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 typede.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.(x z) 0 : S | x C

x:X z:Z (CT-VAR)(CT-VAR)

0 : Nat |ø {}(CT-ZERO)

x : _ |_ _ z : _ |_ _ (CT-APP)

(x z) : _ |_ _ 0 : _ |_ _ (CT-APP)

X |ø {} Z |ø {}

Nat |ø {}V1 |{V1} C1

C1 = {X = ZV1} = x:X,z:Z

(x z) 0 : _ |_ _(CT-ABS)

C2 = C1 {V1 = Nat V2}

V2 |{V1,V2} C2

X Z V2{V1,V2}

C2

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 :: _(K-ABS) 2x

= X:: _ ,Y:: _

\X.\Y.X Y :: K

(K-APP) X :: _ _ Y :: _

(K-TVAR) X :: _ _

(K-TVAR) Y :: _

*

(* *) * *

** *

** *

* * *

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

DATA KindExpr

| KVar

| KStar

| KCon

| KApply

DATA TypeExpr

| TVar

| TCon

| TConProduct

| TProduct

| TPred

| TQuant

| TApply

Generalized AST

• Unify structure

DATA GenExpr

| GVar

| GCon

| GApply

DATA KindExpr

| KGenExpr

gExpr :: GenExpr

| KStar

DATA TypeExpr

| TGenExpr

gExpr :: GenExpr

| TConProduct

| TProduct

| TPred

| TQuant

Kind abstract treeKindExpr

KVar KApply KCon KStar KParen

KindExpr

KGenExpr KStar KParen

GenExpr

GVar GApply GCon

GenExpr

before

unified

KindExpr

KVar

TypeExpr

TVar TApplyKApply

Unified tree

GenExpr

GVar

KindExpr

KGenExpr

TypeExpr

TGenExprGApply

before

unified

GenExpr GenExpr

-- Pass 1, patterns/placeholders

SEM KindExpr

| KVar

loc . (kpuniq,tai,kgam)

= samefun @lhs.patTpTpConstrGam

lhs . patTpTpConstrGam = @kgam

SEM TypeExpr

| TVar

loc . (_,tai,tcgam)

= samefun @lhs.patImTpConstrGam

lhs . patImTpConstrGam = @tcgam

ATTR GenExpr [ | patImGenGam: {TypeAssumptions} | ]

SEM GenExpr

| GVar

loc . (gpuniq,tai,ggam) = samefun @lhs.patImGenGam

lhs . patImGenGam = @ggam

SEM TypeExpr

| TGenExpr

gExpr . patImGenGam = @lhs.patImTpConstrGam

lhs . patImTpConstrGam = @gExpr.patImGenGam

SEM KindExpr

| KGenExpr

gExpr . patImGenGam = @lhs.patTpTpConstrGam

lhs . patTpTpConstrGam = @gExpr.patImGenGam

Conclusion

• Not tested, but this should work

• Draw back: Lot of work now

• Improvement: Future additions/fixes easier