Learning Standard ML

COMP 105

This guide is available both in HTML and PDF.

For someone with a background in COMP 11 and COMP 15, the fastest and easiest way to learn Standard ML is to buy Ullman’s book and work through chapters 2, 3, 5, and 6. But many students choose not to buy Ullman—a move that saves money but costs time. You can recover some of the time by reading this guide; it enumerates the most important concepts, and it tells you where to find key information, not just in Ullman, but also in three other sources:

Know your sources! Mads Tofte and Bob Harper both worked with Robin Milner on the design of Standard ML, and they helped write the Definition of Standard ML. They know what they’re talking about, and they have good taste—though Tofte’s use of the ML modules is considered idiosyncratic. Norman Ramsey at least knows some functional programming. Jeff Ullman, by contrast, got his start in the theory of formal languages and parsing, then switched to databases. He may like ML, but he doesn’t understand it the way the others do.

Key concepts: Algebraic data types, case expressions, and pattern matching

Here is an excerpt from a chapter of Ramsey that is not included in the abridged edition:

S-expressions are great, but if you think carefully, you might notice that S-expressions work as a kind of high-level assembly language on top of which you craft your own data structures. ML, by contrast, offers a powerful tool for expressing data structures directly: the algebraic data type. Don’t let the mathy name worry you; algebraic data types solve real programming problems:

Recursive types and definition by choices enable you to express representations that are very similar to what you might like to define in , and that also give you the benefits of polymorphic type inference that you get with ML. Types help you write good code, and they provide documentation that is checked by the compiler. Pattern matching eliminates the name-space clutter and cognitive overhead associated with choice-identifying and part-extracting functions like null?, car, cdr, fst, and snd. Using pattern matching, you make choices and extract parts by naming the choices and parts directly. The resulting code is shorter and more perspicuous than code without pattern matching. Algebraic data types and pattern matching are ubiquitous in the ML family and in languages derived from it, including Standard ML, OCaml, HAskell, Agda, Coq/Gallina, and Idris.

When you add algebraic data types to a language, you get a new species of value and type, a new expression form for looking at the values, and a new definition form for introducing the types and values.

The new species of value is called a constructed value. A constructed value is made by applying some value constructor to zero or more other values. In the syntax, however, we don’t apply zero-argument value constructors; a zero-argument value constructor constitutes a value all by itself. As examples, in ML, [] is a value constructor for lists, and it expects no arguments, so it constitutes a constructed value all by itself. And cons (spelled ::) is also a value constructor for lists, but it expects arguments, so to make a constructed value, cons must be applied to two other values: an element and a list.

A constructed value is observed or interrogated by a new syntactic form: the case expression. The case expression does pattern matching: a pattern can match a particular value constructor, and when it does, it can name each of the values to which the constructor was applied. For example, the pattern (y :: ys) matches any cons cell, and when it matches, it binds the name y to the car and ys to the cdr. A case expression includes a sequence of choices, each of which comprises a pattern and an associated right-hand side. One reason that programmers really like case expressions is that if your choices don’t cover all possible cases, a compiler can alert you ().

Here’s an example case expression, a definition of null:

fun null xs = (case xs of [] => true
                        | y :: ys => false)

And algebraic data type and its value constructors are created by a new form of definition, which uses the keyword datatype. In a datatype definition, you can name a type whatever you want. For example, you can define a new algebraic data type that is called int, which then hides the built-in int.

Key concept: Types and type inference

The elements of ML

Expressions I: Basic expressions and their types

Ullman is the only resource that goes into simple syntax at any length. If you want more than you find below, search the Web for the “Gentle Introduction to ML.”

Expressions II: Minus signs

For reasons best known to Robin Milner, Standard ML does not use the unary minus sign used in every other known language (except APL). Instead, Standard ML uses the tilde as a minus sign, as in ~1 or ~(n+1). The tilde is in fact an ordinary function and may be treated as such.

Expressions III: Conditionals and short circuits

ML uses if e1 then e2 else e3, as well as short-circuit infix andalso, orelse (never and).

Data I: Tuples

In μML, tuples are ordinary value constructors of ordinary abstract data types (see below). But in Standard ML, they have special syntax:

Ullman pitfall: Jeff Ullman doesn’t understand how to program with tuples. His section 2.4.2 should be torn out of your book and shredded (just kidding). The use of #1 and #3 violates all established customs for writing ML code—in part because #1 and #3 are not really functions, and they don’t have types! The right way to extract an element from a tuple is by pattern matching, like this:1

  fun fst (x, _) = x
  fun snd (_, y) = y

Never write this:

  fun bogus_first  p = #1 p    (* WRONG *)
  fun bogus_second p = #2 p    (* WRONG *)

(For reasons I don’t want to discuss, but will answer in class if asked, these versions don’t even typecheck.) If your pair or tuple is not an argument to a function, use val to do the pattern matching:

  val (x, y) = lookup_pair mumble

But usually you can include matching in ordinary fun matching.

You probably won’t need to extract elements from a bigger tuple, but if you do, try

  fun third (_, _, z) = z

Any uses of #1, #2, and their friends will result in point deductions on homework.

Data II: Lists

The most common mistake I see in list-related code is to write xs = nil (wrong). Never write this. Either use null xs (that’s why null is in the initial basis) or use pattern matching.

You’ll find most of your favorite list functions in the initial basis, either defined at top level or in the List module.

Data III: Constructed values and patterns that match them

Aside from the type system, the big new thing in ML is the system of “constructed values,” which belong to algebraic data types.

Tuples and records should also be considered constructed values. In μML, tuples and records are simulated with ordinary algebraic data types. In Standard ML, tuples and records have their own syntax and their own rules, but the ideas of construction and deconstruction (pattern matching) are the same.

Lists are constructed values that are supported with extra syntactic sugar for constructing and matching lists. Some useful list patterns include these patterns, to match lists of exactly 0, 1, 2, or 3 elements:

[x, y]
[a, b, c]

You can also use the :: (cons) constructor in patterns, where it appears infix. These patterns match lists of at least 0, 1, 2, or 3 elements:

x :: xs
x1 :: x2 :: xs
a :: b :: c :: xs

Inexhaustive or redundant pattern matches

In any case expression or function definition, the patterns you provide must match all cases. If they don’t, the pattern match is considered “inexhaustive” and is rejected by the compiler.

And in any case expression or function definition, every pattern you provide must match some case. If one pattern doesn’t match any case, that pattern is considered “redundant,” and the match is rejected by the compiler.

Types I: Introduction to types

Base types int, string, char, real, bool. Type constructors like 'a list take a type parameter 'a.

Definitions I: Val bindings

ML’s val bindings resemble those from μScheme, although μScheme’s val corresponds to ML’s val rec, which is μML’s val-rec.

What we call a “definition” form is, to Standard ML, a “declaration” form.

Definitions II: Semicolons

It’s a sad fact that if you’re working interactively with an ML compiler, the compiler can’t tell where a definition ends. You have to mark the end with a semicolon. But such a semicolon should never appear in your code. Ullman’s book is full of unnecessary semicolons, and you must learn to ignore him. Emulate the style in Ramsey’s book, which has no unnecessary semicolons. Use a semicolon only to sequence effects in imperative code.

Definitions III: Function definitions

As in μScheme and μML, functions can be defined using lambda with val or val rec. But it is more idiomatic to use fun, which is the analog of the define found in μScheme and μML.

Definitions IV: Clausal (function) definitions

Standard ML’s fun also provides clausal definitions, which in μML are written define*. These definitions look a lot like algebraic laws.

Expressions IV: ML’s let

ML’s let most closely resembles Scheme’s let*, but instead of a sequence of name/expression pairs, it uses a sequence of definition forms. The effect of letrec can be approximated by using a fun definition form with keyword and. Standard ML has nothing corresponding to Scheme’s let form.

Expressions V: ML’s lambda

As noted above, ML’s lambda expressions are written fn (x1, , xn) => e.

Expressions VI: Infix operators and precedence

The initial basis of Standard ML defines the following names as infix identifiers:

  infix  7  * / div mod
  infix  6  + - ^
  infixr 5  :: @
  infix  4  = <> > >= < <=
  infix  3  := o
  infix  0  before

The arithmetic you know, although you may not know that / is for floating point; div and mod are for integers. Here are the others:

Function application has higher precedence than any infix operator. That means a function application underneath an infix operator should never be parenthesized!

Expressions VII: Infix operators as functions

The mechanism that ML uses for infix operators is very different from what you are used to from C and C++.

For details, see

Expressions VIII: Parentheses

It’s easy to be confused about when you need parentheses. Here’s a checklist to tell you when to use parentheses around an expression or a pattern:

  1. Is it an argument to a (possibly Curried) function, and if so, is it more than a single token?
  2. Is it an infix expression that has to be parenthesized because the precedence of another infix operator would do the wrong thing otherwise?
  3. Are you forming a tuple?
  4. Are you parenthesizing an expression involving fn, case, or handle?
  5. Are you parenthesizing an infix operator marked with op?

If the answer to any of these questions is yes, use parentheses. Otherwise, you almost certainly don’t need them—so get rid of them!


Types II: Polymorphic functions

In ML, as in Scheme, you can write polymorphic functions simply by writing functions that are agnostic about some aspects of their arguments. The difference is that in ML, type system infers at compile time the knowledge that the function is polymorphic. You can probably learn all you need by feeding some of your μScheme code to the nano-ML (nml) or μML (uml) interpreters; you can identify a polymorphic function by the forall in the type. (Most unfortunately, Standard ML omits the forall from the type. You’re supposed to imagine it.)

Curried functions

What we call a “partially applied” function, Ullman calls “partially instantiated.” (He’s thinking of a substitution model. Bad Ullman.) There are no new concepts here, but the concrete syntax is radically different from what you’re used to in μScheme.


ML exceptions behave a lot like the Hanson Except_T you may have seen in COMP 40, and somewhat like exceptions in C++ or Java.

Types III: Type abbreviations

Type abbreviations are a leading cause of confusion for beginning ML programmers. A type abbreviation, which begins with the keyword type, creates a new name for an old type. But in its error messages, the compiler may not honor the abbreviation—it may insist on referring to the old type instead.

If you’re asked to “define a type,” you have to decide if you want a type abbreviation with type, or whether you want a “datatype definition” with datatype. Because both have similar effects on the type environment, both count as “define a type.”

A type abbreviation can take type parameters. A type parameter is identified by a name that begins with a tick mark, and the names traditionally used are 'a, 'b, 'c, and so on. On the topic of type abbreviations with type parameters, both Harper and Tofte are unaccountably silent. Ullman at least gives a sketch in section 6.1.3. For an example, I recommend the definition of type env in Ramsey, section 5.1.

Data IV: Datatype definitions

A datatype definition creates a brand new type, which is distinct from any other type—even one that has the same name. If you create multiple types with the same name, you will become confused.

Type pitfall I: “Equality types” (two tick marks)

Some type variables begin with two tick marks, as in

- fun isNothing v = (v = NONE);
> val ''a isNothing = fn : ''a option -> bool

The type variable ''a can be instantiated only by an “equality type.” A type variable like this, called an “equality type variable,” is almost always a programming mistake. The code above should have used pattern matching (compare Ullman, page 215).

For details:

Primitive types exn and real (floating-point numbers) are not equality types.

Type pitfall II: Value polymorphism

In Standard ML, no function is ever trusted to create a polymorphic value. Only value constructors can create polymorphic values. If you violate this so-called “value restriction,” you get a message like this one:

- val empty = rev [];
! Warning: Value polymorphism:
! Free type variable(s) at top level in value identifier empty
> val empty = [] : 'a list

For details:

Basis I: The option type

Let’s suppose you want to represent a value, except the value might not actually be known. For example, I could represent a grade on a homework by an integer, except if a grade hasn’t been submitted. Or the contents of a square on a chessboard is a piece, except the square might be empty. This problem comes up so often that the initial basis for ML has a special type constructor called option, which lets you handle it. The definition of option is

  datatype 'a option = NONE | SOME of 'a

and it is already defined when you start the interactive system. You need not and should not define it yourself. As in a type abbreviation, the type parameter 'a stands for an unknown type—you can substitute any type for the type variable 'a.


Here are some more examples:

- datatype chesspiece = K | Q | R | N | B | P
- type square = chesspiece option
- val empty : square = NONE
- val lower_left : square = SOME R
- fun play piece = SOME piece : square;
> val play = fn : chesspiece -> chesspiece option

- SOME true; 
> val it = SOME true : bool option
- SOME 37;
> val it = SOME 37 : int option

- SOME "fish" = SOME "fowl";
> val it = false : bool
- SOME "fish" = NONE;
> val it = false : bool
- "fish" = NONE;
! Toplevel input:
! "fish" = NONE;
!          ^^^^
! Type clash: expression of type
!   'a option
! cannot be made to have type
!   string

Data V: Record types, values, expressions, and patterns

In addition to tuples, Standard ML has records with named fields. A record is notated by a set of key-value pairs, separated by commas, and enclosed in curly braces. The order of the pairs doesn’t matter. Unfortunately, records come with some special rules and special syntax that can cause pain for beginners.

What’s wrong with the sharp notation? In brief, #name is a piece of syntax—it’s not a function, and it doesn’t have a unique type. What’s wrong with ellipsis patterns? Same thing: an ellipsis pattern doesn’t have a unique type.

Basis II: Access to functions defined in modules

Standard ML includes a sophisticated module language—one of the most expressive module languages ever designed. But at least to start, you’ll use modules in a very stylized way: by selecting components from modules in the initial basis. Such selection uses “dot notation,” with the name of the module followed by the name of a component. Examples include Int.toString and List.filter.

In section 8.2.4, Ullman shows that you can get access to the contents of a module by opening the module, as in open TextIO. Never do this—it is bad enough to open structures in the standard basis, but if you open other structures, your code will be hopelessly difficult to maintain. Instead, abbreviate structure names as needed. For example, after structure T = TextIO, you can use T.openIn, etc., without (much) danger of confusion.

Basis III: Getting to know the Standard Basis

The initial basis of Standard ML is called the “Standard Basis,” or sometimes the “Standard Basis Library.” Get to know it, and use it when you can.

Modules you can learn easily include List, Option, ListPair, Vector, and Array. You may also have some use for TextIO.

Moscow ML ships with an extended version of the standard basis library. Tell Moscow ML help "lib";, and you’ll see what’s there.

ledit mosml -P full

as your interactive top-level loop, it will automatically load almost everything you might want from the standard basis.

Basis IV: Vectors

Although Ullman describes the mutable Array structure in Chapter 7, he doesn’t cover the immutable Vector structure except for a couple of pages deep in Chapter 9. Like an array, a vector offers constant-time access to an array of elements, but a vector is not mutable. Because of its immutability, Vector is often preferred. It is especially flexible when initialized with Vector.tabulate.

The functions to start with include Vector.tabulate, Vector.fromList, Vector.length, and Vector.sub. The Vector structure also includes variations on app, map, foldl, foldr, find, exists, and all.

Unit Testing

Standard ML does not come with built-in unit-testing support. For COMP 105, we have created a special module called Unit, which exports functions that can be used for unit testing. These and related functions are shown in the table.

Functions useful for unit testing
Function Description
An adequate substitute for check-expect. Requires a function to convert result to a string. And because an ML function does not know where in the source coded its definition is, checkExpectWith also requires a test name, which should uniquely identify the test.
Unit.checkAssert An adequate substitute for check-assert. Requires a test name.
Unit.checkExnWith An adequate substitute for check-error. Used when you expect evaluating an expression to raise an exception. Also requires a name and a string-conversion function.
A refined version of Unit.checkExnWith that enables you to control which exceptions are considered to pass the test.
Unit.report Prints a report of testing outcomes.
Prints a report only if some test fails.
Int.toString Convert an integer to a string. Sometimes useful for passing to checkExpectWith, checkExnWith, or checkExnSatisfiesWith.
Bool.toString Convert a Boolean to a string.
Char.toString Convert a character to a string.
Unit.listString A higher-order function. Given a function that converts a value to a string, returns a function that converts a list of values to a string. Useful for passing to checkExpectWith and friends.
Unit.pairString Another higher-order function. Given two string-conversion functions, returns a function that converts a pair of values to a string.

Example unit tests

You write a unit test in the form of a val definition with the empty tuple as the pattern. Here are two examples, only one of which passes:

val () =
    Unit.checkExpectWith Int.toString "2 is third"
    (fn () => List.nth ([1, 2, 3], 2)) 

val () =      (* this test fails *)
    Unit.checkExpectWith Bool.toString "2 is false"
    (fn () => List.nth ([true, false, true], 2)) 

The difference between this module and the check-expect you are used to is that check-expect takes just an expression, but Unit.checkExpectWith requires that expression be wrapped in a function of no arguments. So if you are expecting expression e to evaluated to value v, you must write

Unit.checkExpectWith(fn () => e) v

Functions Unit.checkAssert and Unit.checkExnWith work the same way. Here are two more examples, both of which pass:

val () =
  Unit.checkAssert "has positive"
  (fn () => List.exists (fn n => n > 0) [~1, 0, 1])

val () =
  Unit.checkExnWith (Unit.listString Int.toString)
   "3rd element of empty list"
  (fn () => List.nth ([], 3))

Bureaucracy of getting our module (Moscow ML)

To use the unit tests, you must have a compiled version of the Unit module. You may use ours, or you may compile it yourself.

The batch compiler mosmlc should now be able to compile your code using Unit:

$ mosmlc -toplevel -I /comp/105/lib -c warmup.sml

At this stage, you can load your code into the interactive system:

$ mosml -I /comp/105/lib -P full
Moscow ML version 2.10-3 (Tufts University, April 2012)
Enter `quit();' to quit.
- load "warmup";
> val it = () : unit
- Unit.report();
Both internal Unit tests passed.
> val it = () : unit

If you prefer to bring your source code directly into Moscow ML, without compiling first, you can call ML’s use function, but you will need to load the Unit module first:

$ mosml -P full -I /comp/105/lib
Moscow ML version 2.10-3 (Tufts University, April 2012)
Enter `quit();' to quit.
- load "Unit";
> val it = () : unit
- use "warmup.sml";
[opening file "warmup.sml"]
[closing file "warmup.sml"]
> val it = () : unit

If you forget this step, you’ll get an error message:

nr@homedog /tmp> mosml -P full -I /comp/105/lib
Moscow ML version 2.10-3 (Tufts University, April 2012)
Enter `quit();' to quit.
- use "warmup.sml";
[opening file "warmup.sml"]
File "warmup.sml", line 2, characters 6-22:
!       Unit.checkAssert "has positive"
!       ^^^^^^^^^^^^^^^^
! Cannot access unit Unit before it has been loaded.
[closing file "warmup.sml"]

Just load "Unit"; and you’ll be on your way.

Bureacracy of getting our module (MLton)

Compiling multiple modules in MLton is a hassle: you have to have an “MLB” file that lists all the modules you want, in dependency order, plus the version of the basis you want. This is a hassle. To get the unit module, you include /comp/105/lib/unit.mlb in your own MLB file. Where you need unit tests for the homework, we’ll provide an MLB file you can use.

Documentation of all the functions

Every function in the Unit module is declared with a name and a type. These declaration forms, which are shown below, are part of ML’s modules system. Each declaration is followed by some informal English. To refer to the function from outside the Unit module, in your code, you need “Unit dot function-name.”

Testing functions

The closest analog to check-expect is the function Unit.checkExpectWith:

val checkExpectWith : (''a -> string) ->
                      string ->
                      (unit -> ''a) ->
                      ''a ->

Calling checkExpectWith show name test result evaluates test(), applying test to the empty tuple. If the result equals result, the test passes; otherwise it fails. If the test fails, name is used to identify the failing test.

The closest analog to check-assert is Unit.checkAssert:

val checkAssert : string -> (unit -> bool) -> unit

Calling checkAssert name test evaluates test(), and if the result is true, the test passes; otherwise it fails. If the test fails, name is used to identify the failing test.

The closest analog to check-error is Unit.checkExnWith:

val checkExnWith : ('a -> string) ->
                   string ->
                   (unit -> 'a) ->

Calling checkExn show name test evaluates test(), and if the evaluation raises an exception, the test passes. If test() returns a value, the test fails. If the test fails, name is used to identify the failing test.

The three functions above will serve you in most situations, but if you want to be sure you know what exception is raised, use this one:

val checkExnSatisfiesWith :
                     ('a -> string) ->
                     string ->
                     (unit -> 'a) ->
                     (string * (exn -> bool)) ->


checkExnSatisfiesWith show name test (ename, pred)

evaluates test(), and if the evaluation raises an exception e, the test passes, provided pred e is true. If test() raises a non-satisfying exception, or if test() returns a value, the test fails, complaining that the exception named ename was expected. If the test fails, name is used to identify the failing test.

Reporting functions

Use these functions to confirm that all your unit tests pass.

val report : unit -> unit

If any tests were run, report, on standard output, how many tests were run in total and how many passed.

val reportWhenFailures : unit -> unit

If any tests failed, report, on standard output, how many tests were run in total and how many passed.

String-conversion functions

Use these functions to create string-conversion functions to pass to checkExpectWith and checkExnWith.

val listString : ('a -> string) ->
                 ('a list -> string)

If showA converts a value in A to a string, then listString showA returns a function converts a value in “list of A” to a string.

val pairString : ('a -> string) ->
                 ('b -> string) ->
                 ('a * 'b -> string)

If showA converts a value in A to a string, and showB converts a value in B to a string, pairString showA showB returns a function that converts a pair of type A × B to a string.

val showNothing : 'a -> string

If you need to convert a value of type A to a string, but you don’t have a converter—and you don’t want to write one—you can use showNothing with any type A. Function showNothing converts every value to the string "a value".

An example with a user-defined type

Here is an example of using checkExpectWith:

fun colorString LAVENDER = "LAVENDER"
  | colorString BURGUNDY = "BURGUNDY"
  | colorString VIRIDIAN = "VIRIDIAN"
val () = (* this test fails *)
  Unit.checkExpectWith colorString
  "least favorite color"
  (fn () => List.last favorites) 

This test fails with this error message:

In test ‘least favorite color’, expected value LAVENDER but got VIRIDIAN

Here is an example of using checkExnWith:

val () = (* this test passes *)
  Unit.checkExnWith colorString 
  "last of empty string"
  (fn () => List.last [])     

ML Modules

General information on modules

If you want to review what you heard in class,

Signatures, Structures, and Ascription

The foundations are interfaces, implementations, and matching one to the other—or in the gratuitously mathematical lingo of Standard ML: signatures, structures, and ascription.

Signature refinement or specialization

Quite often, we need to specify the identity of an abstract type after the fact—usually when writing the result signature of a functor. This is done using the where type syntax.

Modules can nest

An excellent aspect of Standard ML’s design is that structures can contain other structures. That’s really all you need to know, but if you want to understand why this feature is valuable, there are plenty of good examples in Harper, chapter 21.


Paraphrased from Harper: an ML functor is a function that operates on the module level. Its formal parameters, if any, are specified by a sequence of declarations, and its actual parameters are given by a sequence of definitions. Its result is a structure. Functors enable a variety of styles of programming, but in 105, we focus on the functor as a mechanism for code reuse.

There is a dirty secret: there is a second mechanism for specifying parameter(s) to a functor, which is to package everything up in a single structure. I much prefer Harper’s “sequence of declarations/definitions” view, and I don’t know why we are stuck with two mechanisms. But you’ll see the second mechanisms from time to time.

Sharing constraints

Don’t ask. In COMP 105 we do our best to avoid putting you in situations where you need sharing constraints. But if you trip over them, Harper’s Chapter 22 is the best source. Follow up with sections 23.2 and 23.3, in which Harper explains when sharing constraints are needed and why they are better than the alternatives. They are also mentioned by Tofte in section 28.

Abstract data types

Harper completely understands the importance of data abstraction and the use of ML modules to enforce it.

  1. The fun is function-definition syntax, like μScheme’s define. It is described below.