System F. Proofs and Types. Bow-Yaw Wang. Academia Sinica. Spring 2012

Transcription

1 Proofs and Types System F Bow-Yaw Wang Academia Sinica Spring 2012

2 The Calculus Types in System F are defined as follows. type variables: X, Y,.... if U and V are types, then U V is a type. if V is a type and X is a type variable, ΠX.V is a type. Terms in System F are defined as follows. variables: x T, y T,... of type T. applications: tu of type V when t and u are of types U V and U respectively. λ-abstraction: λx U.v of type U V when x U is a variable of type U and v is of type V. universal abstraction: if v is a term of type V and the type variable X is not free in the type of any free variable of v, then the term ΛX.v is of type ΠX.V. ΛZ.λx Z.x is of type ΠZ.Z Z; ΛZ.x Z is not allowed. universal application (or extraction): if t is a term of type ΠX.V and U is a type, then tu is of type V[U/X]. (ΛZ.λx Z.x)(ΠZ.Z Z) is of type (ΠZ.Z Z) (ΠZ.Z Z).

3 Conversions Two types of conversions: (λx U.v)u v[u/x] and (ΛX.v)U v[u/x] For example, (ΛZ.λx Z.x)(ΠZ.Z Z) λx ΠZ.Z Z.x. By convention, we write U 1 U 2 U n V for U 1 (U 2 (U n V)) and fu 1 u 2 u n for ( ((fu 1 )u 2 ) )u n

4 Circularity? again the term ΛZ.λx Z.x of type ΠZ.Z Z. Naively, we can say ΛZ.λx Z.x is the identify function λx Z.x for any type Z. (ΛZ.λx Z.x)Int 0 (λx Int.x)0 0. (ΛZ.λx Z.x)Bool F (λx Bool.x)F F. We can define functions of different types. How do you implement a universal identity in C++ or Java? Very easy in modern functional languages (OCaml, Haskell, etc). Our naive semantics may lead to circularity. Does any type include the type of all types? Russell s paradox: A = {x : x A}.

5 Representation of simple types Boolean We can define Booleans in system F. We have Bool T : Bool F : Bool = ΠX.X X X = ΛX.λx X.λy X.x = ΛX.λx X.λy X.y D u v t : U = t U u v (u, v : U; t : Bool) D u v T = (ΛX.λx X.λy X.x) U u v (λx U.λy U.x) u v (λy U.u) v u D u v F = (ΛX.λx X.λy X.y) U u v (λx U.λy U.y) u v

6 Representation of simple types Product Types U V = ΠX.(U V X) X u, v π 1 t π 2 t = ΛX.λx U V X.x u v = t U (λx U.λy U.x) = t U (λx U.λy U.x) We have π 1 u, v = (ΛX.λx U V X.x u v) U (λx U.λy V.x) (λx U V U.x u v)(λx U.λy V.x) (λx U.λy V.x) u v (λy V.u) v u π 2 u, v = (ΛX.λx U V X.x u v) U (λx U.λy V.y) (λx U V U.x u v)(λx U.λy V.y) (λx U.λy V.y) u v

7 Representation of simple types Empty Type If t : Emp, then ɛ U t : U. Emp = ΠX.X ɛ U t = t U

8 Representation of simple types Sum Types U + V = ΠX.(U X) (V X) X ι 1 u = ΛX.λx U X.λy V X.x u ι 2 v = ΛX.λx U X.λy V X.y v δx.u y.v t = t U (λx U.u) (λy V.v) (u, v : U; t : R + S) We have δx.u y.v (ι 1 r) = (ΛX.λx R X.λy S X.x r) U (λx R.u) (λy S.v) (λx R U.λy S U.x r) (λx R.u) (λy S.v) (λy S U.(λx R.u) r) (λy S.v) (λx R.u) r u[r/x] Similarly, δx.u y.v (ι 2 s) v[s/y].

9 Representation of simple types Secondary Conversions the secondary conversions in deductions:. U V. U V U 1E. U V V 2E I U V [U] U V 1I [V] U V 2I E U V converts to converts to. U V. U V Our representations in system F do not reflect the secondary conversions: π 1 t, π 2 t t δx.(ι 1 x) y.(ι 2 y) t t

10 Representation of simple types Existential Types ΣX.V = ΠY.(ΠX.(V Y)) Y U, v X.x.w t = ΛY.λx ΠX.V Y.x U v (v : V[U/X]) = t W (ΛX.λx V.w) ΣX.V means for some predicate X, V holds. This is second-order logic. We will come back to this later. We have ( X.x.w) U, v = (ΛY.λx ΠX.V Y.x U v) W (ΛX.λx V.w) (λx ΠX.V W.x U v) (ΛX.λx V.w) (ΛX.λx V.w) U v (λx V[U/X].w[U/X]) v w[u/x][v/x]

11 Representation of Inductive Types Free Structures a collection Θ of terms generated by functions that builds new Θ-terms from old (constructors). Constructors that do not take old Θ-term are constants. Examples. The Boolean type has two constants: f and t. represented by ΠX.X X X The product type has one constructor:,. represented by ΠX.(U V X) X The sum type has two constructors: ι 1 and ι 2. represented by ΠX.(U X) (V X) X The empty type has nothing. represented by ΠX.X.

12 Representation of Inductive Types Representation of Constructors Suppose a free structure Θ is represented by the type T. A Θ-term is therefore of type T. Representations of its constructors can be derived naturally. Examples. For ΠX.X X X, we have two constants ΛX.λx X.λy X.x and ΛX.λx X.λy X.y. For ΠX.(U V X) X, we have one constructor ΛX.λx U V X.x u v with parameters u U and v V. For ΠX.(U X) (V X) X, we have two constructors: ΛX.λx U X.λy V X.x u with a parameter u U and ΛX.λx U X.λy V X.y v with a parameter v V.

13 Representation of Inductive Types Induction on Free Structures Let a free structure Θ be represented by the type T. the induction h on a Θ-terms t: If t is generated by the constructor ctri from the non-θ-terms s 1, s 2,..., s k, and the Θ-terms t 1, t 2,..., t l, return h i (s 1, s 2,..., s k, h(t 1 ), h(t 2 ),..., h(t l )). Examples. For the Boolean type, return u if t = t; return v if t = f. h 1 = u and h 2 = v. If h 1, h 2 are of type U, t U h 1 h 2 performs the induction. For the sum type, return u[r/x] if t = ι 1 r; return v[s/y] if t = ι 2 s. h 1 = λx R.u and h 2 = λy S.v. If h 1, h 2 are of type R U, S U, t U h 1 h 2 performs the induction.

14 Representation of Inductive Types Integers We now define integers in system F. Int = ΠX.X (X X) X O = ΛX.λx X.λy X X.x S t It u f t = ΛX.λx X.λy X X.y (t X x y) = t U u f (u : U; f : U U) Int has two constructors: O and S. It u f t performs the induction on an Int-term. We have It u f O = (ΛX.λx X.λy X X.x) U u f (λx U.λy U U.x) u f u It u f (St) = (ΛX.λx X.λy X X.y (t X x y)) U u f (λx U.λy U U.y (t U x y)) u f

15 Representation of Inductive Types Lists Let U be a type. We define lists whose elements are of type U. List U = ΠX.X (U X X) X nil cons u t It w f t We have = ΛX.λx X.λy U X X.x = ΛX.λx X.λy U X X.y u (t X x y) = t W w f (w : W; f : U W W) It w f nil w and It w f (cons u t) f u (It w f t)

16 Representation of Inductive Types Lists Conc s t = It t (λx U.λy List U.cons x y) s We have Conc nil t = It t (λx U.λy List U.cons x y) nil = nil (List U) t (λx U.λy List U.cons x y) = (ΛX.λx X.λy U X X.x) (List U) t (λx U.λy List U.cons x y) (λx List U.λy U List U List U.x) t (λx U.λy List U.cons x y) t Conc (cons u s) t = It t (λx U.λy List U.cons x y) (cons u s) = (cons u s) (List U) t (λx U.λy List U.cons x y) = (ΛX.λx X.λy U X X.y u (s X x y)) (List U) t (λx U.λy List U.cons x y) (λx List U.λy U List U List U.y u (s (List U) x y)) t (λx U.λy List U.cons x y) (λy U List U List U.y u (s (List U) t y)) (λx U.λy List U.cons x y) (λx U.λy List U.cons x y) u (s (List U) t (λx U.λy List U.cons x y)) cons u (s (List U) t (λx U.λy List U.cons x y)) U List U

17 Representation of Inductive Types Polymorphic Lists The type of elements in a list can of course be quantified. Nil = ΛX.nil X : ΛX.List X Cons = ΛX.cons X : ΛX.X List X List X

18 Representation of Inductive Types Binary Trees A binary tree is either a node (constant) or a tree with two subtrees (constructor). Bintree = ΠX.X (X X X) X nil = ΛX.λx X.λy X X X.x couple u v = ΛX.λx X.λy X X X.y (u X x y) (v X x y) It w f t = t W w f (w : W; f : W W W) We have It w f nil w and It w f (couple u v) f (It w f u) (It w f v)

19 Representation of Inductive Types Trees of Branching Type U A tree of branching type U is either a node (constant) or a tree with subtrees {t u } u U (constructor). Tree = ΠX.X ((U X) X) X nil collect f It w h t We have = ΛX.λx X.λy (U X) X.x = ΛX.λx X.λy (U X) X.y (λz U.f z X x y) = t W w h (w : W; h : (U W) W) It w h nil w and It w h (collect f ) h (λx U.It w h (f x))