# COMP3161辅导、辅导Python/java编程

- 首页 >> Web COMP3161/9164 22T3 Assignment 2

Type Inference for Polymorphic MinHS

Version 2.1

Marks : 17.5% of the overall mark

Due date: Sunday 20th November 2022, 11:59 PM Sydney time

Overview

In this assignment you will implement a type inference pass for MinHS. The language used in this

assignment differs from the language of the first assignment in two respects: it has a polymorphic

type system, and it has aggregate data structures.

The assignment requires you to:

Implement type synthesis for polymorphic MinHS with sum and product data types.

(0.1% bonus) adjust the type inference pass to include various simple syntax extensions from

Assignment 1.

(0.2% bonus) extend the type inference pass to elaborate mutually recursive letrec bindings.

(0.2% bonus) adjust the type inference pass to allow optional type annotations provided by

the user.

Each of these parts is explained in detail below.

The parser, and an interpreter backend, are provided for you. You do not have to change anything

in any module other than TyInfer.hs (even for the bonus parts).

Your type inference pass should annotate the abstract syntax with type information where it is

missing. The resulting abstract syntax should be fully annotated and correctly typed.

Your assignment will only be tested on correct programs, and will be judged correct if it produces

correctly typed annotated abstract syntax, up to α-renaming of type variables.

Please use the Ed forum for questions about this assignment.

Submission

Submit your (modified) TyInfer.hs using the CSE give system, by typing the command

give cs3161 TyInfer TyInfer.hs

or by using the CSE give web interface.

1

1 Task 1

Task 1 is worth 100% of the marks of this assignment. You are to implement type inference for

MinHS with aggregate data structures. The following cases must be handled:

the MinHS language of the first task of assignment 1 (without n-ary functions or letrecs, or

lists)

product types: the 0-tuple and 2-tuples.

sum types

polymorphic functions

These cases are explained in detail below. The abstract syntax defining these syntactic entities is

in Syntax.hs. You should not need to modify the abstract syntax definition in any way.

Your implementation is to follow the definition of algebraic data types found in the lecture on

data structures, and the rules defined in the lectures on type inference. Additional material can be

found in the lecture notes on polymorphism, and the reference materials. The full set of rules is

outlined in this specification.

2 Bonus Tasks

These tasks are all optional, and are worth a total of an additional 10%.

2.1 Bonus Task 1: Simple Syntax Extensions

This bonus task is worth an additional 3%. In this task, you should implement type inference for

n-ary functions, and multiple bindings in the one let expression, with the same semantics as the

extension tasks for Assignment 1.

You will need to develop the requisite extensions to the type inference algorithm yourself, but

the extensions are very similar to the existing rules.

2.2 Bonus Task 2: Recursive Bindings

This bonus task is worth an additional 3%. In this task you must implement type inference for

letrec constructs.

Each binding will require a fresh unification variable, similar to multiple-binding let expressions,

however for letrec we do not wish types to be generalised.

2.3 Bonus Task 3: User-provided type signatures

This bonus task is worth an additional 4%. In this task you are to extend the type inference pass

to accept programs containing some type information. You need to combine this with the results

of your type inference pass to produce the final type for each declaration. That is, you need to be

able to infer correct types for programs like:

main = let f :: (Int -> Int)

= recfun g x = x;

in f 2;

You must ensure that the type signatures provided are not overly general. For example, the following

program should be a type error, as the user-provided signature is too general:

main :: (forall a. a) = 3;

2

3 Algebraic Data Types

This section covers the extensions to the language of the first assignment. In all other respects

(except lists) the language remains the same, so you can consult the reference material from the

first assignment for further details on the language.

3.1 Product Types

We only have 2-tuples in MinHS, and the Unit type, which could be viewed as a 0-tuple.

Types τ → τ1 * τ2

| 1

Expressions exp → (e1, e2)

| fst e | snd e

| ()

3.2 Sum Types

Sum types in MinHS follow the presentation in the lectures.

Types τ → τ + τ

Expressions exp → Inl e1

| Inr e2

| case e of

Inl x -> e1 ;

Inr y -> e2 ;

3.3 Polymorphism

The extensions to allow polymorphism are relatively minor. Two new type forms have been intro-

duced: the TypeVar t form, and the Forall t e form.

Types τ → forall τ. ..τ..

For example, consider the following code fragment before and after type inference:

main =

let f = recfun g x = x;

in if f True

then f (Inl 1)

else f (Inr ());

main :: (Int + 1) =

let f :: forall a. (a -> a) = recfun g :: (a -> a) x = x;

in if f True

then f (Inl 1)

else f (Inr ());

3

4 Type Inference Rules

Sections coloured in blue can be viewed as inputs to the algorithm, while red text can be viewed

as outputs.

Constants and Variables

Γ ` n : Int

x : ?a1 . . . ?an.τ ∈ Γ

Γ ` x : τ [a1 := β1, . . . , an := βn] βi fresh

(unquantify in TyInfer.hs provides an easy way to implement this)

Constructors and Primops

constructorType(C) = ?a1 . . . ?an.τ

Γ ` C : τ [a1 := β1, . . . , an := βn]

primopType(o) = ?a1 . . . ?an.τ

Γ ` o : τ [a1 := β1, . . . , an := βn] βi fresh

(constructorType and primopType are defined in TyInfer.hs)

Application

TΓ ` e1 : τ1 T ′TΓ ` e2 : τ2 T ′τ1 U～ (τ2 → α)

UT ′TΓ ` Apply e1 e2 : Uα α fresh

If-Then-Else

TΓ ` e : τ τ U～ Bool T1UTΓ ` e1 : τ1 T2T1UTΓ ` e2 : τ2 T2τ1 U

′

～ τ2

U ′T2T1UTΓ ` If e e1 e2 : U ′τ2

Case

TΓ ` e : τ T1T (Γ ∪ {x : αl}) ` e1 : τl

T2T1T (Γ ∪ {y : αr}) ` e2 : τr T2T1T (αl + αr)

U～ T2T1τ UT2τl U

′

～ Uτr

U ′UT2T1TΓ ` Case e x.e1 y.e2 : U ′Uτr αl, αr fresh

Recursive Functions

T (Γ ∪ {x : α1} ∪ {f : α2}) ` e : τ Tα2 U～ Tα1 → τ

UTΓ ` Recfun f.x.e : U(Tα1 → τ) α1, α2 fresh

Let Bindings

TΓ ` e1 : τ T ′(TΓ ∪ {x : Generalise(TΓ, τ)}) ` e2 : τ ′

T ′TΓ ` Let e1 x.e2 : τ ′

TΓ ` e : τ

Γ ` Main e : Generalise(Γ, τ)

where Generalise(Γ , τ) = ? (TV (τ) \ TV (Γ )) . τ

4.1 Implementing the algorithm

The type inference rules imply an algorithm, where the expression and the environment can be seen

as input, and the substitution and the type of the expression can be seen as output, as seen from

the color coding above.

Such an algorithm would probably have a type signature like:

4

inferExp :: Gamma -> Exp -> TC (Type, Subst)

However our goal isn’t just to infer the type of the expression, but to also elaborate the expression to

its explicitly-typed equivalent. In other words, we want to add typing annotations to our expressions,

meaning we must return a new expression as well as the type and substitution.

inferExp :: Gamma -> Exp -> TC (Exp, Type, Subst)

The cases for let and recfun (and letrec, in the bonus) must add a (correct) type signature to

the resultant expression, and all other cases must be sure to return a new expression consisting of

the updated subexpressions.

Because we add annotations to the expressions on the way, those type annotations wouldn’t

contain the information we get from typing successive expressions. Consider the following example:

{+ : Int→ Int→ Int, x : a} ` ( (let z = x in x) , x+1)

When typing let z = x in x, we only know that x is of type a, so we would add the type annota-

tion let z :: a = x in x. Only when we type the second expression, x+1, we know that a has

to be Int (which will be reflected in T ′). That’s why, when we’re done typing the top-level binding,

we have to traverse the whole binding again, applying the substitution to each type annotation

anywhere in the binding.

The function allTypes, defined in Syntax.hs, allows you to update each type annotation in an

expression, making it easy to perform this substitution.

5 Substitution

Substitutions are implemented as an abstract data type, defined in Subst.hs. Subst is an instance

of the Monoid type class, which is defined in the standard library as follows:

class Monoid a where

mappend :: a -> a -> a -- also written as the infix operator <>

mempty :: a

For the Subst instance, mempty corresponds to the empty substitution, and mappend is substi-

tution composition. That is, applying the substitution a <> b is the same as applying a and b

simultaneously. It should be reasonably clear that this instance obeys the monoid laws:

mempty <> x == x -- left identity

x <> mempty == x -- right identity

x <> (y <> z) == (x <> y) <> z -- associativity

It is also commutative (x <> y == y <> x) assuming that the substitutions are disjoint (i.e that

dom(x) ∩ dom(y) = ?). In the type inference algorithm, your substitutions are all applied in order

and thus should be disjoint, therefore this property should hold.

You can use this <> operator to combine multiple substitutions into your return substitution.

You can construct a singleton substitution, which replaces one variable, with the =: operator,

so the substitution ("a" =: TypeVar "b") <> ("b" =: TypeVar "c") is a substitution which

replaces a with c and b with c.

The Subst module also includes a variety of functions for running substitutions on types, quan-

tified types, and environments.

5

6 Unification

The unification algorithm is the structural matching one originally due to Robinson and discussed

in lectures:

input: two type terms t1 and t2, where forall quantified variables have been replaced by fresh,

unique variables

output: the most general unifier of t1 and t2 (if it exists). The unifier is a data structure or

function that specifies the substitutions to take place to unify t1 and t2.

6.1 Unification Cases

For t1 and t2

1. both are type variables v1 and v2:

- if v1 = v2, return the empty substitution

- otherwise, return [v1 := v2]

2. both are primitive types

- if they are the same, return the empty substitution

- otherwise, there is no unifier

3. both are product types, with t1 = (t11 ? t12), t1 = (t21 ? t22)

- compute the unifier S of t11 and t21

- compute the unifier S ′ of S t12 and S t22

- return S <> S′

4. function types and sum types (as for product types)

5. only one is a type variable v, the other an arbitrary type term t

- if v occurs in t , there is no unifier

- otherwise, return [v := t ]

6. otherwise, there is no unifier

Functions in the Data.List library are useful for implementing the occurs check.

Once you generate a unifier (also a substitution), you then need to apply that unifier to your

types, to produce the unified type.

7 Errors and Fresh Names

Thus far, the following type signature would be sufficient for implementing our type inference

function:

inferExp :: Gamma -> Exp -> (Exp, Type, Subst)

Unification is a partial function, however, so we want a principled way to handle the error cases,

rather than just bail out with error calls.

To achieve this, we’ll adjust the basic, pure signature for type inference to include the possibility

of a TypeError:

6

inferExp :: Gamma -> Exp -> Either TypeError (Exp, Type, Subst)

Even this, though, is not sufficient, as we cannot generate fresh, unique type variables for use as

unification variables:

fresh :: Type -- it is impossible for fresh to return a different

fresh = -- value each time!

To solve this problem, we could pass an (infinite) list of unique names around our program, and

fresh could simply take a name from the top of the list, and return a new list with the name

removed:

fresh :: [Id] -> ([Id],Type)

fresh (x:xs) = (xs,TypeVar x)

This is quite awkward though, as now we have to manually thread our list of identifiers throughout

the entire inference algorithm:

inferExp :: Gamma -> Exp -> [Id] -> Either TypeError ([Id], (Exp, Type, Subst))

To resolve this, we bundle both the [Id] state transformer and the Either TypeError x error

handling into one abstract type, called TC (defined in TCMonad.hs)

newtype TC a = TC ([Id] -> Either TypeError ([Id], a))

One can think of TC a abstractly as referring to a stateful action that will, if executed, return a

value of type a, or throw an exception.

As the name of the module implies, TC is a Monad, meaning that it exposes two functions

(return and >>=) as part of its interface.

return :: a -> TC a

return = ...

(>>) :: TC a -> TC b -> TC b

a >> b = a >>= const b

(>>=) :: TC a -> (a -> TC b) -> TC b

a >>= b = ...

The function return is, despite its name, just an ordinary function which lifts pure values into a TC

action that returns that value. The function (>>) (read then), is a kind of composition operator,

which produces a TC action which runs two TC actions in sequence, one after the other, returning

the value of the last one to be executed. Lastly, the function (>>=), more general than (>>), allows

the second executed action to be determined by the return value of the first.

The TCMonad.hs module also includes a few built-in actions:

typeError :: TypeError -> TC a -- throw an error

fresh :: TC Type -- return a fresh type variable

Haskell includes special syntactic sugar for monads, which allow for programming in a somewhat

imperative style. Called do notation, it is simple sugar for (>>) and (>>=).

do e -- e

do e; v -- e >> do v

do p <- e; v -- e >>= \p -> do v

do let x = y; v -- let x = y in do v

This lets us write unification and type inference quite naturally. A simple example of the use of

the TC monad is already provided to you, with the unquantify function, which takes a type with

some number of quantifiers, and replaces all quantifiers with fresh variables (very useful in the type

inference cases for variables, constructors, and primops):

7

unquantify :: QType -> TC Type

unquantify (Ty t) = return t

unquantify (Forall x t) = do x’ <- fresh

unquantify (substQType (x =:x’) t)

To run a TC action in your top level infer function, the runTC function can be used:

runTC :: TC a -> Either TypeError a

Please note: This function runs the TC action with the same source of fresh names each time!

Using it more than once in your program is not likely to give correct results.

8 Program structure

A program in MinHS may evaluate to any non-function type, including an aggregate type. This is

a valid MinHS program:

main = (1,(Inl True, False));

which can be elaborated to the following type:

main :: forall t. (Int * ((Bool + t) * Bool))

= (1,(InL True,False));

8.1 Type information

The most significant change to the language of assignment 1 is that the parser now accepts programs

missing some or all of their type information. Type declarations are no longer compulsory! Unless

you are attempting the bonus parts of the assignment, you can assume that no type information

will be provided in the program. You must reconstruct it all.

You can view the type information after your pass using --dump type-infer.

9 Implementing Type Inference

You are required to implement the function infer. Some stub code has been provided for you,

along with some type declarations, and the type signatures of useful functions you may wish to

implement. You may change any part of TyInfer.hs you wish, as long as it still provides the

function infer, of the correct type. The stub code is provided only as a hint, you are free to ignore

it.

10 Useful interfaces

You need to use environments to a limited extent. This follows the same interface for environments

you used in assignment 1, it is defined in Env.hs, and there are many examples throughout the

program.

11 Testing

Your assignments will be autotested rigorously. You are encouraged to autotest yourself. minhs comes

with a tester script, and you should add your own tests to this. Your assignment will be tested by

comparing the output of minhs --dump type-infer against the expected abstract syntax. Your

8

solution must be α-equivalent to the expected solution. It it up to you to write your own tests for

your submission.

Much like the previous assignment, you are given a suite of tests for Task 1. Unlike the

previous assignment, the tests do not specify the expected results—note the lack of .out files.

Beware: unless you add .out specifying the expected result yourself, the tests will report success

no matter what you return.

In this assignment we make no use of the later phases of the compiler.

12 Building MinHS

Building MinHS is exactly the same as in Assignment 1.

To run the type inference pass and inspect its results, for cabal users type:

$ cabal run minhs-2 -- --dump type-infer foo.mhs

Users of stack should type:

$ stack exec minhs-2 -- --dump type-infer foo.mhs

You may wish to experiment with some of the debugging options to see, for example, how your

program is parsed, and what abstract syntax is generated. Many --dump flags are provided, which

let you see the abstract syntax at various stages in the compiler.

13 Late Penalty

You may submit up to five days (120 hours) late. Each day of lateness corresponds to a 5% reduction

of your total mark. For example, if your assignment is worth 88% and you submit it two days late,

you get 78%. If you submit it more than five days late, you get 0%.

Course staff cannot grant assignment extensions—if you need an extension, you have to apply

for special consideration through the standard procedure. More information here: https://www.

student.unsw.edu.au/special-consideration

14 Plagiarism

Many students do not appear to understand what is regarded as plagiarism. This is no defense.

Before submitting any work you should read and understand the UNSW plagiarism policy https:

//student.unsw.edu.au/plagiarism.

All work submitted for assessment must be entirely your own work. We regard unacknowledged

copying of material, in whole or part, as an extremely serious offence. In this course submission

of any work derived from another person, or solely or jointly written by and or with someone else,

without clear and explicit acknowledgement, will be severely punished and may result in automatic

failure for the course and a mark of zero for the course. Note this includes including unreferenced

work from books, the internet, etc.

Do not provide or show your assessable work to any other person. Allowing another student

to copy from you will, at the very least, result in zero for that assessment. If you knowingly

provide or show your assessment work to another person for any reason, and work derived from

it is subsequently submitted you will be penalized, even if the work was submitted without your

knowledge or consent. This will apply even if your work is submitted by a third party unknown to

you. You should keep your work private until submissions have closed.

If you are unsure about whether certain activities would constitute plagiarism ask us before

engaging in them!

Type Inference for Polymorphic MinHS

Version 2.1

Marks : 17.5% of the overall mark

Due date: Sunday 20th November 2022, 11:59 PM Sydney time

Overview

In this assignment you will implement a type inference pass for MinHS. The language used in this

assignment differs from the language of the first assignment in two respects: it has a polymorphic

type system, and it has aggregate data structures.

The assignment requires you to:

Implement type synthesis for polymorphic MinHS with sum and product data types.

(0.1% bonus) adjust the type inference pass to include various simple syntax extensions from

Assignment 1.

(0.2% bonus) extend the type inference pass to elaborate mutually recursive letrec bindings.

(0.2% bonus) adjust the type inference pass to allow optional type annotations provided by

the user.

Each of these parts is explained in detail below.

The parser, and an interpreter backend, are provided for you. You do not have to change anything

in any module other than TyInfer.hs (even for the bonus parts).

Your type inference pass should annotate the abstract syntax with type information where it is

missing. The resulting abstract syntax should be fully annotated and correctly typed.

Your assignment will only be tested on correct programs, and will be judged correct if it produces

correctly typed annotated abstract syntax, up to α-renaming of type variables.

Please use the Ed forum for questions about this assignment.

Submission

Submit your (modified) TyInfer.hs using the CSE give system, by typing the command

give cs3161 TyInfer TyInfer.hs

or by using the CSE give web interface.

1

1 Task 1

Task 1 is worth 100% of the marks of this assignment. You are to implement type inference for

MinHS with aggregate data structures. The following cases must be handled:

the MinHS language of the first task of assignment 1 (without n-ary functions or letrecs, or

lists)

product types: the 0-tuple and 2-tuples.

sum types

polymorphic functions

These cases are explained in detail below. The abstract syntax defining these syntactic entities is

in Syntax.hs. You should not need to modify the abstract syntax definition in any way.

Your implementation is to follow the definition of algebraic data types found in the lecture on

data structures, and the rules defined in the lectures on type inference. Additional material can be

found in the lecture notes on polymorphism, and the reference materials. The full set of rules is

outlined in this specification.

2 Bonus Tasks

These tasks are all optional, and are worth a total of an additional 10%.

2.1 Bonus Task 1: Simple Syntax Extensions

This bonus task is worth an additional 3%. In this task, you should implement type inference for

n-ary functions, and multiple bindings in the one let expression, with the same semantics as the

extension tasks for Assignment 1.

You will need to develop the requisite extensions to the type inference algorithm yourself, but

the extensions are very similar to the existing rules.

2.2 Bonus Task 2: Recursive Bindings

This bonus task is worth an additional 3%. In this task you must implement type inference for

letrec constructs.

Each binding will require a fresh unification variable, similar to multiple-binding let expressions,

however for letrec we do not wish types to be generalised.

2.3 Bonus Task 3: User-provided type signatures

This bonus task is worth an additional 4%. In this task you are to extend the type inference pass

to accept programs containing some type information. You need to combine this with the results

of your type inference pass to produce the final type for each declaration. That is, you need to be

able to infer correct types for programs like:

main = let f :: (Int -> Int)

= recfun g x = x;

in f 2;

You must ensure that the type signatures provided are not overly general. For example, the following

program should be a type error, as the user-provided signature is too general:

main :: (forall a. a) = 3;

2

3 Algebraic Data Types

This section covers the extensions to the language of the first assignment. In all other respects

(except lists) the language remains the same, so you can consult the reference material from the

first assignment for further details on the language.

3.1 Product Types

We only have 2-tuples in MinHS, and the Unit type, which could be viewed as a 0-tuple.

Types τ → τ1 * τ2

| 1

Expressions exp → (e1, e2)

| fst e | snd e

| ()

3.2 Sum Types

Sum types in MinHS follow the presentation in the lectures.

Types τ → τ + τ

Expressions exp → Inl e1

| Inr e2

| case e of

Inl x -> e1 ;

Inr y -> e2 ;

3.3 Polymorphism

The extensions to allow polymorphism are relatively minor. Two new type forms have been intro-

duced: the TypeVar t form, and the Forall t e form.

Types τ → forall τ. ..τ..

For example, consider the following code fragment before and after type inference:

main =

let f = recfun g x = x;

in if f True

then f (Inl 1)

else f (Inr ());

main :: (Int + 1) =

let f :: forall a. (a -> a) = recfun g :: (a -> a) x = x;

in if f True

then f (Inl 1)

else f (Inr ());

3

4 Type Inference Rules

Sections coloured in blue can be viewed as inputs to the algorithm, while red text can be viewed

as outputs.

Constants and Variables

Γ ` n : Int

x : ?a1 . . . ?an.τ ∈ Γ

Γ ` x : τ [a1 := β1, . . . , an := βn] βi fresh

(unquantify in TyInfer.hs provides an easy way to implement this)

Constructors and Primops

constructorType(C) = ?a1 . . . ?an.τ

Γ ` C : τ [a1 := β1, . . . , an := βn]

primopType(o) = ?a1 . . . ?an.τ

Γ ` o : τ [a1 := β1, . . . , an := βn] βi fresh

(constructorType and primopType are defined in TyInfer.hs)

Application

TΓ ` e1 : τ1 T ′TΓ ` e2 : τ2 T ′τ1 U～ (τ2 → α)

UT ′TΓ ` Apply e1 e2 : Uα α fresh

If-Then-Else

TΓ ` e : τ τ U～ Bool T1UTΓ ` e1 : τ1 T2T1UTΓ ` e2 : τ2 T2τ1 U

′

～ τ2

U ′T2T1UTΓ ` If e e1 e2 : U ′τ2

Case

TΓ ` e : τ T1T (Γ ∪ {x : αl}) ` e1 : τl

T2T1T (Γ ∪ {y : αr}) ` e2 : τr T2T1T (αl + αr)

U～ T2T1τ UT2τl U

′

～ Uτr

U ′UT2T1TΓ ` Case e x.e1 y.e2 : U ′Uτr αl, αr fresh

Recursive Functions

T (Γ ∪ {x : α1} ∪ {f : α2}) ` e : τ Tα2 U～ Tα1 → τ

UTΓ ` Recfun f.x.e : U(Tα1 → τ) α1, α2 fresh

Let Bindings

TΓ ` e1 : τ T ′(TΓ ∪ {x : Generalise(TΓ, τ)}) ` e2 : τ ′

T ′TΓ ` Let e1 x.e2 : τ ′

TΓ ` e : τ

Γ ` Main e : Generalise(Γ, τ)

where Generalise(Γ , τ) = ? (TV (τ) \ TV (Γ )) . τ

4.1 Implementing the algorithm

The type inference rules imply an algorithm, where the expression and the environment can be seen

as input, and the substitution and the type of the expression can be seen as output, as seen from

the color coding above.

Such an algorithm would probably have a type signature like:

4

inferExp :: Gamma -> Exp -> TC (Type, Subst)

However our goal isn’t just to infer the type of the expression, but to also elaborate the expression to

its explicitly-typed equivalent. In other words, we want to add typing annotations to our expressions,

meaning we must return a new expression as well as the type and substitution.

inferExp :: Gamma -> Exp -> TC (Exp, Type, Subst)

The cases for let and recfun (and letrec, in the bonus) must add a (correct) type signature to

the resultant expression, and all other cases must be sure to return a new expression consisting of

the updated subexpressions.

Because we add annotations to the expressions on the way, those type annotations wouldn’t

contain the information we get from typing successive expressions. Consider the following example:

{+ : Int→ Int→ Int, x : a} ` ( (let z = x in x) , x+1)

When typing let z = x in x, we only know that x is of type a, so we would add the type annota-

tion let z :: a = x in x. Only when we type the second expression, x+1, we know that a has

to be Int (which will be reflected in T ′). That’s why, when we’re done typing the top-level binding,

we have to traverse the whole binding again, applying the substitution to each type annotation

anywhere in the binding.

The function allTypes, defined in Syntax.hs, allows you to update each type annotation in an

expression, making it easy to perform this substitution.

5 Substitution

Substitutions are implemented as an abstract data type, defined in Subst.hs. Subst is an instance

of the Monoid type class, which is defined in the standard library as follows:

class Monoid a where

mappend :: a -> a -> a -- also written as the infix operator <>

mempty :: a

For the Subst instance, mempty corresponds to the empty substitution, and mappend is substi-

tution composition. That is, applying the substitution a <> b is the same as applying a and b

simultaneously. It should be reasonably clear that this instance obeys the monoid laws:

mempty <> x == x -- left identity

x <> mempty == x -- right identity

x <> (y <> z) == (x <> y) <> z -- associativity

It is also commutative (x <> y == y <> x) assuming that the substitutions are disjoint (i.e that

dom(x) ∩ dom(y) = ?). In the type inference algorithm, your substitutions are all applied in order

and thus should be disjoint, therefore this property should hold.

You can use this <> operator to combine multiple substitutions into your return substitution.

You can construct a singleton substitution, which replaces one variable, with the =: operator,

so the substitution ("a" =: TypeVar "b") <> ("b" =: TypeVar "c") is a substitution which

replaces a with c and b with c.

The Subst module also includes a variety of functions for running substitutions on types, quan-

tified types, and environments.

5

6 Unification

The unification algorithm is the structural matching one originally due to Robinson and discussed

in lectures:

input: two type terms t1 and t2, where forall quantified variables have been replaced by fresh,

unique variables

output: the most general unifier of t1 and t2 (if it exists). The unifier is a data structure or

function that specifies the substitutions to take place to unify t1 and t2.

6.1 Unification Cases

For t1 and t2

1. both are type variables v1 and v2:

- if v1 = v2, return the empty substitution

- otherwise, return [v1 := v2]

2. both are primitive types

- if they are the same, return the empty substitution

- otherwise, there is no unifier

3. both are product types, with t1 = (t11 ? t12), t1 = (t21 ? t22)

- compute the unifier S of t11 and t21

- compute the unifier S ′ of S t12 and S t22

- return S <> S′

4. function types and sum types (as for product types)

5. only one is a type variable v, the other an arbitrary type term t

- if v occurs in t , there is no unifier

- otherwise, return [v := t ]

6. otherwise, there is no unifier

Functions in the Data.List library are useful for implementing the occurs check.

Once you generate a unifier (also a substitution), you then need to apply that unifier to your

types, to produce the unified type.

7 Errors and Fresh Names

Thus far, the following type signature would be sufficient for implementing our type inference

function:

inferExp :: Gamma -> Exp -> (Exp, Type, Subst)

Unification is a partial function, however, so we want a principled way to handle the error cases,

rather than just bail out with error calls.

To achieve this, we’ll adjust the basic, pure signature for type inference to include the possibility

of a TypeError:

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inferExp :: Gamma -> Exp -> Either TypeError (Exp, Type, Subst)

Even this, though, is not sufficient, as we cannot generate fresh, unique type variables for use as

unification variables:

fresh :: Type -- it is impossible for fresh to return a different

fresh = -- value each time!

To solve this problem, we could pass an (infinite) list of unique names around our program, and

fresh could simply take a name from the top of the list, and return a new list with the name

removed:

fresh :: [Id] -> ([Id],Type)

fresh (x:xs) = (xs,TypeVar x)

This is quite awkward though, as now we have to manually thread our list of identifiers throughout

the entire inference algorithm:

inferExp :: Gamma -> Exp -> [Id] -> Either TypeError ([Id], (Exp, Type, Subst))

To resolve this, we bundle both the [Id] state transformer and the Either TypeError x error

handling into one abstract type, called TC (defined in TCMonad.hs)

newtype TC a = TC ([Id] -> Either TypeError ([Id], a))

One can think of TC a abstractly as referring to a stateful action that will, if executed, return a

value of type a, or throw an exception.

As the name of the module implies, TC is a Monad, meaning that it exposes two functions

(return and >>=) as part of its interface.

return :: a -> TC a

return = ...

(>>) :: TC a -> TC b -> TC b

a >> b = a >>= const b

(>>=) :: TC a -> (a -> TC b) -> TC b

a >>= b = ...

The function return is, despite its name, just an ordinary function which lifts pure values into a TC

action that returns that value. The function (>>) (read then), is a kind of composition operator,

which produces a TC action which runs two TC actions in sequence, one after the other, returning

the value of the last one to be executed. Lastly, the function (>>=), more general than (>>), allows

the second executed action to be determined by the return value of the first.

The TCMonad.hs module also includes a few built-in actions:

typeError :: TypeError -> TC a -- throw an error

fresh :: TC Type -- return a fresh type variable

Haskell includes special syntactic sugar for monads, which allow for programming in a somewhat

imperative style. Called do notation, it is simple sugar for (>>) and (>>=).

do e -- e

do e; v -- e >> do v

do p <- e; v -- e >>= \p -> do v

do let x = y; v -- let x = y in do v

This lets us write unification and type inference quite naturally. A simple example of the use of

the TC monad is already provided to you, with the unquantify function, which takes a type with

some number of quantifiers, and replaces all quantifiers with fresh variables (very useful in the type

inference cases for variables, constructors, and primops):

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unquantify :: QType -> TC Type

unquantify (Ty t) = return t

unquantify (Forall x t) = do x’ <- fresh

unquantify (substQType (x =:x’) t)

To run a TC action in your top level infer function, the runTC function can be used:

runTC :: TC a -> Either TypeError a

Please note: This function runs the TC action with the same source of fresh names each time!

Using it more than once in your program is not likely to give correct results.

8 Program structure

A program in MinHS may evaluate to any non-function type, including an aggregate type. This is

a valid MinHS program:

main = (1,(Inl True, False));

which can be elaborated to the following type:

main :: forall t. (Int * ((Bool + t) * Bool))

= (1,(InL True,False));

8.1 Type information

The most significant change to the language of assignment 1 is that the parser now accepts programs

missing some or all of their type information. Type declarations are no longer compulsory! Unless

you are attempting the bonus parts of the assignment, you can assume that no type information

will be provided in the program. You must reconstruct it all.

You can view the type information after your pass using --dump type-infer.

9 Implementing Type Inference

You are required to implement the function infer. Some stub code has been provided for you,

along with some type declarations, and the type signatures of useful functions you may wish to

implement. You may change any part of TyInfer.hs you wish, as long as it still provides the

function infer, of the correct type. The stub code is provided only as a hint, you are free to ignore

it.

10 Useful interfaces

You need to use environments to a limited extent. This follows the same interface for environments

you used in assignment 1, it is defined in Env.hs, and there are many examples throughout the

program.

11 Testing

Your assignments will be autotested rigorously. You are encouraged to autotest yourself. minhs comes

with a tester script, and you should add your own tests to this. Your assignment will be tested by

comparing the output of minhs --dump type-infer against the expected abstract syntax. Your

8

solution must be α-equivalent to the expected solution. It it up to you to write your own tests for

your submission.

Much like the previous assignment, you are given a suite of tests for Task 1. Unlike the

previous assignment, the tests do not specify the expected results—note the lack of .out files.

Beware: unless you add .out specifying the expected result yourself, the tests will report success

no matter what you return.

In this assignment we make no use of the later phases of the compiler.

12 Building MinHS

Building MinHS is exactly the same as in Assignment 1.

To run the type inference pass and inspect its results, for cabal users type:

$ cabal run minhs-2 -- --dump type-infer foo.mhs

Users of stack should type:

$ stack exec minhs-2 -- --dump type-infer foo.mhs

You may wish to experiment with some of the debugging options to see, for example, how your

program is parsed, and what abstract syntax is generated. Many --dump flags are provided, which

let you see the abstract syntax at various stages in the compiler.

13 Late Penalty

You may submit up to five days (120 hours) late. Each day of lateness corresponds to a 5% reduction

of your total mark. For example, if your assignment is worth 88% and you submit it two days late,

you get 78%. If you submit it more than five days late, you get 0%.

Course staff cannot grant assignment extensions—if you need an extension, you have to apply

for special consideration through the standard procedure. More information here: https://www.

student.unsw.edu.au/special-consideration

14 Plagiarism

Many students do not appear to understand what is regarded as plagiarism. This is no defense.

Before submitting any work you should read and understand the UNSW plagiarism policy https:

//student.unsw.edu.au/plagiarism.

All work submitted for assessment must be entirely your own work. We regard unacknowledged

copying of material, in whole or part, as an extremely serious offence. In this course submission

of any work derived from another person, or solely or jointly written by and or with someone else,

without clear and explicit acknowledgement, will be severely punished and may result in automatic

failure for the course and a mark of zero for the course. Note this includes including unreferenced

work from books, the internet, etc.

Do not provide or show your assessable work to any other person. Allowing another student

to copy from you will, at the very least, result in zero for that assessment. If you knowingly

provide or show your assessment work to another person for any reason, and work derived from

it is subsequently submitted you will be penalized, even if the work was submitted without your

knowledge or consent. This will apply even if your work is submitted by a third party unknown to

you. You should keep your work private until submissions have closed.

If you are unsure about whether certain activities would constitute plagiarism ask us before

engaging in them!