Inductive Types for Free

Transcription

1 Inductive Types for Free Representing Nested Inductive Types using W-types Michael Abbott (U. Leicester) Thorsten Altenkirch (U. Nottingham) Neil Ghani (U. Leicester) Inductive Types for Free p.1/22

2 Ideology Inductive Types for Free p.2/22

3 Ideology David Turner: Elementary Strong Functional Programming Inductive Types for Free p.2/22

4 Ideology David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Inductive Types for Free p.2/22

5 Ideology David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions. Inductive Types for Free p.2/22

6 Ideology David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions. Dependent types to avoid accidental partiality (e.g. hd). Inductive Types for Free p.2/22

7 Ideology David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions. Dependent types to avoid accidental partiality (e.g. hd). E.g.: Conor McBride s Epigram system. Inductive Types for Free p.2/22

8 Plan of the talk Inductive and coinductive types. Container types for dummies. Properties of container types. W-types are sufficent for inductive types. Further work and applications Related work Inductive Types for Free p.3/22

9 Inductive and coinductive types Widespread in functional programming (e.g. Haskell) Lam Int Lam Lam Lam Inductive Types for Free p.4/22

10 Inductive and coinductive types Widespread in functional programming (e.g. Haskell) Lam Int Lam Lam Lam Categorically: Initial algebra of a functor Lam Inductive Types for Free p.4/22

11 vs. In a total setting (terminal coalgebras): Inductive Types for Free p.5/22

12 vs. In a total setting (terminal coalgebras): finite lists Inductive Types for Free p.5/22

13 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists Inductive Types for Free p.5/22

14 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists Inductive Types for Free p.5/22

15 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists Inductive Types for Free p.5/22

16 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists Inductive Types for Free p.5/22

17 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists can be nested Inductive Types for Free p.5/22

18 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists can be nested Inductive Types for Free p.5/22

19 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists can be nested Inductive Types for Free p.5/22

20 vs. In a total setting (terminal coalgebras): finite lists potentially infinite lists infinite lists can be nested Inductive Types for Free p.5/22

21 Axiomatically? How do we say? All inductive or coinductive datatypes Inductive Types for Free p.6/22

22 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Inductive Types for Free p.6/22

23 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Inductive Types for Free p.6/22

24 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Inductive Types for Free p.6/22

25 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Strictly positive types, e.g. Inductive Types for Free p.6/22

26 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Strictly positive types, e.g. ordinal notations Inductive Types for Free p.6/22

27 Axiomatically? How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g. Strictly positive types, e.g. ordinal notations How do we say? Strictly positive Inductive Types for Free p.6/22

28 Containers and -types Container types are a syntax-free representation of strictly positive types. Inductive Types for Free p.7/22

29 Containers and -types Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types ( ) are sufficent to show that container types are closed under Inductive Types for Free p.7/22

30 Containers and -types Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types ( ) are sufficent to show that container types are closed under (here) Inductive Types for Free p.7/22

31 Containers and -types Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types ( ) are sufficent to show that container types are closed under (here) (journal paper, submitted) Inductive Types for Free p.7/22

32 Containers and -types Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types ( ) are sufficent to show that container types are closed under (here) (journal paper, submitted) Application: Generic Programming e.g. see our work on derivatives of datatypes Inductive Types for Free p.7/22

33 Containers and -types Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types ( ) are sufficent to show that container types are closed under (here) (journal paper, submitted) Application: Generic Programming e.g. see our work on derivatives of datatypes Application: Small trusted cores e.g. for Epigram Inductive Types for Free p.7/22

34 Containers for dummies Inductive Types for Free p.8/22

35 Containers for dummies A container type is given by Inductive Types for Free p.8/22

36 Containers for dummies A container type is given by A collection of shapes {, e.g.,, } Inductive Types for Free p.8/22

37 Containers for dummies A container type is given by A collection of shapes {, e.g.,, An assignment of positions to shapes }, e.g. Inductive Types for Free p.8/22

38 Containers for dummies... We can use a container by Inductive Types for Free p.9/22

39 Containers for dummies... We can use a container by Choosing a shape, e.g. Inductive Types for Free p.9/22

40 Containers for dummies... We can use a container by Choosing a shape, e.g. Filling the positions with payload (here natural numbers), e.g Inductive Types for Free p.9/22

41 Container for scientists Inductive Types for Free p.10/22

42 Container for scientists A container type a set of shapes is given by a family of sets of position, e.g. for any. is a set Inductive Types for Free p.10/22

43 Container for scientists A container type a set of shapes is given by a family of sets of position, e.g. for any. The extension endofunctor is a set of a container is the Inductive Types for Free p.10/22

44 Container for scientists A container type a set of shapes is given by a family of sets of position, e.g. for any. The extension endofunctor is a set of a container is the Straightforward extension to. -ary containers Inductive Types for Free p.10/22

45 Example: Lists Inductive Types for Free p.11/22

46 Inductive Types for Free p.11/22 Example: Lists

47 Inductive Types for Free p.11/22 Example: Lists

48 Inductive Types for Free p.11/22 Example: Lists

49 Morphisms of containers Given containers and a morphism is given by Inductive Types for Free p.12/22

50 Morphisms of containers Given containers and a morphism is given by Inductive Types for Free p.12/22

51 Morphisms of containers Given containers and a morphism is given by its extension is the natural transformation Inductive Types for Free p.12/22

52 Representation theorem Theorem (AAG,FOSSACS 03) The extension functor is full and faithful. Inductive Types for Free p.13/22

53 Representation theorem Theorem (AAG,FOSSACS 03) The extension functor is full and faithful. Consequence: All polymorphic functions (i.e. natural transformations) are given by Inductive Types for Free p.13/22

54 Representation theorem Theorem (AAG,FOSSACS 03) The extension functor is full and faithful. Consequence: All polymorphic functions (i.e. natural transformations) are given by A length transformer, Inductive Types for Free p.13/22

55 Representation theorem Theorem (AAG,FOSSACS 03) The extension functor is full and faithful. Consequence: All polymorphic functions (i.e. natural transformations) are given by A length transformer A where-did-you-come-from function., Inductive Types for Free p.13/22

56 Closure properties Containers are closed under (*) Constant functors, Coproducts ( ) Products ( Constant exponentation Composition of functors ) initial algebras ( ) [ICALP 04] terminal coalgebras ( ) [Journal paper] Inductive Types for Free p.14/22

57 Closure properties Containers are closed under (*) Constant functors, Coproducts ( ) Products ( Constant exponentation Composition of functors ) initial algebras ( ) [ICALP 04] terminal coalgebras ( ) [Journal paper] (*) In any Martin-Löf category = LCCC (locally cartesian closed category) + W-types. Inductive Types for Free p.14/22

58 Coproducts of containers Given containers Inductive Types for Free p.15/22

59 Coproducts of containers Given containers Inductive Types for Free p.15/22

60 Inductive Types for Free p.15/22 Coproducts of containers Given containers

61 Products of containers Given containers Inductive Types for Free p.16/22

62 Products of containers Given containers Inductive Types for Free p.16/22

63 Inductive Types for Free p.16/22 Products of containers Given containers

64 Initial algebras ( ) Given a 2-ary container Inductive Types for Free p.17/22

65 Initial algebras ( ) Given a 2-ary container Inductive Types for Free p.17/22

66 Initial algebras ( ) Given a 2-ary container Inductive Types for Free p.17/22

67 Initial algebras ( ) Given a 2-ary container Inductive Types for Free p.17/22

68 Initial algebras ( ) Given a 2-ary container Inductive Types for Free p.17/22

69 Reply to referee comment... Now, the above is a strictly positive definition so should have a least as well as a greatest solution which are not in general isomorphic. Thus the corollary mentioned in the proof of 4.1 would be wrong and as a result the entire argument collapses. I thus fear that the paper must be rejected;... Inductive Types for Free p.18/22

70 Reply to referee comment... Now, the above is a strictly positive definition so should have a least as well as a greatest solution which are not in general isomorphic. Thus the corollary mentioned in the proof of 4.1 would be wrong and as a result the entire argument collapses. I thus fear that the paper must be rejected;... Reply: Since there are no infinite paths in a finite tree, there is only one solution to this isomorphism, the initial one. This is reflected in the proof of proposition 4.1! Inductive Types for Free p.18/22

71 Constructing the iso Inductive Types for Free p.19/22

72 Inductive Types for Free p.19/22 Constructing the iso

73 Inductive Types for Free p.19/22 Constructing the iso

74 Terminal coalgebras [Journal submiss Inductive Types for Free p.20/22

75 Terminal coalgebras [Journal submiss The same construction works for the -case. Inductive Types for Free p.20/22

76 Terminal coalgebras [Journal submiss The same construction works for the Now we have to show that the solution is initial! -case. Inductive Types for Free p.20/22

77 Terminal coalgebras [Journal submiss The same construction works for the Now we have to show that the solution is initial! We need -types, the dual of -types. -case. Inductive Types for Free p.20/22

78 Terminal coalgebras [Journal submiss The same construction works for the Now we have to show that the solution is initial! We need -types, the dual of -types. However, -types. -case. -types can be constructed from Inductive Types for Free p.20/22

79 Terminal coalgebras [Journal submiss The same construction works for the Now we have to show that the solution is initial! We need -types, the dual of -types. However, -types. -case. -types can be constructed from See: Containers - Constructing Strictly Positive Types on my publication page. Inductive Types for Free p.20/22

80 Further work Quotient containers to model types like bags. First steps, see our MPC paper. Constructing Polymorphic Programs with Quotient Types Dependent containers Work in progress. Inductive Types for Free p.21/22

81 Related work Joyal 86 Foncteurs Analytiques et Espèces de Structures Jay 95 A semantics for shape Dybjer 97 Representing inductively defined sets by wellorderings in Martin-Löf s type theory Hoogendijk and de Moor 00 Container Types Categorically Moerdijk and Palmgren 00 Wellfounded Trees in Categories Hasegawa 02 Two applications of analytic functors Gambino and Hyland 03 Wellfounded Trees and Dependent Polynomial Functors Inductive Types for Free p.22/22