Parametricity and GADTs

Transcription

1 Dimitrios Vytiniotis Stephanie Weirich Computer and Information Science Department University of Pennsylvania Boston, July 2006

2 A very simple GADT example data R :: * -> * where Rint :: R Int Rbool :: R Bool inc :: forall a. R a -> a -> a inc Rint x = x + 1 inc Rbool x = True

3 A very simple GADT example inc :: forall a. R a -> a -> a inc Rint x = x + 1 inc Rbool x = True This is a strange function: Can t apply inc to all types. The argument of type a is not treated parametrically. So, what does parametricity mean in this language?

4 Overview 1. System F + this GADT 2. Parametricity theorem for this language 3. Free theorems 4. Other GADTs

5 Overview 1. System F + this GADT 2. Parametricity theorem for this language 3. Free theorems 4. Other GADTs This is all work in progress.

6 System F τ, σ ::= int bool α σ σ a.σ e ::= i b λx.e e 1 e 2 Λα.e e[σ]... v ::= i λx.e

7 System F + R τ, σ ::= int bool α σ σ a.σ R τ e ::= i b λx.e e 1 e 2 Λα.e e[σ]... R int R bool case e e int e bool v ::= i λx.e R int R bool

8 System F + R τ, σ ::= int bool α σ σ a.σ R τ e ::= i b λx.e e 1 e 2 Λα.e e[σ]... R int R bool case e e int e bool v ::= i λx.e R int R bool inc :: a.r α α α inc = λx.case x (λy.y + 1) (λz.true)

9 Typing rules Γ R int : R int Γ R bool : R bool Γ e : R τ Γ e int : σ{int/α} Γ e bool : σ{bool/α} Γ case e e int e bool : σ{τ/α}

10 Bigstep, CBN Operational Semantics v v e 1 λx.e 1 e 1 {e 2/x} v e 1 e 2 v e R int e int v case e e int e bool v e 1 Λa.e 1 e 1 {σ/α} v e 1 [σ] v e R bool e bool v case e e int e bool v

11 Some Definitions Definition (Typed value relations) Let V(τ 1, τ 2 ) be the set of relations between closed values of closed type τ 1 and τ 2.

12 Some Definitions Definition (Typed value relations) Let V(τ 1, τ 2 ) be the set of relations between closed values of closed type τ 1 and τ 2. Definition (Type substitution) A type substitution η is a map from type variables to (τ 1, τ 2, r) where τ 1 and τ 2 are closed types and r V(τ 1, τ 2 ). If η(α) = (τ 1, τ 2, r), then let η 1 (α) = τ 1, η 2 (α) = τ 2 and η r (α) = r.

13 Some Definitions Definition (Typed value relations) Let V(τ 1, τ 2 ) be the set of relations between closed values of closed type τ 1 and τ 2. Definition (Type substitution) A type substitution η is a map from type variables to (τ 1, τ 2, r) where τ 1 and τ 2 are closed types and r V(τ 1, τ 2 ). If η(α) = (τ 1, τ 2, r), then let η 1 (α) = τ 1, η 2 (α) = τ 2 and η r (α) = r. Definition (Computational closure) If r V(τ 1, τ 2 ), then define r as {(e 1, e 2 ) e 1 : τ 1 e 2 : τ 2 e 1 v 1 e 2 v 2 (v 1, v 2 ) r}.

14 Logical Relation (System F) [int ] η = {(i, i)} [bool ] η = {(b, b)} [σ 1 σ 2 ] η = {(v 1, v 2 ) v 1 : η 1 (σ 1 σ 2 ) v 2 : η 2 (σ 1 σ 2 ) (e 1, e 2 ) [σ 1 ] η (v 1 e 1, v 2 e 2 ) [σ 2 ] η } [ α.σ] η = {(v 1, v 2 ) v 1 : η 1 ( α.σ) v 2 : η 2 ( α.σ) τ 1, τ 2, r V(τ 1, τ 2 ), (v 1 [τ 1 ], v 2 [τ 2 ]) [σ] η,α (τ1,τ 2,r) } [α] η = η r (α)

15 Parametricity Theorem Definition (Related substitution) Let γ be a mapping from term variables to pairs of closed expressions. Say Γ, η γ iff x : σ Γ, (γ 1 (x), γ 2 (x)) [σ] η.

16 Parametricity Theorem Definition (Related substitution) Let γ be a mapping from term variables to pairs of closed expressions. Say Γ, η γ iff x : σ Γ, (γ 1 (x), γ 2 (x)) [σ] η. Theorem (Fundamental theorem) If Γ e : σ and ftv(γ, e, σ) = dom(η) and Γ, η γ then (γ 1 (e), γ 2 (e)) [σ] η.

17 Relation for R types [R int ] η = {(R int, R int )} [R bool ] η = {(R bool, R bool )} [Rτ] when η 1 (α) = η 2 (α) = τ and η [Rα] η = r (α) = [τ] and τ is a closed monotype otherwise [R τ] η = otherwise

18 Relation for R types [R int ] η = {(R int, R int )} [R bool ] η = {(R bool, R bool )} C [Rτ] when η 1 (α) = η 2 (α) = τ and η [Rα] η = r (α) = C [τ] and τ is a closed monotype otherwise [R τ] η = otherwise

19 Closed relation C [int ] = {(i, i)} C [bool ] = {(b, b)} C [σ 1 σ 2 ] = {(v 1, v 2 ) v 1 : η 1 (σ 1 σ 2 ) v 2 : η 2 (σ 1 σ 2 ) (e 1, e 2 ) C [σ 1 ] (v 1 e 1, v 2 e 2 ) C [σ 2 ] } C [R int ] = {(R int, R int )} C [R bool ] = {(R bool, R bool )} C [σ] = otherwise Lemma If τ is a closed monotype then [τ] = C [τ]

20 A free theorem Consider a closed expression f of type α.α α. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (x, y) r (f [τ 1 ]x, f [τ 2 ]y) r

21 A free theorem Consider a closed expression f of type α.α α. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (x, y) r (f [τ 1 ]x, f [τ 2 ]y) r We can use this theorem to show that forall values v of type τ, f [τ]v v.

22 A free theorem Consider a closed expression f of type α.α α. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (x, y) r (f [τ 1 ]x, f [τ 2 ]y) r We can use this theorem to show that forall values v of type τ, f [τ]v v. Let r be the relation {(v, v)}.

23 A free theorem Consider a closed expression f of type α.α α. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (x, y) r (f [τ 1 ]x, f [τ 2 ]y) r We can use this theorem to show that forall values v of type τ, f [τ]v v. Let r be the relation {(v, v)}. Now, (x, y) r (f [τ]x, f [τ]y) r.

24 A free theorem Consider a closed expression f of type α.α α. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (x, y) r (f [τ 1 ]x, f [τ 2 ]y) r We can use this theorem to show that forall values v of type τ, f [τ]v v. Let r be the relation {(v, v)}. Now, (x, y) r (f [τ]x, f [τ]y) r. So (v, v) r (f [τ]v, f [τ]v) r.

25 Free theorem for inc Now consider a closed expression f of type α.rα α α.

26 Free theorem for inc Now consider a closed expression f of type α.rα α α. τ 1, τ 2, r V(τ 1, τ 2 ),

27 Free theorem for inc Now consider a closed expression f of type α.rα α α. τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (v, w) [Rτ], (x, y) r (f [τ 1 ] v x, f [τ 2 ] w y) r )

28 Free theorem for inc Now consider a closed expression f of type α.rα α α. τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (v, w) [Rτ], (x, y) r (f [τ 1 ] v x, f [τ 2 ] w y) r ) (τ 1 τ 2 r [τ 1 ] (v, w), (x, y) r, (f [τ 1 ] v x, f [τ 2 ] w y) r )

29 You get what you pay for Now consider a closed expression f of type α.rα Rα, which is an identity function. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (x, y) [Rτ 1 ], (f [τ 1 ] x, f [τ 2 ] y) [Rτ] ) (τ 1 τ 2 r [τ 1 ] (x, y), (f [τ 1 ] x, f [τ 2 ] y) ) This theorem is also uninteresting all it says is that when given equal arguments, f will produce equal results.

30 Not always useless Consider a closed expression f of type α.rα. The free theorem for this type is:

31 Not always useless Consider a closed expression f of type α.rα. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ),

32 Not always useless Consider a closed expression f of type α.rα. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (f [τ 1 ], f [τ 2 ]) [Rτ 1 ] )

33 Not always useless Consider a closed expression f of type α.rα. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (f [τ 1 ], f [τ 2 ]) [Rτ 1 ] ) (τ 1 τ 2 r [τ 1 ] (f [τ 1 ], f [τ 2 ]) )

34 Not always useless Consider a closed expression f of type α.rα. The free theorem for this type is: τ 1, τ 2, r V(τ 1, τ 2 ), (τ 1 = τ 2 r = [τ 1 ] (f [τ 1 ], f [τ 2 ]) [Rτ 1 ] ) (τ 1 τ 2 r [τ 1 ] (f [τ 1 ], f [τ 2 ]) ) By this theorem, (f [int], f [bool]). So there cannot be any such f.

35 Alternative reasoning Lemma (Canonical forms) 1. If v : R int then v = R int. 2. If v : R bool then v = R bool. 3. There are no closed values of type R σ, when σ is not int or bool.

36 Alternative reasoning Lemma (Canonical forms) 1. If v : R int then v = R int. 2. If v : R bool then v = R bool. 3. There are no closed values of type R σ, when σ is not int or bool. Using this this lemma, we can show that if f : α.rα Rα then for all v : Rτ, f [τ] v v.

37 Vector GADT Consider another GADT. data Z :: * data S :: * -> * data Vec :: * -> * -> * where Nil :: Vec Z a Cons :: a -> Vec n a -> Vec (S n) a

38 More formally Γ Nil : α.vec Z α Γ Cons : αβ.α Vec β α Vec (S β)α Γ e : Vec σ ind σ Γ e n : σ {Z/α} Γ e c : β.σ σ {β/α} σ {S β/α} Γ case e e n e c : σ {n/α}

39 Logical relation [Z ] η = [Sσ] η = [Vec Z σ] η = {(Nil, Nil)} [Vec (S σ i ) σ] η = {(Cons[η 1 (σ)][η 1 (σ i )] x 1 y 1, Cons[η 2 (σ)][η 2 (σ i )] x 2 y 2 ) { (x 1, x 2 ) [σ] η, (y 1, y 2 ) [Vec σ i σ] η } [Vec τ σ]η when η [Vec α σ] η = 1 (α) = η 2 (α) = τ otherwise [Vec σ i σ] η = otherwise

40 Logical relation [Z ] η = [Sσ] η = [Vec Z σ] η = {(Nil, Nil)} [Vec (S σ i ) σ] η = {(Cons[η 1 (σ)][η 1 (σ i )] x 1 y 1, Cons[η 2 (σ)][η 2 (σ i )] x 2 y 2 ) { (x 1, x 2 ) [σ] η, (y 1, y 2 ) [Vec σ i σ] η } [Vec τ σ]η when η [Vec α σ] η = 1 (α) = η 2 (α) = τ otherwise [Vec σ i σ] η = otherwise Note: Because the index type is empty, don t need to restrict η r (α).

41 Where to next? More free theorems. Leave the pure world. Parametricity for general GADTs. Mechanize everything in a theorem prover. Dimitrios has a good start in Isabelle/HOL.