a brief introduction to (dependent) type theory

Transcription

1 a brief introduction to (dependent) type theory Cory Knapp January 14, 2015 University of Birmingham

2 what is type theory?

3 What is type theory? formal language of terms with types x : A x has type A

4 What is type theory? formal language of terms with types x : A x has type A Idealized (functional) language type datatype. x : A x is a program (or value) of type A. 2

5 What is type theory? formal language of terms with types x : A x has type A Idealized (functional) language type datatype. x : A x is a program (or value) of type A. (+) :: Int -> Int -> Int map (+1) [1,2,3] :: [Int] 2

6 What is type theory? formal language of terms with types x : A x has type A 2

7 What is type theory? formal language of terms with types x : A x has type A Logic with book-keeping x : A A is a proposition and x is a proof of A. 2

8 What is type theory? formal language of terms with types x : A x has type A Logic with book-keeping x : A A is a proposition and x is a proof of A. λ(x : A). x : A A Given a proof x of A, we can find a proof x of A 2

9 What is type theory? formal language of terms with types x : A x has type A 2

10 What is type theory? formal language of terms with types x : A x has type A Alternative to set theory x : A A is a set and x A 2

11 What is type theory? formal language of terms with types x : A x has type A Alternative to set theory x : A A is a set and x A 0 : N 0 N 2

12 What is type theory? formal language of terms with types x : A x has type A 2

13 Contexts Everything happens in a context environment implicitly bound identifiers. 3

14 Contexts Everything happens in a context environment implicitly bound identifiers. Object foo(int x, int y){ //... int z = x + y; //... } x : int, y : int z : int 3

15 Contexts Everything happens in a context environment implicitly bound identifiers. Object foo(int x, int y){ //... int z = x + y; //... } x : int, y : int z : int 3

16 Contexts Everything happens in a context environment implicitly bound identifiers. Object foo(int x, int y){ //... int z = x + y; //... } x : int, y : int z : int 3

17 Rules and judgments Judgment = Result 4

18 Rules and judgments Judgment = Result Typing judgments 4

19 Rules and judgments Judgment = Result Typing judgments Γ A : Type Γ x : A 4

20 Rules and judgments Judgment = Result Typing judgments Γ A : Type Γ x : A Equality judgments 4

21 Rules and judgments Judgment = Result Typing judgments Γ A : Type Γ x : A Equality judgments Γ A B : Type Γ x y : A 4

22 Rules and judgments Judgment = Result Rule = Step Typing judgments Γ A : Type Γ x : A Equality judgments Γ A B : Type Γ x y : A 4

23 Rules and judgments Judgment = Result Rule = Step Typing judgments Γ A : Type Γ x : A Equality judgments Γ A B : Type Γ x y : A formally: J 1... J n RULE J 4

24 Rules and judgments Judgment = Result Rule = Step Typing judgments Γ A : Type Γ x : A Equality judgments Γ A B : Type Γ x y : A formally: conditional J 1... J n RULE J Γ b : bool Γ c 1 : C Γ c 2 : C Γ if b then c 1 else c 2 : C 4

25 Rules and judgments Judgment = Result Rule = Step Typing judgments Γ A : Type Γ x : A Equality judgments Γ A B : Type Γ x y : A formally: conditional J 1... J n RULE J Γ b : bool Γ c 1 : C Γ c 2 : C Γ if b then c 1 else c 2 : C Type isn t a type (but we can pretend) 4

26 simple types

27 Defining types Types are defined by rules Formation rules When does the type exist? 6

28 Defining types Types are defined by rules Formation rules When does the type exist? Γ A : Type Γ B : Type Γ A B : Type function type 6

29 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? 6

30 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? Γ B : Type Γ, x : A b : B Γ λx.b : A B lambda abstraction 6

31 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? Elimination rules What is the basic way to use a term? 6

32 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? Elimination rules What is the basic way to use a term? Γ m : A Γ f : A B Γ f(m) : B application 6

33 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? Elimination rules What is the basic way to use a term? Computation rules How do we reduce expressions? 6

34 Defining types Types are defined by rules Formation rules When does the type exist? Introduction rules What are the basic terms? Elimination rules What is the basic way to use a term? Computation rules How do we reduce expressions? Γ B : Type Γ, x : A b : B Γ m : A Γ (λx.b)(m) b[x/m] : B substitution 6

35 Inductive types 7

36 Inductive types Introduction rules Constructors 7

37 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching 7

38 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching data List = Cons Int List Empty sum :: List -> Int sum (Cons i l) = i + sum l sum Empty = 0 7

39 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching data List = Cons Int List Empty sum :: List -> Int sum (Cons i l) = i + sum l sum Empty = 0 7

40 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching data List = Cons Int List Empty sum :: List -> Int sum (Cons i l) = i + sum l sum Empty = 0 7

41 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching data List = Cons Int List Empty sum :: List -> Int sum (Cons i l) = i + sum l sum Empty = 0 7

42 Inductive types Introduction rules Constructors Elimination/Computation rules Pattern-matching data List = Cons Int List Empty sum :: List -> Int sum (Cons i l) = i + sum l sum Empty = 0 7

43 The simple types A B (a, b) Pair A B 8

44 The simple types A B (a, b) Pair A B A + B inla; inrb Inl A Inr B 8

45 The simple types A B (a, b) Pair A B A + B inla; inrb Inl A Inr B empty no constructors! 8

46 The simple types A B (a, b) Pair A B A + B inla; inrb Inl A Inr B empty no constructors! unit () 8

47 The simple types A B (a, b) Pair A B A + B inla; inrb Inl A Inr B empty no constructors! unit () others? Bool, Int, Nat, well-founded trees... 8

48 Interpretation Types Sets Propositions 9

49 Interpretation Types A B Sets A B Propositions A B 9

50 Interpretation Types A B A + B Sets A B A B Propositions A B A B 9

51 Interpretation Types A B A + B A B Sets A B A B B A Propositions A B A B B A 9

52 Interpretation Types A B A + B A B unit Sets A B A B B A { } Propositions A B A B B A 9

53 Interpretation Types A B A + B A B unit empty Sets A B A B B A { } Propositions A B A B B A 9

54 Interpretation Types A B A + B A B unit empty Sets A B A B B A { } Propositions A B A B B A What about quantifiers? 9

55 dependent types

56 Dependent types Γ, x : A B : Type 11

57 Dependent types Γ, x : A B : Type B depends on x : A 11

58 Dependent types Γ, x : A B : Type B depends on x : A Vectors of length n : nat. Days of month m : Months. 11

59 Dependent types Γ, x : A B : Type B depends on x : A Vectors of length n : nat. Days of month m : Months. λ-abstraction: Γ λx.b : A Type A type family 11

60 Dependent products dependent functions 12

61 Dependent products dependent functions Output type depends on input. 12

62 Dependent products dependent functions Output type depends on input. lastday(january) : daysof(january) lastday(april) : daysof(april) 12

63 Dependent products dependent functions Output type depends on input. lastday(january) : daysof(january) lastday(april) : daysof(april) x:a B dependent product 12

64 Dependent products dependent functions Output type depends on input. lastday(january) : daysof(january) lastday(april) : daysof(april) x:a B lastday : m:month daysof(m) dependent product 12

65 Dependent elimination Dependent functions are generalized functions generalize elimination 13

66 Dependent elimination Dependent functions are generalized functions generalize elimination C : A B Type f : p:a B C(p) f((a, b)) := something 13

67 Dependent elimination Dependent functions are generalized functions generalize elimination C : A B Type f : p:a B C(p) f((a, b)) := something Induction principle 13

68 Dependent sums dependent pair second coordinate depends on first 14

69 Dependent sums dependent pair second coordinate depends on first red-black tree: binary tree T with color assignment f : color(t) 14

70 Dependent sums dependent pair second coordinate depends on first red-black tree: binary tree T with color assignment f : color(t) (T, f) 14

71 Dependent sums dependent pair second coordinate depends on first red-black tree: binary tree T with color assignment f : color(t) (T, f) x:a B dependent sum 14

72 Dependent sums dependent pair second coordinate depends on first red-black tree: binary tree T with color assignment f : color(t) (T, f) x:a B RBTree := T:BinTree color(t) dependent sum 14

73 Interpretation Types Sets Propositions 15

74 Interpretation Types B : A Type Sets {B a a A} Propositions predicate B on A 15

75 Interpretation Types B : A Type x:a B(x) Sets {B a a A} x:a B(x) Propositions predicate B on A x A(B(x)) 15

76 Interpretation Types B : A Type x:a B(x) x:a B(x) Sets {B a a A} x:a B(x) x:a B(x) Propositions predicate B on A x A(B(x)) x A(B(x)) 15

77 Equality Propositions are types x = y is a proposition. Identity types 16

78 Equality Propositions are types x = y is a proposition. Identity types Inductive family = A : A A Type 16

79 Equality Propositions are types x = y is a proposition. Identity types Inductive family = A : A A Type Constructor refl a : a = A a 16

80 Equality Propositions are types x = y is a proposition. Identity types Inductive family = A : A A Type Constructor refl a : a = A a Eliminate by pattern-match on refl 16

81 Equality Propositions are types x = y is a proposition. Identity types Inductive family = A : A A Type Constructor refl a : a = A a Eliminate by pattern-match on refl sym : x,y:x(x = y y = x) sym x,x (refl x ) = refl x 16

82 Two notions of equality Is the same as =? 17

83 Two notions of equality Is the same as =? Extensional 17

84 Two notions of equality Is the same as =? Extensional Intensional 17

85 Two notions of equality Is the same as =? Extensional Too rigid Intensional 17

86 Two notions of equality Is the same as =? Extensional Too rigid Intensional Too flexible 17

87 Two notions of equality Is the same as =? Extensional Too rigid Intensional Too flexible Can we find a middle ground? 17

88 homotopy type theory

89 Homotopy interpretation Type theory: formal language for homotopy theory Voevdsky (2006); Awodey & Warren (2007) 19

90 Homotopy interpretation Type theory: formal language for homotopy theory Voevdsky (2006); Awodey & Warren (2007) types are spaces terms are points equalities are paths 19

91 Homotopy interpretation Type theory: formal language for homotopy theory Voevdsky (2006); Awodey & Warren (2007) types are spaces terms are points equalities are paths homotopy theory Use paths to characterize spaces Natural interpretations 19

92 univalence homotopy theory Equivalent spaces are identified A B if there is an invertible function A B 20

93 univalence homotopy theory Equivalent spaces are identified A B if there is an invertible function A B Univalence Axiom (A B) (A = Type B) 20

94 Higher Inductive types Paths are part of a space. 21

95 Higher Inductive types Paths are part of the definition of a space. 21

96 Higher Inductive types Paths are part of the definition of a space. Allow path constructors. 21

97 Higher Inductive types Paths are part of the definition of a space. Allow path constructors. Higher-inductive types 21

98 Higher Inductive types Paths are part of the definition of a space. Allow path constructors. Higher-inductive types Point constructors x : A 21

99 Higher Inductive types Paths are part of the definition of a space. Allow path constructors. Higher-inductive types Point constructors x : A path constructors p : x = A y 21

100 Higher Inductive types Paths are part of the definition of a space. Allow path constructors. Higher-inductive types Point constructors x : A path constructors p : x = A y Quotients, pushouts, suspensions, more. 21

101 More Information homotopytypetheory.org Blog, book, articles. github.com/hott Computer implementations Questions?