Implementing nML: Hindley-Milner Type Inference

Due Friday, April 10 at 5:59PM.

In this assignment you will implement Hindley-Milner type inference, which represents the current ``best practice'' for flexible static typing. The assignment has two purposes:

Complete Exercises S and T below, and from Chapter 7 in Ramsey, complete Exercises 1, 2, 15, 16, and  17.

Getting the code

To get the code,
  git clone
The code you need is in bare/nml/ml.sml.

Two exercises to do by yourself (10%)

Complete Exercises 1  and  2 on page 346 of Ramsey. These exercises explore some implications of type inference.
In your answer to question 1, do not use letrec.
Question 2 in the text is worded in a somewhat confusing way. To clarify: the typing rules of nML permit the declaration (val-rec n n). To preserve soundness, however, the declaration is rejected in evaldef. If the declaration weren't rejected, what type scheme σ would be inferred for n?
The answers to both exercises should go into file 1-2.nml; your answer to Exercise 2 should appear in a comment.

Five exercises to do with a partner (90%)

Complete Exercises 15, 16 and 17 from pages 347–350 of Ramsey, and the two exercises S and T below.
For the coding exercises you'll be modifying the interpreter in build-prove-compare/bare/nml/ml.sml.

S. Test cases for the solver.
Submit three test cases for the constraint solver. At least two of these test cases should be constraints that have no solution. Assuming that we provide a function constraintTest : con -> answer, put your test cases in file stest.sml as three successive calls to constraintTest. Do not define constraintTest yourself.

Here is a sample stest.sml file:

val _ = constraintTest (TYVAR "a" ~ TYVAR "b")
val _ = constraintTest (CONAPP (TYCON "list", [TYVAR "a"]) ~ TYCON "int")
val _ = constraintTest (TYCON "bool" ~ TYCON "int")
Naturally, you will supply your own test cases.

T. Test cases for type inference.
Submit three test cases for type inference. At least two of these test cases should be for terms that fail to type check. Each test case should be a definition written in nML. Put your test cases in a file ttest.nml. Here is a sample ttest.nml file:

(val weird (lambda (x y z) (cons x y z)))
(+ 1 #t)
(lambda (x) (cons x x))
Naturally, you will supply your own test cases.

For the remaining exercises, here are some additional remarks and suggestions.

This is one assignment where it pays to run a lot of tests, of both good and bad definitions. The most effective test of your algorithm is not that it properly assign types to correct terms, but that it reject ill-typed terms. This assignment is your best chance to earn the large bonuses available by finding bugs in the instructor's code. I have posted a functional topological sort that makes an interesting test case.

Incidentally, if you call your interpreter ml.sml, you can build a standalone version in a.out by running mosmlc ml.sml or a faster version in ml by running mlton -output a.out ml.sml.

Hints, guidelines, and test code

To help you with the solver, once you have implemented solve, the following code redefines solve into a version that checks itself for sanity (ie, idempotence). It is a good idea to check that the substitution returned by your solver is idempotent before using it in your type inferencer.
fun isStandard pairs =
    let fun distinct a' (a, tau) = a <> a' andalso not (member a' (freetyvars tau))
        fun good (prev', (a, tau)::next) =
              List.all (distinct a) prev' andalso List.all (distinct a) next
              andalso good ((a, tau)::prev', next)
          | good (_, []) = true
    in  good ([], pairs)

val solve =
    fn c => let val theta = solve c
            in  if isStandard theta then theta
                else raise BugInTypeInference "non-standard substitution"
The type-inference code will be easier to write with the aid of a summary of the nano-ML type system which I have placed online. With your solver in place, the type inference should be straightforward, with two exceptions: let and letrec. You can emulate the implementations for val and val-rec, but you must split the constraint. The splitting is covered in detail in the book, on pages 324–327.

Extra Credit

For extra credit, you may complete any of the following:
  • Mutation, as in Exercise 21(a)(b) and possibly (c)
  • Explicit types, as in Exercise 22
  • Better error messages, as in Exercise 18(a)(b) and possibly (c)
  • Tuples, as in Exercise 19
  • Generative types, as in Exercise 20
  • Of these exercises the most interesting are probably Mutation (easy) and Explicit types (not easy).


    The course interpreter is located in /comp/105/bin/nml. If your interpreter can process the initial basis and infer correct types, you are doing OK.

    The real test of your interpreter is that it should reject incorrect definitions. You should prepare a dozen or so definitions that should not type check, and make sure they don't. For example:

    (val bad (lambda (x) (cons x x)))
    (val bad (lambda (x) (cdr (pair x x))))
    Pick your toughest three test cases to submit for Exercise T.

    Avoid common mistakes

    Here some common mistakes:

    There are also some common assumptions which are mistaken:

    What to submit

    For your solo work, run submit105-ml-inf-solo to submit file 1-2.nml.

    For your work with a partner, run submit105-ml-inf-pair to submit these files:

    Your solutions are going to be evaluated automatically. We must be able to compile your solution in Moscow ML by typing, e.g.,

    mosmlc ml.sml
    If there are errors or warnings in this step, your work will earn No Credit for functional correctness.

    How your work will be evaluated

    We will focus most of our evaluation on your constraint solving and type inference.

    Exemplary Satisfactory Must improve

    • The code has no offside violations.

    Or, the code has just a couple of minor offside violations.

    • Indentation is consistent everywhere.

    • The submission has no bracket faults.

    • The submission has a few minor bracket faults.

    Or, the submission has no bracketed names, but a few bracketed conditions or other faults.

    • The code has several offside violations, but course staff can follow what's going on without difficulty.

    • In one or two places, code is not indented in the same way as structurally similar code elsewhere.

    • The submission has some redundant parentheses around function applications that are under infix operators (not checked by the bracketing tool)

    Or, the submission contains a handful of bracketing faults.

    Or, the submission contains more than a handful of bracketing faults, but just a few bracketed names or conditions.

    • Offside violations make it hard for course staff to follow the code.

    • The code is not indented consistently.

    • The submission contains more than a handful of parenthesized names as in (x)

    • The submission contains more than a handful of parenthesized if conditions.


    • Type variables have names beginning with a; types have names beginning with t or tau; constraints have names beginning with c; substitutions have names beginning with theta; lists of things have names that begin conventionally and end in s.

    • Types, type variables, constraints, and substitutions mostly respect conventions, but there are some names like x or l that aren't part of the typical convention.

    • Some names misuse standard conventions; for example, in some places, a type variable might have a name beginning with t, leading a careless reader to confuse it with a type.


    • The nine cases of simple type equality are handled by these five patterns: TYVAR/any, any/TYVAR, CONAPP/CONAPP, TYCON/TYCON, other.

    • The code for solving α ∼ τ has exactly three cases.

    • The constraint solver is implemented using an appropriate set of helper functions, each of which has a good name and a clear contract.

    • Type inference for list literals has no redundant case analysis.

    • Type inference for expressions has no redundant case analysis.

    • In the code for type inference, course staff see how each part of the code is necessary to implement the algorithm correctly.

    • Wherever possible appropriate, submission uses map, filter, foldr, and exists, either from List or from ListPair

    • The nine cases are handled by nine patterns: one for each pair of value constructors for ty

    • The code for α ∼ τ has more than three cases, but the nontrivial cases all look different.

    • The constraint solver is implemented using too many helper functions, but each one has a good name and a clear contract.

    • The constraint solver is implemented using too few helper functions, and the course staff has some trouble understanding the solver.

    • Type inference for list literals has one redundant case analysis.

    • Type inference for expressions has one redundant case analysis.

    • In some parts of the code for type inference, course staff see some code that they believe is more complex than is required by the typing rules.

    • Submission sometimes uses a fold where map, filter, or exists could be used.

    • The case analysis for a simple type equality does not have either of the two structures on the left.

    • The code for α ∼ τ has more than three cases, and different nontrivial cases share duplicate or near-duplicate code.

    • Course staff cannot identify the role of helper functions; course staff can't identify contracts and can't infer contracts from names.

    • Type inference for list literals has more than one redundant case analysis.

    • Type inference for expressions has more than one redundant case analysis.

    • Course staff believe that the code is significantly more complex than what is required to implement the typing rules.

    • Submission includes one or more recursive functions that could have been written without recursion by using map, filter, List.exists, or a ListPair function.