R H H

=

57

Braided Hopf algebras

Every braiding R =

• i

si ⊗ ti

• n the Hopf algebra H induces a braiding on the category Mod(H)

cV,W : V ⊗ W −→ W ⊗ V defined as v ⊗ w →

• i

(ti ⊲ w) ⊗ (si ⊲ v)

58

Braided Hopf algebras

The braiding on Mod(H) induces a map of left H-module ⊥ V −→ V ⊸ ⊥ where ⊥ = k denotes the base field, defined as f → v → f(u ⊲ v) where the vector u is itself defined as u =

• i

S(ti) ⊗ si

59

The element u of a braided Hopf algebra

H

S

R

60

Ribbon Hopf algebras

A braided Hopf algebra H equipped with an invertible element θ ∈ H which is central in H and satisfies the three equations ∆(θ) = (R21R)−1(θ ⊗ θ) ε(θ) = 1 S(θ) = θ

61

Ribbon Hopf algebras

θ

=

θ S θ

H H

= 1

θ

H H

θ

H H

=

θ

R -1 R -1

62

The grouplike element σ

An important point is that the element σ = θ−1 · u ∈ H is group-like: ∆(σ) = σ ⊗ σ.

63

Ribbon Hopf algebras

When the Hopf algebra is ribbon, the category Mod(H) is balanced. The twist map θV : V −→ V is defined as: v → θ−1 ⊲ v

• Theorem. The subcategory of finite dimensional modules is ribbon.

64

A functorial bridge from proofs to knots

Balanced dialogue categories

65

Dialogue categories

Definition. A dialogue category is a monoidal category C equipped with ⊲ an object ⊥ ⊲ two natural bijections ϕA,B :

C (A ⊗ B, ⊥)

−→

C (B, A ⊸ ⊥)

ψA,B :

C (A ⊗ B, ⊥)

−→

C (A, ⊥ B)

66

Cyclic dialogue categories

A dialogue category equipped with a family of bijections wheel A,B :

C (A ⊗ B, ⊥)

−→

C (B ⊗ A, ⊥)

natural in A and B making the diagram

C ((B ⊗ C) ⊗ A, ⊥)

associativity

C (A ⊗ (C ⊗ B), ⊥)

wheel B,C⊗A

• C (A ⊗ (B ⊗ C))

wheel A,B⊗C

• associativity
• C ((C ⊗ A) ⊗ B, ⊥)

C ((A ⊗ B) ⊗ C, ⊥)

wheel A⊗B,C

C (C ⊗ (A ⊗ B), ⊥)

associativity

• commute.

67

Balanced dialogue categories

A balanced category C equipped with ⊲ an object ⊥ ⊲ two natural bijections ϕA,B :

C (A ⊗ B, ⊥)

−→

C (B, A ⊸ ⊥)

ψA,B :

C (A ⊗ B, ⊥)

−→

C (A, ⊥ B)

Equivalently, it is a dialogue category equipped with a braiding and a twist.

68

From balanced to cyclic dialogue categories

Property. Every balanced dialogue category C is cyclic. The function wheel A,B :

C (A ⊗ B, ⊥)

−→

C (B ⊗ A, ⊥)

defining the cyclic structure transports every map A ⊗ B

f

⊥

to the map B ⊗ A

B⊗θA

B ⊗ A

γB,A

A ⊗ B

f

⊥

The twist is fundamental, the property would not hold with just a braiding.

69

The bad wheel

x y z

f

x y z

f

=

70

The good wheel

wheelx,y :

x y

f

→

x y

f 71

The twist ensures the topological equality

f B A ⊥

=

f B A ⊥ 72

Back to finite dimensional categories

Suppose given a ribbon Hopf algebra H. In that case, the category Mod(H) is a balanced dialogue category. This comes from the fact that σ = θ−1 · u is group-like.

• Corollary. The category of finite dimensional H-modules is ribbon.

73

A bridge between proofs and knots

Every category C induces a functor [−] such that free-dialogue(C )

[−]

free-ribbon(C ∗)

C

• where C ∗ is the category C extended with an object ∗.

Main theorem. The functor [−] is faithful. Equality of proofs reduces to equality of tangles modulo deformation

74

A basic illustration

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 75

A basic illustration

The left-to-right proof of the sequent ¬¬A ⊗ ¬¬B ⊢ ¬¬(A ⊗ B) is transported to a tangle A ⊗ ⊥ ⊗ ⊥ ⊗ B ⊗ ⊥ ⊗ ⊥ −→ A ⊗ B ⊗ ⊥ ⊗ ⊥ A topological account of game semantics

76

Another illustration

In every cyclic dialogue category ⊥ (⊥ x)

⊥turn x

⊥ (x ⊸ ⊥)

(⊥ x) ⊸ ⊥

turn ⊥x

• ⊥ (x ⊸ ⊥)

twist (x⊸⊥)

• x

η′

• η
• 77

Another illustration

η x *

=

x * x x η’

78

The free dialogue category

A topological point of view on game semantics

79

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. In the case of the free symmetric dialogue category

80

The free dialogue category

The objects of the category free-dialogue(C ) are the formulas

• f ribbon logic:

A, B ::= X | A ⊗ B | ¬A | 1 where X is an object of the category C . The morphisms are the proofs of the logic modulo equality. In the case of the free balanced dialogue category

81

Ribbon logic

Axiom A ⊢ A Γ ⊢ A Υ1, A, Υ2 ⊢ B Υ1, Γ, Υ2 ⊢ B Cut Right ⊗ Γ ⊢ A ∆ ⊢ B Γ, ∆ ⊢ A ⊗ B Υ1, A, B, Υ2 ⊢ C Υ1, A ⊗ B, Υ2 ⊢ C Left ⊗ Right I ⊢ I Υ1, Υ2 ⊢ A Υ1, I, Υ2 ⊢ A Left I

82

Ribbon logic

Right Γ, A ⊢ ⊥ Γ ⊢ ⊥ A Γ ⊢ A ⊥ A, Γ ⊢ ⊥ Left Right ⊸ A, Γ ⊢ ⊥ Γ ⊢ A ⊸ ⊥ Γ ⊢ A Γ, A ⊸ ⊥ ⊢ ⊥ Left ⊸

83

The exchange rule

Main ingredient: the exchange rule Exchange [g] A1, . . . , An ⊢ B Ag∗1, . . . , Ag∗n ⊢ B is parametrized by the elements g ∈ Gn of the ribbon group.

84

The ribbon group

The ribbon group is presented by the generators σi 1 ≤ i ≤ n − 1 θj 1 ≤ j ≤ n together with the equations below: σi ◦ σi+1 ◦ σi = σi+1 ◦ σi ◦ σi+1 σi ◦ σj = σj ◦ σi when |j − i| ≥ 2. σi ◦ θi = θi+1 ◦ σi σi ◦ θi+1 = θi ◦ σi σi ◦ θj = θj ◦ σi when j < i or when j ≥ j + 2.

85

Equalities between proofs

π · · · A1, . . . , An ⊢ B [h] Ah∗1, . . . , Ah∗n ⊢ B [g] Ag∗(h∗1), . . . , Ag∗(h∗n) ⊢ B

• π

· · · A1, . . . , An ⊢ B [g ◦ h] Ag◦h∗1, . . . , Ag◦h∗n ⊢ B π · · · A1, . . . , An ⊢ B [e] Ae∗1, . . . , Ae∗n ⊢ B

• π

· · · A1, . . . , An ⊢ B

86

Equalities between proofs

An equality reflecting the coherence diagram for braidings:

π · · · Υ1, A, B, C, Υ2 ⊢ D Left ⊗ Υ1, A ⊗ B, C, Υ2 ⊢ D [p ⊗ σ ⊗ q] Υ1, C, A ⊗ B, Υ2 ⊢ D

• π

· · · Υ1, A, B, C, Υ2 ⊢ D [p ⊗ 1 ⊗ σ ⊗ q] Υ1, A, C, B, Υ2 ⊢ C [p ⊗ σ ⊗ 1 ⊗ q] Υ1, C, A, B, Υ2 ⊢ C Left ⊗ Υ1, C, A ⊗ B, Υ2 ⊢ C

where p and q are the respective lengths of Υ1 and Υ2.

87

Equalities between proofs

An equality reflecting the coherence diagram for twists:

π · · · Υ1, A, B, Υ2 ⊢ C Left ⊗ Υ1, A ⊗ B, Υ2 ⊢ C [p ⊗ θ ⊗ q] Υ1, A ⊗ B, Υ2 ⊢ C

• π

· · · Υ1, A, B, Υ2 ⊢ C [p ⊗ θ ⊗ θ ⊗ q] Υ1, A, B, Υ2 ⊢ C [p ⊗ σ ⊗ q] Υ1, B, A, Υ2 ⊢ C [p ⊗ σ ⊗ q] Υ1, A, B, Υ2 ⊢ C Left ⊗ Υ1, A ⊗ B, Υ2 ⊢ C

where p and q are the respective lengths of Υ1 and Υ2.

88

A bridge between proofs and knots

Every category C induces a functor [−] such that free-dialogue(C )

[−]

free-ribbon(C ∗)

C

• where C ∗ is the category C extended with an object ∗.

Main theorem. The functor [−] is faithful. Equality of proofs reduces to equality of tangles modulo deformation

89