Univalent multisets. V through the eyes of the identity type. Håkon Robbestad Gylterud. August 2014

Transcription

1 Univalent multisets V through the eyes of the identity type Håkon Robbestad Gylterud August 2014 Håkon Robbestad Gylterud Univalent multisets Stockholm University 1 / 25

2 Outline of the talk 1 Present common intuition about multisets Håkon Robbestad Gylterud Univalent multisets Stockholm University 2 / 25

3 Outline of the talk 1 Present common intuition about multisets 2 Give a model of multisets in type theory Håkon Robbestad Gylterud Univalent multisets Stockholm University 2 / 25

4 Outline of the talk 1 Present common intuition about multisets 2 Give a model of multisets in type theory 3 A result about W-types Håkon Robbestad Gylterud Univalent multisets Stockholm University 2 / 25

5 Outline of the talk 1 Present common intuition about multisets 2 Give a model of multisets in type theory 3 A result about W-types 4 Apply this result to the model Håkon Robbestad Gylterud Univalent multisets Stockholm University 2 / 25

6 Outline of the talk 1 Present common intuition about multisets 2 Give a model of multisets in type theory 3 A result about W-types 4 Apply this result to the model 5 Outline of current and future work Håkon Robbestad Gylterud Univalent multisets Stockholm University 2 / 25

7 Mathematical context In this talk... We work in Martin-Löf type theory. Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

8 Mathematical context In this talk... We work in Martin-Löf type theory. The notion of set is that of a type in type theory Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

9 Mathematical context In this talk... We work in Martin-Löf type theory. The notion of set is that of a type in type theory (or rather element in the type Set in the logical framework). Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

10 Mathematical context In this talk... We work in Martin-Löf type theory. The notion of set is that of a type in type theory (or rather element in the type Set in the logical framework). We will use the term iterative set to refer to the notion of set which is studied in Set Theory. Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

11 Mathematical context In this talk... We work in Martin-Löf type theory. The notion of set is that of a type in type theory (or rather element in the type Set in the logical framework). We will use the term iterative set to refer to the notion of set which is studied in Set Theory. Juxtaposition denotes (left associative) function application. That is, f x denotes f applied to x, and f x y := (f x) y Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

12 Mathematical context In this talk... We work in Martin-Löf type theory. The notion of set is that of a type in type theory (or rather element in the type Set in the logical framework). We will use the term iterative set to refer to the notion of set which is studied in Set Theory. Juxtaposition denotes (left associative) function application. That is, f x denotes f applied to x, and f x y := (f x) y The technical parts are formalized in Agda. Håkon Robbestad Gylterud Univalent multisets Stockholm University 3 / 25

13 What are multisets? Håkon Robbestad Gylterud Univalent multisets Stockholm University 4 / 25

14 What are multisets? Our intuition is that multisets... Consists of elements. Håkon Robbestad Gylterud Univalent multisets Stockholm University 4 / 25

15 What are multisets? Our intuition is that multisets... Consists of elements. Elements are considered to be unordered. Håkon Robbestad Gylterud Univalent multisets Stockholm University 4 / 25

16 What are multisets? Our intuition is that multisets... Consists of elements. Elements are considered to be unordered. For each element the number of occurences is relevant. Håkon Robbestad Gylterud Univalent multisets Stockholm University 4 / 25

17 What are multisets? Our intuition is that multisets... Consists of elements. Elements are considered to be unordered. For each element the number of occurences is relevant. The first two points are applies to sets as well. The third point distinguishes the two notions. Håkon Robbestad Gylterud Univalent multisets Stockholm University 4 / 25

18 Examples The roots of a polynomial is a multiset if we count multiplicity. x 3 2x 2 + x has roots {0, 1, 1}. Håkon Robbestad Gylterud Univalent multisets Stockholm University 5 / 25

19 Examples The roots of a polynomial is a multiset if we count multiplicity. x 3 2x 2 + x has roots {0, 1, 1}. Sequent calculus. A, A φ Håkon Robbestad Gylterud Univalent multisets Stockholm University 5 / 25

20 Examples The roots of a polynomial is a multiset if we count multiplicity. x 3 2x 2 + x has roots {0, 1, 1}. Sequent calculus. A, A φ Bags in computer science. Håkon Robbestad Gylterud Univalent multisets Stockholm University 5 / 25

21 Related work Blizzard (1989), develops a classical, two sorted, first order theory of multisets which, when restricted to sets, is equivalent to ZFC. Håkon Robbestad Gylterud Univalent multisets Stockholm University 6 / 25

22 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

23 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Instead of a ternary relation, we will keep the -relation binary and invoke the propositions-as-sets attitude of Martin-Löf type theory. Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

24 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Instead of a ternary relation, we will keep the -relation binary and invoke the propositions-as-sets attitude of Martin-Löf type theory. Our notation x y denotes the set of occurences of x in y. Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

25 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Instead of a ternary relation, we will keep the -relation binary and invoke the propositions-as-sets attitude of Martin-Löf type theory. Our notation x y denotes the set of occurences of x in y. Example (1 {0, 0, 1, 1, 1}) = 3 Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

26 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Instead of a ternary relation, we will keep the -relation binary and invoke the propositions-as-sets attitude of Martin-Löf type theory. Our notation x y denotes the set of occurences of x in y. Example (1 {0, 0, 1, 1, 1}) = 3 ( 2 Roots(x 3 2x 2 + x) ) =. Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

27 Elementhood in multisets Blizzard and others use the notation: Notation (Blizzard) x n y denotes that x occurs in y exactly n times. Instead of a ternary relation, we will keep the -relation binary and invoke the propositions-as-sets attitude of Martin-Löf type theory. Our notation x y denotes the set of occurences of x in y. Example (1 {0, 0, 1, 1, 1}) = 3 ( 2 Roots(x 3 2x 2 + x) ) =. (3 {3, 3, 3, }) = N. Håkon Robbestad Gylterud Univalent multisets Stockholm University 7 / 25

28 Exensionality In set theory Given two iterative sets x and y, if for each z we have that z x iff z y, then x and y are equal. Håkon Robbestad Gylterud Univalent multisets Stockholm University 8 / 25

29 Exensionality In set theory Given two iterative sets x and y, if for each z we have that z x iff z y, then x and y are equal. The principle of extensionality for multisets Two multisets x and y are considered equal iff for any z, the number of occurences of z in x and the number of occurences of z in y are in bijective correspondence (in our symbolism: (z x) = (z y)). Håkon Robbestad Gylterud Univalent multisets Stockholm University 8 / 25

30 Classical vs Constructive Classically, one can model a multiset as a set X, called the domain, and a function, e : X N. Or if extended into the infinite, a function e : X Card. Håkon Robbestad Gylterud Univalent multisets Stockholm University 9 / 25

31 Classical vs Constructive Classically, one can model a multiset as a set X, called the domain, and a function, e : X N. Or if extended into the infinite, a function e : X Card. Constructively, there might not be many interesting functions into N, and the notion of cardinals is problematic. Håkon Robbestad Gylterud Univalent multisets Stockholm University 9 / 25

32 Classical vs Constructive Classically, one can model a multiset as a set X, called the domain, and a function, e : X N. Or if extended into the infinite, a function e : X Card. Constructively, there might not be many interesting functions into N, and the notion of cardinals is problematic. A solution is to consider a multiset as a family. m : X Set, Håkon Robbestad Gylterud Univalent multisets Stockholm University 9 / 25

33 Classical vs Constructive Classically, one can model a multiset as a set X, called the domain, and a function, e : X N. Or if extended into the infinite, a function e : X Card. Constructively, there might not be many interesting functions into N, and the notion of cardinals is problematic. A solution is to consider a multiset as a family. m : X Set, or m : I X. Håkon Robbestad Gylterud Univalent multisets Stockholm University 9 / 25

34 Iterative multisets Is it possible to parallell the construction of iterative sets? Håkon Robbestad Gylterud Univalent multisets Stockholm University 10 / 25

35 Iterative multisets Is it possible to parallell the construction of iterative sets? For iterative sets, we consider the totality V, consisting of sets where all elements of the sets, them selves are sets. Håkon Robbestad Gylterud Univalent multisets Stockholm University 10 / 25

36 Iterative multisets Is it possible to parallell the construction of iterative sets? For iterative sets, we consider the totality V, consisting of sets where all elements of the sets, them selves are sets. One may then wish for a totality M, consistsing of multisets of multisets, all with with domain M it self. Håkon Robbestad Gylterud Univalent multisets Stockholm University 10 / 25

37 Trees It is well known that (wellfounded) trees can serve as models of (wellfounded) iterative sets. Håkon Robbestad Gylterud Univalent multisets Stockholm University 11 / 25

38 Trees It is well known that (wellfounded) trees can serve as models of (wellfounded) iterative sets. Example The iterative set {{{ }, }, { }} is represented by Håkon Robbestad Gylterud Univalent multisets Stockholm University 11 / 25

39 Trees It is well known that (wellfounded) trees can serve as models of (wellfounded) iterative sets. Example The iterative set {{{ }, }, { }} is represented by but also by Håkon Robbestad Gylterud Univalent multisets Stockholm University 11 / 25

40 Trees It is well known that (wellfounded) trees can serve as models of (wellfounded) iterative sets. Example The iterative set {{{ }, }, { }} is represented by but also by For iterative multisets, we want to keep these two distinct. Håkon Robbestad Gylterud Univalent multisets Stockholm University 11 / 25

41 The W-type Definition Given a family A : Set, B : A Set, the set of all well founded trees with branchings in this family, denoted W a:a Ba is inductively generated by the rule: Håkon Robbestad Gylterud Univalent multisets Stockholm University 12 / 25

42 The W-type Definition Given a family A : Set, B : A Set, the set of all well founded trees with branchings in this family, denoted W a:a Ba is inductively generated by the rule: For each a : A and f : Ba W a:a Ba, there is a unique element (sup a f ) : W a:a Ba. Håkon Robbestad Gylterud Univalent multisets Stockholm University 12 / 25

43 Aczel s model of iterative sets in type theory Håkon Robbestad Gylterud Univalent multisets Stockholm University 13 / 25

44 Aczel s model of iterative sets in type theory Definition (Aczel) Given en a universe U : Set with decoding familty T : U Set, define a setoid (V, = V ) by Håkon Robbestad Gylterud Univalent multisets Stockholm University 13 / 25

45 Aczel s model of iterative sets in type theory Definition (Aczel) Given en a universe U : Set with decoding familty T : U Set, define a setoid (V, = V ) by V : Set V := W a:u Ta Håkon Robbestad Gylterud Univalent multisets Stockholm University 13 / 25

46 Aczel s model of iterative sets in type theory Definition (Aczel) Given en a universe U : Set with decoding familty T : U Set, define a setoid (V, = V ) by V : Set V := W a:u Ta = V : V V Set (sup a f ) = V (sup b g) := i) = V (g j) i:ta j:tb(f (f i) = V (g j) j:tb i:ta Håkon Robbestad Gylterud Univalent multisets Stockholm University 13 / 25

47 Aczel s model of iterative sets in type theory Lemma = V is equivalent to = V : V V Set (sup a f ) = V (sup b g) := (f x) = V (g (α x)) (f (βy)) = V α:ta Tb x:ta β:tb Ta y:tb (g y) Håkon Robbestad Gylterud Univalent multisets Stockholm University 14 / 25

48 Aczel s model of iterative sets in type theory Lemma = V is equivalent to = V : V V Set (sup a f ) = V (sup b g) := (f x) = V (g (α x)) (f (βy)) = V α:ta Tb x:ta β:tb Ta y:tb (g y) Proof. W-induction on V and apply the (type theoretical) axiom of choice twice. Håkon Robbestad Gylterud Univalent multisets Stockholm University 14 / 25

49 Aczel s model of iterative sets in type theory Diagramatically, (sup a f ) is equal, according to = V, to (sup b g) if the diagrams Ta α Tb Ta β Tb f g f g V V commutes up to = V. Håkon Robbestad Gylterud Univalent multisets Stockholm University 15 / 25

50 Aczel s model of iterative sets in type theory Diagramatically, (sup a f ) is equal, according to = V, to (sup b g) if the diagrams Ta α Tb Ta β Tb f g f g V V commutes up to = V. The natural change to make is to require that α and β form an equivalence of types. Håkon Robbestad Gylterud Univalent multisets Stockholm University 15 / 25

51 The model Definition M : Set M := W a:u Ta = M : M M Set (sup a f ) = M (sup b g) := (f x) = M (g (α x)) α:ta =Tb x:ta Håkon Robbestad Gylterud Univalent multisets Stockholm University 16 / 25

52 The model Definition M : Set M := W a:u Ta = M : M M Set (sup a f ) = M (sup b g) := (f x) = M (g (α x)) α:ta =Tb x:ta Ta α = Tb f g V Håkon Robbestad Gylterud Univalent multisets Stockholm University 16 / 25

53 The model Definition Elementhood between multisets is defined by : M M Set x (sup a f ) := i:ta (f a = M x) Håkon Robbestad Gylterud Univalent multisets Stockholm University 17 / 25

54 The identity type and Equivalence In Martin-Löf type theory, every A : Set is equipped with a type Id A : A A Set, which is inductively generated by If a : A then (refl a) : Id A a a. Håkon Robbestad Gylterud Univalent multisets Stockholm University 18 / 25

55 The identity type and Equivalence In Martin-Löf type theory, every A : Set is equipped with a type Id A : A A Set, which is inductively generated by If a : A then (refl a) : Id A a a. This induces a notion of extensional equality on functions, and a notion of equivalence between types, which are essential in Homotopy Type Theory. Håkon Robbestad Gylterud Univalent multisets Stockholm University 18 / 25

56 The identity type and Equivalence In Martin-Löf type theory, every A : Set is equipped with a type Id A : A A Set, which is inductively generated by If a : A then (refl a) : Id A a a. This induces a notion of extensional equality on functions, and a notion of equivalence between types, which are essential in Homotopy Type Theory. If A, B : Set we denote by A = B the type of equivalences from A to B. And if f, g : A B, we denote by f g the type of extensional equalities (homotopies) from f to g. Håkon Robbestad Gylterud Univalent multisets Stockholm University 18 / 25

57 A result on the identity type of W types Håkon Robbestad Gylterud Univalent multisets Stockholm University 19 / 25

58 A result on the identity type of W types Lemma For any A : Set and B : A Set, and all (sup a f ), (sup b g) : W A B, there is an equivalence Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Håkon Robbestad Gylterud Univalent multisets Stockholm University 19 / 25

59 A result on the identity type of W types Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Proof. There is a map going from left to right by induction on Id WA B. That is, for each (sup a f ) the element (refl a, refl f ) works. Call this map φ. Håkon Robbestad Gylterud Univalent multisets Stockholm University 20 / 25

60 A result on the identity type of W types Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Proof. There is a map going from left to right by induction on Id WA B. That is, for each (sup a f ) the element (refl a, refl f ) works. Call this map φ. To show that this map is an equivalence, we need to show that the inverse images of each element is a singleton. Håkon Robbestad Gylterud Univalent multisets Stockholm University 20 / 25

61 A result on the identity type of W types Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Proof. There is a map going from left to right by induction on Id WA B. That is, for each (sup a f ) the element (refl a, refl f ) works. Call this map φ. To show that this map is an equivalence, we need to show that the inverse images of each element is a singleton. So assume that p : α:id A a b Id f (Bα g). Håkon Robbestad Gylterud Univalent multisets Stockholm University 20 / 25

62 A result on the identity type of W types Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Proof. There is a map going from left to right by induction on Id WA B. That is, for each (sup a f ) the element (refl a, refl f ) works. Call this map φ. To show that this map is an equivalence, we need to show that the inverse images of each element is a singleton. So assume that p : α:id A a b Id f (Bα g). By induction (on the Σ-type and the two Id-types), it is enough to consider the case where p (refl a, refl f ). Håkon Robbestad Gylterud Univalent multisets Stockholm University 20 / 25

63 A result on the identity type of W types Id WA B (sup a f ) (sup b g) = α:id A a b Id f (Bα g) Proof. There is a map going from left to right by induction on Id WA B. That is, for each (sup a f ) the element (refl a, refl f ) works. Call this map φ. To show that this map is an equivalence, we need to show that the inverse images of each element is a singleton. So assume that p : α:id A a b Id f (Bα g). By induction (on the Σ-type and the two Id-types), it is enough to consider the case where p (refl a, refl f ). We check that φ refl (sup a f ) p, by the above definiton. And by induction on Id, we can show that every element in the inverse image of p is equal to refl (sup a f ). Håkon Robbestad Gylterud Univalent multisets Stockholm University 20 / 25

64 The univalence axiom Definition The axiom of extensionality states that for each f, g : A B, the obvious function is an equivalence of types. Id f g f g Definition The axiom of univalence for a universe U : Set with decoding family T : U Set, states that for each a, b : U, the obvious function is an equivalence of types. Id a b Ta = Tb Håkon Robbestad Gylterud Univalent multisets Stockholm University 21 / 25

65 Id is equivalent to = M Theorem The univalence axiom implies that for any m, m : M we have that Id m m = m =M m Håkon Robbestad Gylterud Univalent multisets Stockholm University 22 / 25

66 Proof. By W-induction. Assume a, b : U and f : Ta M and g : Tb M. Then (sup a f ) = M (sup b g) Inducion hypotheis = Definition of α:ta =Tb x:ta α:ta =Tb x:ta α:ta =Tb Extensionality = Univalence α:ta =Tb = α:a=b (fx) = M (g(αx)) Id (f x) (g(αx)) f g α Id f (g α) Id f (g T α) Previous lemma = Id (sup a f ) (sup b g) Håkon Robbestad Gylterud Univalent multisets Stockholm University 23 / 25

67 Axiomatisation of multiset theory Håkon Robbestad Gylterud Univalent multisets Stockholm University 24 / 25

68 Axiomatisation of multiset theory Extensionality xy x = y = z (z x = z y) Håkon Robbestad Gylterud Univalent multisets Stockholm University 24 / 25

69 Axiomatisation of multiset theory Extensionality xy x = y = z (z x = z y) x,y:m (Id x y) = z:m (z x = z y) Håkon Robbestad Gylterud Univalent multisets Stockholm University 24 / 25

70 Axiomatisation of multiset theory Extensionality xy x = y = z (z x = z y) x,y:m (Id x y) = z:m (z x = z y) Pairing xy u z z u = (z = x z = y)) Håkon Robbestad Gylterud Univalent multisets Stockholm University 24 / 25

71 Axiomatisation of multiset theory Extensionality xy x = y = z (z x = z y) x,y:m (Id x y) = z:m (z x = z y) Pairing xy u z z u = (z = x z = y)) Restricted separation x u z z u = (z x φ(z)) Håkon Robbestad Gylterud Univalent multisets Stockholm University 24 / 25

72 Conclusion This is work in progress, but the result on the identity type of M indicates that it is a good model of multisets in type theory. The current project is to give this more substance to this claim by giving an axiomatisation of iterative multiset theory. Håkon Robbestad Gylterud Univalent multisets Stockholm University 25 / 25