hindley-milner type inference cse 340 principles of programming languages fall 2015 adam doup...

Hindley-Milner Type Inference

CSE 340 – Principles of Programming LanguagesFall 2015

Adam DoupéArizona State Universityhttp://adamdoupe.com

2Adam Doupé, Principles of Programming Languages

Type Systems

• In what we have seen so far, the programmer must declare the types of the variables

array [0..5] of int a;int i;

a[i] = 1;


Type Systems


array [0..5] of int a;string i;

a[i] = 1;


Type Systems


array [0..5] of int a;int i;

a[i] = "testing";


Parameterized Types

• Some languages allow the programmer to declare parameterized types– Instead of being specific to a given type, the

specific type is given as a parameter• Generics in Java and C#, templates in C+

+


import java.util.Random;public class Chooser{ static Random rand = new Random(); public static <T> T choose(T first, T second) { return ((rand.nextInt() % 2) == 0)? first: second; }}class ParameterizedTypes{ public static void main(String [] args) { int x = 100; int y = 999; System.out.println(Chooser.choose(x, y)); String a = "foo"; String b = "bar"; System.out.println(Chooser.choose(a, b)); }}


Explicit Polymorphism

• Note that in the previous example, the programmer must declare the parameterized types explicitly

• Slightly different polymorphism than what is used in the object orientation context

• The compiler/interpreter will allow a function to be called with different types (while still checking for type compatibility)


Implicit Polymorphism

• The programmer does not need to specify the type parameters explicitly– Dynamic languages have this property too

• However, the type checker will, statically, attempt to assign the most general type to every construct in the program


Implicit Polymorphismfun foo(x) = x

• What is the type of foo?– Function of T returns T– (T) -> T

fun foo(x) = x;fun bar(y) = foo(y);• What is the type of bar and foo?

– foo: Function of T returns T• (T) -> T

– bar: Function of T returns T• (T) -> T



fun max(x, y) = if x < y thenyelsex

• What is the type of max?– Function of (int, int) returns int– (int,int) -> int


Implicit Polymorphismfun max(cmp, x, y)

= if cmp(x,y) thenyelsex

• What is the type of max?– Function of (Function of (T, T) returns bool, T, T) returns T– ((T, T) -> bool, T, T) -> T

• max(<, 10, 200)• max(strcmp, "foo", "bar")



fun foo(a, b, c) = c(a[b])

• What is the type of foo?– Function of (Array of T, int, Function of (T)

returns U) returns U– (Array of T, int, (T -> U)) -> U



fun foo(a, b, c) = a = 10;a(b[c]);

• What is the type of foo?– Type error!


Hindley-Milner Type Checking

• Hindley-Milner type checking is a general type inference approach– It infers the types of constructs that are not explicitly

declared– It leverages the constraints of the various constructs– It applies these constraints together with type

unification to find the most general type for each construct (or can find a type error if there is one)

• Full Hindley-Milner type checking is used in OCaml, F#, and Haskell


Type Constraints• To apply Hindley-Milner, we must first define the type constraints• Constant integers

– …, -1, 0, 1, 2, ...– Type = int

• Constant real numbers– ..., 0.1, 2.2, ... other floating point numbers– Type = real

• Constant booleans– true or false– Type = boolean

• Constant strings– "foo", "bar", ...– Type = string


Operators

• Relational Operators a op b

• op is <, <=, >, >=, !=, ==• T1 = boolean• T2 = T3 = numeric type

(T1)op

(T2)a

(T3)b


Operators

• Arithmetic Operators a op b

• op is +, -, *, /• T1 = T2 = T3 = numeric type

(T1)op

(T2)a

(T3)b


Operators

• Array Access Operator a[b]

• T2 = array of T1

• T3 = int

(T1)[]

(T2)a

(T3)b


Function Application

• foo(x1, x2, …, xk)

• F = (T1, T2, …, Tk) -> R

(R)apply

(F)foo

(T1)x1

(T2)x2

(Tk)xk

…


Function Definition

• fun foo(x1, x2, …, xk) = expr

• F = (T1, T2, …, Tk) -> E

fun

F T1, T2, …, Tk

foo (x1, x2, …, xk)

(E)expr


If Expression

• if (cond) then expr1 else expr2

• T1 = boolean• T2 = T3 = T4

(T4)if

(T1)cond

(T2)expr

1

(T3)expr

2


Type Unification

• Type unification is the process by which the constraints are propagated

• Basic idea is simple– Start from the top of the tree– Every time you see a construct with

unconstrained types, create a new type– If a construct is found to have type T1 and also

to have type T2, then T1 and T2 must be the same type


fun foo(a, b, c) = c(a[b])(1)def

foo (a, b, c)(2)apply

(3)c

(4)[]

(5)a

(6)b

fooabc(1)(2)(3)(4)(5)(6)




(3)c

(4)[]

(5)a

(6)b

fooa T1

b T2

c T3

(1)(2)(3)(4)(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,T3) -> T4

a T1

b T2

c T3

(1)(2)(3)(4)(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,T3) -> T4

a T1

b T2

c T3

(1)(2) T4

(3)(4)(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,T3) -> T4

a T1

b T2

c T3

(1)(2) T4

(3)(4) T5

(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,T3) -> T4

a T1

b T2

c T3

(1)(2) T4

(3) T5 -> T4

(4) T5

(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,T3) -> T4

a T1

b T2

c T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,(T5->T4)) -> T4

a T1

b T2

c T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5)(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,(T5->T4)) -> T4

a T1

b T2

c T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5) Array of T5

(6)




(3)c

(4)[]

(5)a

(6)b

foo (T1,T2,(T5->T4)) -> T4

a T1

b T2

c T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5) Array of T5

(6) int




(3)c

(4)[]

(5)a

(6)b

foo (Array of T5 ,T2,(T5->T4)) -> T4

a Array of T5

b T2

c T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5) Array of T5

(6) int




(3)c

(4)[]

(5)a

(6)b

foo (Array of T5, T2,(T5->T4)) -> T4

a Array of T5

b intc T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5) Array of T5

(6) int




(3)c

(4)[]

(5)a

(6)b

foo (Array of T5, int,(T5->T4)) -> T4

a Array of T5

b intc T5 -> T4

(1)(2) T4

(3) T5 -> T4

(4) T5

(5) Array of T5

(6) int

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