String Diagrams Paul-Andr Mellis Oregon Summer School in - - PowerPoint PPT Presentation

Programming Languages in String Diagrams [

• ne

]

String Diagrams

Paul-André Melliès Oregon Summer School in Programming Languages June 2011

String diagrams

A diagrammatic account of logic and programming

2

Semantics in the 1980’s

Cartesian-Closed Categories Intuitionistic Logic λ-calculus

3

Semantics in the 1990’s

Monoidal-Closed Categories Linear Logic Proof-Nets

4

Where is the flow of logic?

• Ideography (Frege)
• Sequent calculus (Gentzen)
• Natural deduction (Prawitz)
• Proof nets (Girard)

Looking for a purely mathematical notation for proofs...

5

Guide : the algebra of Feynman diagrams

We want to see the flow inside our proofs and programs !

String Diagrams

Categorical Algebra Logic and Language String Diagrams

An algebraic investigation of logic and programming

7

String Diagrams

Categorical Algebra Logic and Language String Diagrams

A bridge between algebra, physics and low level languages

8

Compositionality in logic

The reason for introducing categories

9

Sequent calculus of linear logic

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

10

Modus Ponens

If the property A is true and the property A ⇒ B is true, then the property B is true.

11

Modus ponens in the sequent calculus π1 . . . ⊢ A π2 . . . A ⊢ B Cut ⊢ B

12

Modus ponens in the type system π1 . . . ⊢ P : A π2 . . . x : A ⊢ M : B ⊢ (λx.M) P : B

13

Modus Ponens

If the property A ⇒ B is true and the property B ⇒ C is true, then the property A ⇒ C is true.

14

Modus ponens in the sequent calculus π1 . . . A ⊢ B π2 . . . B ⊢ C Cut A ⊢ C

15

Modus ponens in the type system π1 . . . x : A ⊢ P : B π2 . . . y : B ⊢ M : C x : A ⊢ (λy.M) P : C

16

The logical graph

Its vertices are the logical formulas and its edges A

π

• B

are the logical proofs π

. . .

A ⊢ B

17

Modus ponens in the logical graph

Every pair of edges A

π1

• B

π2

• C

induces an edge A

π3

• C

which may be denoted as π3 = π2

• π1

18

Associativity

Given a path A

π1

• B

π2

• C

π3

• D
• ne would like that the order of composition does not matter:

π3 ◦ ( π2 ◦ π1 ) = ( π3 ◦ π2 ) ◦ π1

19

Associativity in the sequent calculus

π1

. . .

A ⊢ B π2

. . .

B ⊢ C A ⊢ C π3

. . .

C ⊢ D A ⊢ D = π1

. . .

A ⊢ B π2

. . .

B ⊢ C π3

. . .

C ⊢ D B ⊢ D A ⊢ D

20

The identity proof Axiom A ⊢ A

21

Identity

Given A

idA

• A

π

• B

idB

• B
• ne would like that

idA ◦ π = π = π ◦ idB

22

Identity in the sequent calculus

Axiom A ⊢ A π

. . .

A ⊢ B Cut A ⊢ B is equal to π

. . .

A ⊢ B is equal to π

. . .

A ⊢ B Axiom B ⊢ B Cut A ⊢ B

23

The logical graph (bis)

Its vertices are the logical formulas and its edges A

π

• B

are the logical proofs π

. . .

A ⊢ B modulo symbolic execution.

24

Definition of a category

A graph equipped with a composition operation A

f

B

g

C

→ A

g◦f

C

and an identity edge for every vertex of the graph A

idA

A

satisfying the associativity h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f and identity equations: idA ◦ f = f = f

• idB

25

What about the conjunction?

Given two proofs π1

. . .

A1 ⊢ A2 π2

. . .

B1 ⊢ B2

• ne is able to construct the proof:

π1

. . .

A1 ⊢ A2 π2

. . .

B1 ⊢ B2 Right ⊗ A1, B1 ⊢ A2 ⊗ B2 Left ⊗ A1 ⊗ B1 ⊢ A2 ⊗ B2

26

Tensor product

So, every pair of edges A1

π1

B1

A2

π1

B2

induces an edge A1 ⊗ A2

π3

• B1 ⊗ B2

which may be denoted as π3 = π1 ⊗ π2

27

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.

28

String Diagrams

A notation by Roger Penrose

29

String Diagrams

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

f A B C D E 30

Composition

The morphism A

f

−→ B

g

−→ C is depicted as

A A C g ◦ f

=

g f A C B

Vertical composition

31

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

32

Example

f A B D D

f ⊗ idD

33

Example

g f A B C D

( f ⊗ idD) ◦ (idA ⊗ g)

34

Example

g f A B C D

(idB ⊗ g) ◦ (f ⊗ idC)

35

Meaning preserved by deformation

g f A B C D

=

g f A B C D

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

36

Application to knot theory

A tensor algebra of knots

37

Braided categories

A monoidal category equipped with braid maps A ⊗ B

γA,B

−→ B ⊗ A

B B A A

B ⊗ A

γ−1

A,B

−→ A ⊗ B

A A B B

38

Coherence diagram for braids

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

• 39

Topological deformation in string diagrams

C B A A C B

=

B A A C B C

40

Coherence diagram for braids

(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

• 41

Topological deformation in string diagrams

C B A B A C

=

B A B A C C

42

Duality

A duality A ⊣ B is a pair of morphisms I

η

−→ A ⊗ B

A A∗ ηA

B ⊗ A

ǫ

−→ I

A A∗ ǫA

43

Duality

satisfying the two “zig-zag” equalities:

A A

=

A A B B

=

B B

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

44

Ribbon category

A braided category in which every object A has a dual A∗ satisfying:

A A∗ A∗∗ A∗ A

=

A A

=

A A∗ A∗∗ A∗ A

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

A compositional semantics of knots

47

Can we apply this to proofs and programs?

The topic of the next session...

48

Monoidal categories with feedback

A definition by Joyal, Street and Verity (1996)

49

Trace operator

A trace in a symmetric monoidal category C is an operator

A ⊗ U −→ B ⊗ U Tr U

A, B

A −→ B

depicted as feedback in string diagrams:

50

Trace operator

(

)

f f = A A U B B U U TrU

A,B

51

Sliding (naturality in U)

u u f f = A A B B V U U V

52

Tightening (naturality in A, B)

a b a b f f =

53

Vanishing (monoidality in U)

f f = U ⊗ V V U f f = I 54

Superposing

g f f g f g = =

55

Yanking

U U U = = 56

Fixpoint operator

Given a map f : A ⊗ U −→ U we want to compute a map Fix [f] : A −→ U such that f ( a , Fix[f] (a) ) = Fix[f] (a) Enables to compute recursive definitions

57

Fixpoints computed from traces

Suppose given a diagonal map U −→ U ⊗ U Then, Fix :

f A U U

→

∆ f A U U

defines a well-behaved fixpoint operator.

58

Traces = parametric fixpoints (1)

∆ f A U U

59

Traces = parametric fixpoints (2)

∆ ∆ f f A U U

60

Traces = parametric fixpoints (3)

∆ ∆ f f A A U U U

61

Traces = parametric fixpoints (4)

∆ f ∆ f A A U U U

62

Traces = parametric fixpoints (5)

∆ f ∆ f A A A U U

63

Traces = parametric fixpoints (6)

∆ f A U U

=

∆ f ∆ f A U U

64