Nonuniform (Co)datatypes

Transcription

1 Foundational Nonuniform (Co)datatypes for Higher-Order Logic Jasmin Blanchette Fabian Meier Andrei Popescu Dmitriy Traytel

2 uniform datatype 'a list = Nil Cons 'a ('a list) 1 3 4

3 uniform datatype 'a list = Nil Cons 'a ('a list) 1 3 4

4 uniform datatype 'a list = Nil Cons 'a ('a list) uniform codatatype 'a stream = SCons 'a ('a stream) 1 3 4

5 uniform datatype 'a list = Nil Cons 'a ('a list) uniform codatatype 'a stream = SCons 'a ('a stream) nonuniform datatype 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7))

6 uniform datatype 'a list = Nil Cons 'a ('a list) uniform codatatype 'a stream = SCons 'a ('a stream) nonuniform datatype 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) nonuniform codatatype 'a pstream = PSCons 'a (('a list) pstream) 1 [,3,4] [[5],[6,7,8],[9,10]]

7 What are nonuniform types good for? Mycroft Okasaki pioneering: optimization techniques bootstrapping implicit recursive slowdown theory: data structures finger trees generalized folds advanced (co)iteration Bird Paterson Hinze Matthes Abel Uustalu Abbott Altenkirch Ghani Benton Hur Kennedy McBride Danielsson Hirschowitz Maggesi Naves Spiwack Sozeau 3 practice: proof assistants binders balancing lists finger trees complexity

8 Contribution: enable users of to 1 define nonuniform (co)datatypes define primitively (co)recursive functions 3 prove theorems by nonuniform (co)induction 4

9 Contribution: enable users of to 1 define nonuniform (co)datatypes 'a tm = Var 'a App ('a tm) ('a tm) Lam (('a option) tm) define primitively (co)recursive functions 3 prove theorems by nonuniform (co)induction 4

10 Contribution: enable users of to 1 define nonuniform (co)datatypes 'a tm = Var 'a App ('a tm) ('a tm) Lam (('a option) tm) define primitively (co)recursive functions join :: 'a tm tm => 'a tm join (Var t) = t join (App t u) = App (join t) (join u) join (Lam u) = Lam (join (maptm (λx. case x of None => Var None Some y => maptm Some y) u)) subst σ = join maptm σ 3 prove theorems by nonuniform (co)induction 4

11 Contribution: enable users of to 1 define nonuniform (co)datatypes 'a tm = Var 'a App ('a tm) ('a tm) Lam (('a option) tm) define primitively (co)recursive functions join :: 'a tm tm => 'a tm join (Var t) = t join (App t u) = App (join t) (join u) join (Lam u) = Lam (join (maptm (λx. case x of None => Var None Some y => maptm Some y) u)) subst σ = join maptm σ 3 prove theorems by nonuniform (co)induction subst τ (subst σ s) = subst (subst τ σ) s 4

12 But Coq and Agda Contribution: enable users of to 1 3 define nonuniform (co)datatypes 'a tm = Var 'a App ('a tm) ('a tm) Lam (('a option) tm) have had this built define primitively (co)recursive functions into their logics join :: 'a tm tm => 'a tm join (Var t) = t join (App t u) = App (join t) (join u) join (Lam u) = Lam (join (maptm (λx. case x of None => Var None Some y => maptm Some y) u)) for decades! subst σ = join maptm σ prove theorems by nonuniform (co)induction subst τ (subst σ s) = subst (subst τ σ) s 4

13 But Coq and Agda Contribution: enable users of to 1 define nonuniform (co)datatypes 'a tm = Var 'a App ('a tm) ('a tm) Lam (('a option) tm) foundational have had this built 3 define primitively (co)recursive functions into their logics join :: 'a tm tm => 'a tm join (Var t) = t join (App t u) = App (join t) (join u) join (Lam u) = Lam (join (maptm (λx. case x of None => Var None Some y => maptm Some y) u)) for decades! subst σ = join maptm σ prove theorems by nonuniform (co)induction subst τ (subst σ s) = subst (subst τ σ) s Our approach is new features are reduced to existing features 4

14 Foundations 5

15 Simple Theory of Types Alonzo Church 1940 types: T = ο ι T => T terms: simply typed λ-calculus + few built-in constants 6

16 Higher-Order Logic Mike Gordon 1988 types: T = ο ι T => T 'a (T,,T)κ + nonrecursive type definitions new type T Rep Abs A U existing type terms: simply typed λ-calculus + few built-in constants + Hilbert Choice + nonrecursive constant definitions 7

17 Foundational Uniform (Co)datatypes for Higher-Order Logic Jasmin Blanchette Andrei Popescu Dmitriy Traytel et al. LICS 01 ITP 014 ESOP 015 ICFP 015 ESOP 017 FroCoS 017 8

18 Blanchette, Hölzl, Lochbihler, Panny, Popescu, Traytel ITP 014 9

19 Blanchette, Hölzl, Lochbihler, Panny, Popescu, Traytel ITP 014 9

20 Blanchette, Hölzl, Lochbihler, Panny, Popescu, Traytel ITP 014 9

21 Blanchette, Hölzl, Lochbihler, Panny, Popescu, Traytel ITP 014 9

22 Foundational Nonuniform (Co)datatypes for Higher-Order Logic LICS

23 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7))

24 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem)

25 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem)

26 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7))

27 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6)

28 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r))

29 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0)

30 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0)

31 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0) 1 (,3) ((4,5),(6,7)) ((4,5),(6,7)) (,3)

32 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0) 1 (,3) ((4,5),(6,7)) ((4,5),(6,7)) (,3) ok n PNil0 full n x ok (n + 1) xs ok n (PCons0 x xs)

33 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0) 1 (,3) ((4,5),(6,7)) ((4,5),(6,7)) (,3) ok n PNil0 full n x ok (n + 1) xs ok n (PCons0 x xs) 3 carve out ok powerlists 4 'a plist0 'a plist Rep {xs ok 0 xs} Abs 11

34 'a plist = PNil PCons 'a (('a 'a) plist) 1 (,3) ((4,5),(6,7)) 1 overapproximate the elements of a powerlist 'a elem = Leaf 'a Node ('a elem 'a elem) 1 (,3) ((4,5),(6,7)) ((4,5),6) full 0 (Leaf x) full n l full n r full (n + 1) (Node (l, r)) overapproximate the set of all powerlists 'a plist0 = PNil0 PCons0 ('a elem) ('a plist0) 1 (,3) ((4,5),(6,7)) ((4,5),(6,7)) (,3) ok n PNil0 full n x ok (n + 1) xs ok n (PCons0 x xs) 3 carve out ok powerlists 4 'a plist0 'a plist Rep {xs ok 0 xs} Abs 11 lift constructors PCons x xs = Abs (PCons0 (Leaf x) ( (Rep xs)))

35 General construction supports: multiple recursive occurrences 'a biplist = Nil Cons1 'a (('a list) biplist) Cons 'a (('a 'a) biplist) 1

36 General construction supports: multiple recursive occurrences 'a biplist = Nil Cons1 'a (('a list) biplist) Cons 'a (('a 'a) biplist) multiple type arguments ('a, 'b) tplist = Nil 'b Cons 'a (('a 'a, 'b option) tplist) 1

37 General construction supports: multiple recursive occurrences 'a biplist = Nil Cons1 'a (('a list) biplist) Cons 'a (('a 'a) biplist) multiple type arguments ('a, 'b) tplist = Nil 'b Cons 'a (('a 'a, 'b option) tplist) mutual definitions 'a ptree = Node 'a ('a pforest) 'a pforest = Nil Cons ('a ptree) (('a 'a) pforest) 1

38 General construction supports: multiple recursive occurrences 'a biplist = Nil Cons1 'a (('a list) biplist) mutual definitions Cons 'a (('a 'a) biplist) multiple type arguments ('a, 'b) tplist = Nil 'b Cons 'a (('a 'a, 'b option) tplist) 'a ptree = Node 'a ('a pforest) 'a pforest = Nil Cons ('a ptree) (('a 'a) pforest) codatatypes 'a pstream = PSCons 'a (('a list) pstream) 1

39 General construction supports: multiple recursive occurrences 'a biplist = Nil Cons1 'a (('a list) biplist) mutual definitions Cons 'a (('a 'a) biplist) multiple type arguments ('a, 'b) tplist = Nil 'b Cons 'a (('a 'a, 'b option) tplist) 'a ptree = Node 'a ('a pforest) 'a pforest = Nil Cons ('a ptree) (('a 'a) pforest) codatatypes 'a pstream = PSCons 'a (('a list) pstream) arbitrary bounded natural functors 'a crazy = Crazy 'a ((((('a pstream) fset) crazy) multiset) list) 1

40 Take Home Messages It is tempting to introduce a new hot logic/language for each new feature. But this is not always necessary. The foundational path requires work. But it also saves work elsewhere. (keyword: consistency) 13

41 Foundational Nonuniform (Co)datatypes Bedankt! Higher-Order Logic Vragen? for Jasmin Blanchette Fabian Meier Andrei Popescu Dmitriy Traytel