String diagrams. a functorial semantics of proofs and programs. Paul-André Melliès. CNRS, Université Paris Denis Diderot.

Transcription

1 String diagrams a functorial semantics of proofs and programs Paul-André Melliès CNRS, Université Paris Denis Diderot Hopf-in-Lux Luxembourg 17 July

2 Proof-knots Aim: formulate an algebra of these logical knots 2

3 The purpose of semantics A mathematical study of programming languages and compilation schemes. Both functional and imperative languages, based on a kernel of λ-calculus: PCF Algol ML OCAML JAVA λ-calcul states exceptions modules concurrency higher order references objects synchronization typing threads recursion Languages designed between academic and industrial circles, between semantics (denotational or operational) and implementations. 2

4 The purpose of semantics A mathematical theory of programming languages from conception to compilation. Advanced programming language compilation Elementary commands of the machine A prerequisite to the complete verification of a given implementation. 3

5 The simply-typed λ-calculus Variable Abstraction Application Weakening Contraction Permutation x :X $ x :X Γ, x :A $ P :B Γ $ λx.p :A ñ B Γ $ P :A ñ B $ Q :A Γ, $ PQ :B Γ $ P :B Γ, x :A $ P :B Γ, x :A, y :A $ P :B Γ, z :A $ Prx, y Ð zs :B Γ, x :A, y :B, $ P :C Γ, y :B, x :A, $ P :C 4

6 Key idea Compilation itself is the ultimate semantics! Advanced programming language semantics Elementary transition systems There remains to describe the grammar of an idealized compilation. 5

7 Starting point: 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 6

8 The formal proof of the drinker s formula Axiom Apx 0 q $ Apx 0 q Right Weakening Apx 0 q $ Apx 0 Right ñ $ Apx 0 q, Apx 0 q Right D $ Apx 0 q, Dy.tApyq Dy.tApyq Left Weakening Apy 0 q Dy.tApyq Right ñ $ Apy 0 q Dy.tApyq Right D $ Dy.tApyq Dy.tApyq Contraction $ Dy.tApyq 7

9 Game semantics Simply typed λ-calculus Games game semantics Games typically defined as 2-player decision trees 8

10 Sequential game semantics A proof π alternating sequences of moves A proof π Game semantics: an interleaving semantics of proofs. 9

11 Sequential games A sequential game pm, P, λq consists of M P M λ : M Ñ t 1, 1u a set of moves, a set of plays, a polarity function on moves. Alternatively, a sequential game is a polarized decision tree. 10

12 Strategies A strategy σ is a set of alternating plays of even-length s m 1 m 2k starting by an Opponent move such that: σ contains the empty play, σ is closed by n P M, s m n P σ ñ s P σ σ is P n 1, n 2 P M, s m n 1 P σ and s m n 2 P σ ñ n 1 n 2. 11

13 An interleaving semantics The boolean game B: V true q F false question Player in red Opponent in blue 12

14 Three algebraic operations on sequential games the negation A the tensor the sum A b B A ` B The same algebraic structure as in linear algebra! 13

15 Negation Proponent Program plays the game A Opponent Environment plays the game A Negation permutes the rôles of Proponent and Opponent 14

16 Negation Opponent Environment plays the game A Proponent Program plays the game A Negation permutes the rôles of Opponent and Proponent 15

17 Tensor product b Play the two games in parallel 16

18 Sum ` Proponent selects one component 17

19 A category G of games and strategies its objects A, B are the sequential games its morphisms A ÝÑ B are the strategies playing on A b B 18

20 A typical strategy R L R R L R L R L L 19

21 Composition of strategies The composite strategy of the two strategies A A ÝÑ C ÝÑ σ B B ÝÑ τ C is obtained by letting the strategies σ and τ interact on the game B. Describes the evaluation of a program and its procedure 20

22 A typical composition R L 21

23 The identity strategy The identity strategy A ÝÑ A is defined as the copycat strategy on the game A b A Clarifies the logical principle that : A or not A 22

24 An interleaving semantics The tensor product of two boolean games B 1 et B 2 : false 2 true 1 q 2 q 1 true 1 false 2 q 1 q 2 23

25 A step towards true concurrency: bend the branches! true 1 false 2 q 2 q 1 true 1 q2 q 1 false2 24

26 Asynchronous games: tile the diagram! V b q 2 true 1 V b q q b false 2 V b F q b q true 1 true 1 false 2 2 q q 1 b q 1 q 2 q b F b q q 1 false 2 b F 25

27 Asynchronous game semantics A proof π 1 trajectories in asynchronous transition spaces A proof π 2 Main result: innocent strategies are positional. 26

28 Illustration: the strategy (true b false) V b q 2 true 1 V b q q b false 2 V b F q b q true 1 true 1 false 2 2 q q 1 b q 1 q2 q b F b q q 1 false 2 b F Strategies seen as closure operators on complete lattices 27

29 Asynchronous games There are two sequential implementations of the conjunction operator ^ B b B ÝÑ B q q true q true true Left implementation B b B ÝÑ B q q true q true true Right implementation 28

30 Asynchronous games The game associated to B b B ÝÑ B looks like a flower with eight petals output cri question R input L input R input L question L question L inputr questionr question Left implementation one for each terminal position pinput L b input R q ÝÑ output of the game. 29

31 Asynchronous games true aie question R true L true R true L question L true R Left implementation question L questionr question 30

32 Asynchronous games true cri true R true L question R question L true L true R question L question R question Right implementation 31

33 Asynchronous games true cri question R true L true R true L question L question L true R questionr question Left implementation 32

34 Logical dualities in categories From orders to categories 33

35 Duality in a boolean algebra Negation defines a bijection negation B K B op negation between the boolean algebra B and its opposite boolean algebra B op. 34

36 Duality in a category Negation should define an equivalence negation C K C op negation between a cartesian closed category C and its opposite category C op. 35

37 However, in a cartesian closed category... Suppose that the category has an initial object 0. Then, Every morphism f : A ÝÑ 0 is an isomorphism. Its inverse is the unique map h making the diagram commute: id π 1 A h A 0 f π 2 A 0 36

38 In a self-dual cartesian closed category... Homp A, B q Homp A 1, B q Homp 1, A ñ B q Homp pa ñ Bq, 1 q Homp pa ñ Bq, 0 q Hence, every such self-dual category C is a preorder! 37

39 Duality in a heyting algebra Every object K defines an adjunction negation H K H op negation between the heyting algebra H and its opposite algebra H op. a H K b ðñ b H a K ðñ a K H op b 38

40 Duality in a dialogue category Negation defines an adjunction negation C K C op negation between the dialogue category C and its opposite category C op. Apa, K bq Apb, a Kq A op pa K, bq 39

41 String Diagrams A notation by Roger Penrose (1970) 40

42 Monoidal Categories A monoidal category is a category C equipped with a functor: b : C C ÝÑ C an object: I and three natural transformations: pa b Bq b C α ÝÑ A b pb b Cq I b A λ ÝÑ A A b I ρ ÝÑ A satisfying a series of coherence properties. 41

43 String Diagrams A morphism f : A b B b C ÝÑ D b E is depicted as: D E f A B C 42

44 Composition The morphism A f ÝÑ B g ÝÑ C is depicted as C C g g f = B f A A Vertical composition 43

45 The morphism pa f ÝÑ Bq b pc Tensor product g ÝÑ Dq is depicted as B D B D f g = g f A C A C Horizontal tensor product 44

46 Example B D f A D f b id D 45

47 Example B D f g A C p f b id D q pid A b gq 46

48 Example B g D f A C pid B b gq p f b id C q 47

49 Meaning preserved by deformation B D B D f g = f g A C A C p f b id D q pid A b gq = pid B b gq p f b id C q 48

50 Functorial boxes F B F B F f = f B F A F A F A F : C ÝÑ D A window on the blue C inside the ambiant D. 49

51 Functorial equalities F C F C F A F A g C g C A F B B = F B = f f B F A F A F A F A F A F A F A 50

52 Lax monoidal functor A lax monoidal functor is a functor F : C ÝÑ D equipped with morphisms m ra,bs : FA b FB ÝÑ FpA b Bq m r s : I ÝÑ FI satisfying a series of coherence relations. A strong monoidal functor is lax monoidal with invertible coercions. 51

53 The purpose of coercions F (A 1 A 2 A 3 ) F I A 1 A 2 A 3 I F A 1 A 2 A 3 F F A 1 F A 2 F A 3 m ra1,a 2,A 3 s m r s 52

54 Lax monoidal functor A lax monoidal functor is a box with many inputs - one output. F B f B F A 1 A k F A 1 F A k Fp f q m ra1,,a k s : FA 1 b b FA k ÝÑ FB 53

55 Functorial equalities (on lax functors) F C F C g C g C A 1 A k F B B = F B B f f F A 1 A i A j A k F A i A j F A 1 F A i F A j F A k F A 1 F A i F A j F A k 54

56 Strong monoidal functors A strong monoidal functor is a box with many inputs - many outputs 55

57 Functorial equalities (on strong functors) F C1 F C k F C 1 F C k C 1 C k C 1 C k g g B 1 B j = F F B 1 F B j f f F A 1 A i F A 1 A i F A 1 F A i F A 1 F A i 56

58 Functorial equalities (on strong functors) F B 1 F B k F D 1 F D l F B 1 F B k F D 1 F D l B 1 f B j D 1 g D l = B 1 f B j D 1 g D l F A 1 A i C 1 C k F A 1 A i F C 1 C k F A 1 F A i F C 1 F C k F A 1 F A i F C 1 F C k 57

59 Natural transformations A natural transformation θ : F ÝÑ G : C ÝÑ D satisfies the pictorial equality: GB GB θ F B B B f f F A = G A GA F A θ F A 58

60 Monoidal natural transformations A monoidal natural transformation θ : F ÝÑ G : C ÝÑ D satisfies the pictorial equality: GB GB θ F B B B f f F A 1 A k = G A 1 A k GA 1 GA k θ θ F A 1 F A k F A 1 F A k 59

61 The algebraic nature of knots Functorial knot theory in ribbon categories 60

62 Braided categories A monoidal category equipped with braid maps A b B γ A,B ÝÑ B b A B b A γ 1 A,B ÝÑ A b B B A A B A B B A 61

63 Coherence diagram for braids A b pb b Cq γ pb b Cq b A α α pa b Bq b C B b pc b Aq γbc pb b Aq b C α B b pa b Cq Bbγ 62

64 Topological deformation in string diagrams B C A B C A = A B C A B C 63

65 Coherence diagram for braids α 1 pa b Bq b C γ C b pa b Bq α 1 A b pb b Cq pc b Aq b B Abγ A b pc b Bq α 1 pa b Cq b B γbb 64

66 Topological deformation in string diagrams C A B C A B = A B C A B C 65

67 Duality A duality A % B is a pair of morphisms I η ÝÑ A b B B b A ɛ ÝÑ I A B A B 66

68 Duality satisfying the two zig-zag equalities: A A B B = = A A B B In that case, A is called a left dual of B. 67

69 Ribbon category A braided category in which every object A has a right dual A, satisfying: A A A A A A = = A A A A A A 68

70 Knot invariants Every ribbon category D induces a knot invariant free-ribbonpc q r s D C The free ribbon category is a category of framed tangles 69

71 Jones polynomial invariant 2 1 y2 x 2 x 4 x 2 2x 2 x 4 x 2 y 2 A compositional semantics of knots 70

72 The topological nature of negation At the interface between topology, algebra and logic 71

73 Cartesian closed categories A cartesian category C is closed when there exists a functor ñ : C op C ÝÑ C and a natural bijection ϕ A,B,C : C pa B, Cq C pa, B ñ Cq A B C A B C 72

74 The free cartesian closed category The objects of the category free-ccc(c ) are the formulas A, B :: X A B A ñ B 1 where X is an object of the category C. The morphisms are the simply-typed λ-terms, modulo βη-conversion. 73

75 The simply-typed λ-calculus Variable Abstraction Application Weakening Contraction Permutation x :X $ x :X Γ, x :A $ P :B Γ $ λx.p :A ñ B Γ $ P :A ñ B $ Q :A Γ, $ PQ :B Γ $ P :B Γ, x :A $ P :B Γ, x :A, y :A $ P :B Γ, z :A $ Prx, y Ð zs :B Γ, x :A, y :B, $ P :C Γ, y :B, x :A, $ P :C 74

76 Proof invariants Every ccc D induces a proof invariant r s modulo execution. free-cccpc q r s D C Hence the prejudice that proof theory is intrinsically syntactical... 75

77 Dialogue categories A symmetric monoidal category C equipped with a functor and a natural bijection : C op ÝÑ C ϕ A,B,C : C pa b B, Cq C pa, p B b C q q A B C A B C 76

78 The free dialogue category The objects of the category free-dialogue(c ) are dialogue games constructed by the grammar A, B :: X A b 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... 77

79 A presentation of logic by generators and relations Negation defines a pair of adjoint functors L C K C op R witnessed by the series of bijection: C pa, Bq C pb, Aq C op p A, Bq 78

80 The 2-dimensional topology of adjunctions The unit and counit of the adjunction L % R are depicted as η : Id ÝÑ R L ε : L R ÝÑ Id η R L L R ε Opponent move = functor R Proponent move = functor L 79

81 A typical proof R L R R L R L R L L Reveals the algebraic nature of game semantics 80

82 A purely diagrammatic cut elimination R L 81

83 The 2-dimensional dynamic of adjunction L η ε L = L L R ε η R = R R Recovers the usual way to compose strategies in game semantics 82

84 Interesting fact There are just as many canonical proofs 2p hkkikkj hkkikkj R A $ 2q R A as there are maps rps ÝÑ rqs between the ordinals rps t0 1 p 1u and rqs. This fragment of logic has the same combinatorics as simplices. 83

85 The two generators of a monad Every increasing function is composite of faces and degeneracies: η : r0s $ r1s µ : r2s $ r1s Similarly, every proof is composite of the two generators: η : A $ A µ : A $ A The unit and multiplication of the double negation monad 84

86 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 85

87 The two generators in string diagrams The unit and multiplication of the monad R L are depicted as η : Id ÝÑ R L µ : R L R L ÝÑ R L R R η R L L R µ L L 86

88 Tensor and negation An atomist approach to proof theory 87

89 Four primitive components of logic r1s the negation r2s the linear conjunction b r3s the repetition modality! r4s the existential quantification D Logic = Data Structure + Duality 88

90 Tensor vs. negation A well-known fact: the continuation monad is strong p Aq b B ÝÑ pa b Bq The starting point of the algebraic theory of side effects 89

91 Tensor vs. negation Proofs are generated by a parametric strength κ X : px b Aq b B ÝÑ px b pa b Bqq which generalizes the usual notion of strong monad : κ : A b B ÝÑ pa b Bq 90

92 Proofs as 3-dimensional string diagrams The left-to-right proof of the sequent is depicted as A b B $ pa b Bq κ + κ + R ε L R L A R L B A B 91

93 Tensor vs. negation : conjunctive strength B R A 2 L A 1 κ ÝÑ B R L A 1 A 2 Linear distributivity in a continuation framework 92

94 Tensor vs. negation : disjunctive strength A L R B 1 B 2 κ ÝÑ A L B 2 R B 1 Linear distributivity in a continuation framework 93

95 A factorization theorem The four proofs η, ɛ, κ and κ generate every proof of the logic. Moreover, every such proof X ɛ ÝÑ κ ÝÑ ɛ ÝÑ ɛ ÝÑ η ÝÑ η ÝÑ κ ÝÑ ɛ ÝÑ η ÝÑ ɛ ÝÑ κ ÝÑ η ÝÑ η ÝÑ Z factors uniquely as X ɛ ÝÑÝÑ κ ÝÑÝÑ ÝÑÝÑ ÝÑÝÑ κ Z η Corollary: two proofs are equal iff they are equal as strategies 94

96 Evaluation of a strategy against a counter-strategy true η κ A κ B f actorisation κ X ε Y η Z κ C ε RLptrueq the formula X is -free and the formula Z is -free hence, the formula Y is both -free and -free hence, the formula Y it is a play! 95

97 Define a distributivity law Categorical combinatorics (Russ Harmer, Martin Hyland, PAM)!? λ ÝÑ?! 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 ÝÑ? B Big step instead of small step 96

98 Multi-threaded strategies (Samuel Mimram, PAM) κr κ L R L R L Additional hypothesis that negation defines a monoidal functor 97

99 Revisiting the negative translation A rational reconstruction of linear logic 98

100 The algebraic point of view (in the style of Boole) The negated elements of a Heyting algebra form a Boolean algebra. 99

101 The algebraic point of view (in the style of Frege) A double negation monad is commutative iff it is idempotent. This amounts to the following diagrammatic equations: R R R R L L L L L L = L L R R = R R R R R R L L L L In that case, the negated elements form a -autonomous category. 100

102 Tensor logic tensor logic = a logic of tensor and negation = linear logic without A A = the very essence of polarized logic Offers a synthesis of linear logic, games and continuations Research program: recast every aspect of linear logic in this setting 101

103 A lax monoidal structure Typically, the family of n-ary connectives pa 1 A n q : p A 1 b b A k q : R p LA 1 LA k q is still associative, but all together, and in the lax sense. 102

104 A general phenomenon of adjunctions Given a 2-monad T and an adjunction L A K B every lax T-algebraic structure TB induces a lax T-algebraic structure R b ÝÑ B TA TL ÝÑ TB b ÝÑ B R ÝÑ A 103

105 A general phenomenon of adjunctions Consequently, every adjunction with a monoidal category B L A K B R induces a lax action of B on the category A B A B L ÝÑ B B b B ÝÑ B R ÝÑ A Enscopes the double negation monad and the arrow. 104

106 Distributivity laws seen as lax bimodules The distributivity law RpA LBq C κ ÝÑ RpA LpB Cqq may be seen as a lax notion of bimodule: pa Bq C ÝÑ A pb Cq A useful extension of the notion of strong monad (case A ) 105

107 Complementary objects in dialogue categories An involutive duality in continuations 106

108 Linearly distributive categories A category C equipped with a symmetric monoidal structure and true, a symmetric monoidal structure and false with a distributivity law δ L A,B,C δ R A,B,C : A pb Cq ÝÑ pa Bq C : pa Bq C ÝÑ A pb Cq 107

109 Pentagonal diagram ppa Bq Cq D α pa Bq pc Dq δ R A pb pc Dqq δ R D Aα pa pb Cqq D δ R A ppb Cq Dq 108

110 Pentagonal diagram ppa Bq Cq D δ R pa Bq pc Dq α A pb pc Dqq α D Aδ R pa pb Cqq D δ R A ppb Cq Dq 109

111 Pentagonal coherence diagram ppa Bq Cq D α pa Bq pc Dq δ L A pb pc Dqq δ L D Abα pa pb Cqq D δ L A ppb Cq Dq 110

112 Pentagonal coherence diagram pa Bq pc Dq δ L ppa Bq Cq D α A pb pc Dqq α D Abδ L pa pb Cqq D δ L A ppb Cq Dq 111

113 Pentagonal diagram ppa Bq Cq D δ L pa Bq pc Dq δ R A pb pc Dqq δ R D Aδ L pa pb Cqq D α A ppb Cq Dq 112

114 Pentagonal diagram ppa Bq Cq D δ R pa Bq pc Dq δ L A pb pc Dqq δ L D Abδ R pa pb Cqq D α A ppb Cq Dq 113

115 Triangular diagrams true pa Bq λ δ L A B λ B ptrue Aq B pa Bq b true ρ δ R A B Aρ A pb b trueq 114

116 Triangular diagrams A pb falseq Abρ δ L A B ρ pa Bq false pfalse Aq B λ B δ R λ A B false pa Bq 115

117 Right complementary A right complementary of the object A is an object B together with ax : true ÝÑ A A cut : A A ÝÑ false 116

118 Complementary pair making the diagrams A true Aax A pa Aq true A axa pa Aq A δ L δ R ρ pa A q A λ A pa A q A λ cuta false A A ρ A cut A false commute. 117

119 Fact A -autonomous category is the same thing as a linearly distributive category with a complementary object A for every object A. Established by Blute, Cockett and Seely 118

120 Linearly distributive adjunction A specific notion of adjunction L A K B R between two monoidal categories pa,, trueq and pb,, falseq. 119

121 Right complementary A right complementary of the object A is an object B together with ax : true ÝÑ RpA LAq cut : LpA RA q ÝÑ false 120

122 Theorem A dialogue category is the same thing as a linearly distributive adjunction with a complementary functor pa ÞÑ A q : A ÝÑ B 121

123 An amusing illustration Free monoids in games 122

124 Monoid A monoid is a game A equipped with two strategies 1 u m A A b A such that the diagrams A b A b A mba A b A 1 b A uba A b A Abu A b 1 Abm m λ m ρ A b A m A A A A commute. 123

125 Free commutative monoids Tensor algebra: Symmetric algebra: SA TA à n à n A bn A bn { Games: SA» n A bn Computed as an inductive limit in the category G of games 124

126 Computing the free commutative monoid Step 1. Compute the quotient A n A bn symmetry symmetry A bn quotient A n Step 2. The free commutative monoid SA is the inductive limit A 0 A 1 A 2 A n A n 1 The game SA is the game A where Proponent may backtrack for ever 125

127 Revisiting the drinker s formula The expressive power of resource modalities 126

128 Four primitive components of logic r1s the negation r2s the linear conjunction b r3s the repetition modality! r4s existential quantification D A theory of witnesses what remains of intuitionism

129 The classical formula translated as a game! Dy b!! Apyq Dx Apxq The repetition modality! gives an explicit account of the fact that Proponent is allowed to change her witness. 128

130 The classical formula translated as a game! Dy b!! Apyq Dx Apxq The formula explicates the exact rules of the game (and space of logical interaction) 129

131 The formal proof in tensor logic Axiom Apx 0 q $ Apx 0 q Dereliction!Apx 0 q $ Apx 0 q Left!Apx 0 q, Apx 0 q $ Weakening!Apx 0 q,! Dx Apxqq, Apx 0 q $ Right Apx 0 q $ p!apx 0 q b! Dx Apxqq Right D Apx 0 q $ Dy p!apyq b! Dx Apxqq Left Dy p!apyq b! Dx Apxqq, Apx 0 q $ Left D Dy p!apyq b! Dx Apxqq, Dx Apxq $ Dereliction Dy p!apyq b! Dx Apxqq,! Dx Apxq $ Weakening Dy p!apyq b! Dx Apxqq,!Apy 0 q,! Dx Apxq $ Right Dy p!apyq b! Dx Apxqq $ p!apy 0 q b! Dx Apxqq Right D Dy p!apyq b! Dx Apxqq $ Dy p!apyq b! Dx Apxqq Left Dy p!apyq b! Dx Apxqq, Dy p!apyq b! Dx Apxqq $ Contraction! Dy p!apyq b! Dx Apxqq $ Right $! Dy p!apyq b! Dx Apxqq 130

132 The intuitionistic formula translated as a game Dy b!! Apyq Dx Apxq This formula is not valid, because Proponent has to choose the witness y immediately and cannot change its mind later. 131

133 Conclusion Logic = Data Structure + Duality This point of view is accessible thanks to algebra 132