Unifying Kind and Type Inference

Unifying Kind and Type Inference Remko van Beusekom & Jeroen Gordijn

Overview • Type inferencing • Kind inferencing • Example • Similarities kind and type inferencing

Overview (2) • Implementation problem in uhc • Our solution to the problem • Conclusion

Approach • Investigate type and kind inferencing • Check the similarities • Investigate the implementations in UHC • Check whether unification is possible

Type inferencing • Check if an (untyped) term can be typed e.g.: (\x.x+1) true can’t be typed (\x.x+1) 2 can be typed.

Kind inferencing • Kinds are “the types of types” • Check if a type definition can be kinded e.g.: (Bool)(Nat)Can’t be kinded (Nat->Bool)(Nat)Can be kinded

Type Inferencing example

 = x:X,z:Z C1 = {X = ZV1} C2 = C1 {V1 = Nat V2} x:X   z:Z   (CT-VAR) (CT-VAR)  x : _ |_ _ X |ø {}  z : _ |_ _ Z |ø {} (CT-APP) 0 : Nat |ø {} V1 |{V1} C1 (CT-ZERO)   (x z) : _ |_ _  0 : _ |_ _ Nat |ø {} (CT-APP)  (x z) 0 : _ |_ _ V2 |{V1,V2} C2 (CT-ABS) X Z V2 C2 {V1,V2}  \x:X.\z:Z.(x z) 0 : S | x C

Unifying the constraints • Unify({X = Z V1}, {V1 = Nat V2}) • Unification fails or succeeds • If succeeds then typeable •  = [X  Z  V1, V1Nat  V2] • (X  Z  V2) = (Z  (Nat  V2))  Z  V2

Kind Inferencing example

 = X:: _ ,Y:: _ *  * * Y :: _   X :: _  _   *  * * (K-TVAR) (K-TVAR)   X :: _  _ *  *   Y :: _ * (K-APP)   X Y :: _ (K-ABS) 2x *  \X.\Y.X Y :: K (*  *)  *  *

Similarities • Walk over the tree. (BOTH) • Introduce placeholders for types/kinds (BOTH) • Introduce placeholders for Constraints (TYPE) • Fill the environment (BOTH) • Fill in the placeholders (BOTH)

The challenge • Type inferencing implemented first • Kind inferencing added by copying parts from type inferencing • Duplicate code • Q: Can we unify these implementations?

Solution to the problem • Generalize the AST • put common constructors in general data type • extra general constructor • Move inferencing code into general code • Problem: extra node in the generalised AST

Implementation in UHC • Different structure

Generalized AST • Unify structure

before unified GenExpr KindExpr KGenExpr KStar KParen GVar GApply GCon GenExpr Kind abstract tree KindExpr KVar KApply KCon KStar KParen

before unified KindExpr TypeExpr GenExpr KGenExpr TGenExpr GVar GApply GenExpr GenExpr Unified tree TypeExpr KindExpr KVar KApply TVar TApply

Conclusion • Not tested, but this should work • Draw back: Lot of work now • Improvement: Future additions/fixes easier

Unifying Kind and Type Inference