Programming Languages in String Diagrams. [ one ] String Diagrams. Paul-André Melliès. Oregon Summer School in Programming Languages June 2011

Transcription

1 Programming Languages in String Diagrams [ one ] String Diagrams Paul-ndré Melliès Oregon Summer School in Programming Languages June 2011

2 String diagrams diagrammatic account o logic and programming 2

3 Semantics in the 1980 s Cartesian-Closed Categories λ-calculus Intuitionistic Logic 3

4 Semantics in the 1990 s Monoidal-Closed Categories Proo-Nets Linear Logic 4

5 Where is the low o logic? o o o o Ideography (Frege) Sequent calculus (Gentzen) Natural deduction (Prawitz) Proo nets (Girard) Looking or a purely mathematical notation or proos... 5

6 Guide : the algebra o Feynman diagrams We want to see the low inside our proos and programs!

7 String Diagrams Categorical lgebra String Diagrams Logic and Language n algebraic investigation o logic and programming 7

8 String Diagrams Categorical lgebra String Diagrams Logic and Language bridge between algebra, physics and low level languages 8

9 Compositionality in logic The reason or introducing categories 9

10 Sequent calculus o linear logic xiom Cut Let Right Γ Υ 1,, Υ 2 Υ 1, Γ, Υ 2 Υ 1,,, Υ 2 C Υ 1,, Υ 2 C Γ Γ, Let Γ Υ 1,, Υ 2 C Υ 1, Γ,, Υ 2 C Right, Γ Γ 10

11 Modus Ponens I the property is true and the property is true, then the property is true. 11

12 Modus ponens in the sequent calculus π 1. π 2. Cut 12

13 Modus ponens in the type system π 1. P : π 2. x : M : (λx.m) P : 13

14 Modus Ponens I the property is true and the property C is true, then the property C is true. 14

15 Modus ponens in the sequent calculus π 1. C π 2. C Cut 15

16 Modus ponens in the type system π 1. x : P : π 2. y : M : C x : (λy.m) P : C 16

17 The logical graph Its vertices are the logical ormulas and its edges π are the logical proos π. 17

18 Modus ponens in the logical graph Every pair o edges π 1 π 2 C induces an edge π 3 C which may be denoted as π 3 = π 2 π 1 18

19 ssociativity Given a path π 1 π 2 C π 3 D one would like that the order o composition does not matter: π 3 ( π 2 π 1 ) = ( π 3 π 2 ) π 1 19

20 ssociativity in the sequent calculus π 1. C π 2. π 3 C. C D D = π 1. π 2. C D D π 3. C D 20

21 The identity proo xiom 21

22 Identity Given id π id one would like that id π = π = π id 22

23 Identity in the sequent calculus π. xiom is equal to Cut π. is equal to π. Cut xiom 23

24 The logical graph (bis) Its vertices are the logical ormulas and its edges π are the logical proos π. modulo symbolic execution. 24

25 Deinition o a category graph equipped with a composition operation g C g C and an identity edge or every vertex o the graph id satisying the associativity h ( g ) = ( h g ) and identity equations: id = = id 25

26 What about the conjunction? Given two proos π π one is able to construct the proo: π 1. π , Right Let 26

27 Tensor product So, every pair o edges induces an edge 1 π π 1 2 which may be denoted as 1 2 π π 3 = π 1 π 2 27

28 Monoidal Categories monoidal category is a category C equipped with a unctor: an object: : C C C and three natural transormations: ( ) C I α ( C) I λ I ρ satisying a series o coherence properties. 28

29 String Diagrams notation by Roger Penrose 29

30 String Diagrams morphism : C D E is depicted as: D E C 30

31 Composition The morphism g C is depicted as C C g g = Vertical composition 31

32 Tensor product The morphism ( ) (C g D) is depicted as D D g = g C C Horizontal tensor product 32

33 Example D D id D 33

34 Example D g C ( id D ) (id g) 34

35 Example g D C (id g) ( id C ) 35

36 Meaning preserved by deormation D D g = g C C ( id D ) (id g) = (id g) ( id C ) 36

37 pplication to knot theory tensor algebra o knots 37

38 raided categories monoidal category equipped with braid maps γ, γ 1, 38

39 Coherence diagram or braids α ( C) γ ( C) α ( ) C (C ) γ C ( ) C α ( C) γ 39

40 Topological deormation in string diagrams C C = C C 40

41 Coherence diagram or braids α 1 ( ) C γ C ( ) α 1 ( C) (C ) γ (C ) α 1 ( C) γ 41

42 Topological deormation in string diagrams C C = C C 42

43 Duality duality is a pair o morphisms I η ɛ I ɛ η 43

44 Duality satisying the two zig-zag equalities: = = In that case, is called a let dual o. 44

45 Ribbon category braided category in which every object has a dual satisying: = = 45

46 Knot invariants Every ribbon category D induces a knot invariant ree-ribbon(c) [[ ]] D C The ree ribbon category is a category o ramed tangles 46

47 Jones polynomial invariant 2 x x 4 + y2 x 2 2x 2 x 4 + x 2 y 2 compositional semantics o knots 47

48 Can we apply this to proos and programs? The topic o the next session... 48

49 Monoidal categories with eedback deinition by Joyal, Street and Verity (1996) 49

50 Trace operator trace in a symmetric monoidal category C is an operator Tr U, U U depicted as eedback in string diagrams: 50

51 Trace operator U Tr U, ( ) = U U 51

52 Sliding (naturality in U) V u = U u V U 52

53 Tightening (naturality in, ) b b = a a 53

54 Vanishing (monoidality in U) U V U I V = = 54

55 Superposing g = g = g 55

56 Yanking = = U U U 56

57 Fixpoint operator Given a map : U U we want to compute a map Fix [ ] : U such that ( a, Fix[ ] (a) ) = Fix[ ] (a) Enables to compute recursive deinitions 57

58 Fixpoints computed rom traces Suppose given a diagonal map Then, U U U U U Fix : U U deines a well-behaved ixpoint operator. 58

59 Traces = parametric ixpoints (1) U U 59

60 Traces = parametric ixpoints (2) U U 60

61 Traces = parametric ixpoints (3) U U U 61

62 Traces = parametric ixpoints (4) U U U 62

63 Traces = parametric ixpoints (5) U U 63

64 Traces = parametric ixpoints (6) U U U = U 64