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COMP3161 Type Inference for Polymorphic MinHS
创建日期：2024-05-16 16:21:37
标签： COMP3161
Type Inference for Polymorphic MinHS

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COMP3161/9164 22T3 Assignment 2

Type Inference for Polymorphic MinHS
Version 2.1
Marks : 17.5% of the overall mark
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%.

14 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!
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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