Notes on ML


ML is case-sensitive.

ML is interactive; you start it up, then you type expressions (at the prompt) to be evaluated. Every expression ends with a semicolon. ML responds with both the value and the type of the result. Example:

- 3 * 4;
> val it = 12 : int

In this example, "it" stands for the English word "it." You can read this response as: Its value is 12, and its type is int.

To define a value, use the form:

val pi = 3.1416;

ML is an expression-oriented language, not a statement-oriented language. That is, everything in it is considered to be an expression and to have a value. A few expressions that are executed for their side effects rather than their value (mostly output expressions) return the unit, (), as their value&emdash;this is like void in C or Java.

To define a function, use the form:

fun average (x, y) = (x + y) / 2.0;

ML is strongly typed; however, it almost always figures out the types for itself. If you need to help it along, you can specify the type of any identifier by following it by a colon and the name of the type. For example, in the function

fun max (x, y) = if x > y then x else y;

the variables x and y could be int, real, char, or string; if you don't specify, ML assumes that they are int. To define this function to use char values, give ML a hint by attaching :char to any one of the variables, anyplace in the function. For example,

fun max (x:char, y) = if x > y then x else y;

To execute the expressions in file myFile (usually definitions that you are testing), use

use "myFile";

To load a library unit, such as Int, use

load "Int";

Once loaded, the functions in the library unit can be used, but you have to prefix them with the name of the library and a dot, as in Int.max (5, 3); To "open" a library unit means to make the functions in it available without prefixing them; open Int like this:

open Int;

(* This is a comment (* and comments may be nested. *) *)



There are two kinds of identifiers.

Alphanumeric identifiers beginning with a ' (single quote) are used only for type identifiers. Often ML will use 'a to indicate a variable (unknown or arbitrary) type. For example, 'a list means a list whose elements are of type 'a.

The variable _ (an underscore all by itself) is used as a "wildcard" or "don't care" variable.



There are several primitive types in ML; the following table gives the most important ones.

Primitive type




0, 5, 42, ~17, 0x00FF

~ is used for unary minus,
0x starts a hexadecimal number


0.0, ~5.3, 1.7e14, 1e~10

Can't start or end with a decimal point


true, false

These are the only boolean values.


"", "One\nTwo"

"\n" is newline,
"\t" is tab,
"\\" is backslash


#"a", #"\n"

# before a string of length 1



This is the only value of this type, and is often used to mean "no value" (much like void in C).

There are two special real constants in ML:

There are three families of constructed types in ML: lists, tuples, and functions. Functions are first-class objects: they can be created, manipulated, passed as parameters, and otherwise treated like other kinds of values. (However, when ML prints the result of an expression, and that result is a function, ML doesn't print out the entire function; it just prints the word fn.) Every function takes exactly one parameter as input and returns one value as its result; however, that parameter and that result may each be of a constructed type, such as a tuple.

The following table gives a few examples of constructed types. Pay special attention to the second column, which shows how ML expresses type information.

Example expression

Expression type


[2, 3, 5, 7, 11]

int list

Lists may be of any length, but all elements must be of the same type.


'a list

The empty list can be represented by [] or by nil. The type of this list might be unknown.

(5, "hello", ~16)

int * string * int

The type of a tuple depends on its length and the types in each position.

("abc", [1, 2, 3])

string * int list

Tuples can contain lists, and vice versa.



A tuple with one element is the same as that one element.

fun double x = 2.0 * x;

real -> real

All functions take exactly one parameter.

fun sum (x, y) = x + y;

int * int -> int

In this example the parameter is a tuple.

fun hi () =
  print "hello\n";

unit -> unit

In this example the parameter is the "unit," and so is the result.

(double, [sum])

(real -> real) *
(int * int -> int) list

Functions are values, and can be put into lists and tuples.


Built-In Functions

Standard boolean operators:




not : bool -> bool

not true, not (i = j)

(Prefix) Unary negation.

andalso : bool * bool -> bool

(i = j) andalso (j = k)

(Infix, left associative) Conjunction. "and" is a keyword with a different meaning.

orelse : bool * bool -> bool

(i = j) orelse (j = k)

(Infix, left associative) Disjunction.


Standard arithmetic operators:




~ : int -> int
~ : real -> real

~5, ~limit
~1e10, ~average

(Prefix) Unary negation.

* : int * int -> int
* : real * real -> real

2 * limit,
3.1416 * r * r

(Infix, left associative) Multiplication; operands and result are all of same type.

/ : real * real -> real

7.0 / 3.5, score / average

(Infix, left associative) Division of real numbers.

div : int * int -> int

limit div 2

(Infix, left associative) Integer division with truncation.

mod : int * int -> int

limit mod 2

(Infix, left associative) Modulus.

+ : int * int -> int
+ : real * real -> real

2 + 2, limit + 1
score + 1.0

(Infix, left associative) Addition of reals or of integers.

- : int * int -> int
- : real * real -> real

2 - 2, limit - 1
score - 1.0

(Infix, left associative) Subtraction of reals or of integers.





real : int -> real

real i, real (limit)

Convert integer to real.

round : real -> int

round (average)

Numbers ending in .5 are rounded to an even number.

trunc : real -> int

trunc average

Fractional part is discarded.

floor : real -> int

floor average

Largest integer not greater than argument.

ceil : real -> int

ceil (average + 3.0)

Smallest integer not less than argument.

ord : char -> int

ord #"a"

ASCII value of character.

chr : int -> char

chr 97

Character corresponding to ASCII value; argument must be in range 0..255.

str : char -> string

str initialLetter

Convert single character to string of length 1.





< : 'a * 'a -> bool

i < 0

Less than. a' can be int, real, char, or string.

<= : 'a * 'a -> bool

x <= 0.0

Less than or equal to. a' can be int, real, char, or string.

= : 'a * 'a -> bool

s = "abc"

Equals. a' can be int, char, or string, but not real.

<> : 'a * 'a -> bool

ch <> #"\n"

Compares two ints for inequality. a' can be int, char, or string, but not real.

>= : 'a * 'a -> bool

i >= j

Greater than or equal to. a' can be int, real, char, or string.

> : 'a * 'a -> bool

x > y

Greater than. a' can be int, real, char, or string.

Operations with strings

The operators <  <=  =  <>  >=  > can be applied to strings for lexical comparisons.




^ : string * string -> string

"Hello, " ^ name

Infix concatenation of two strings.

concat : string list -> string

concat ["ab", "c", "de"]

String concatenation.

size : string -> int

size "hello"

Number of characters in string.

explode : string -> char list

explode inputString

Convert string to list of characters.

implode : char list -> string

implode [#"h", #"i"]

Convert list of characters to string.

Int.toString : int -> string

print (Int.toString 5);

Convert int to string. Load Int library before using.

Real.toString : real -> string

print (Real.toString 3.1416);

Convert real to string. Load Real library before using.

Bool.toString : bool -> string

print (Int.toString (5 < 3));

Convert bool to string. Load Bool library before using.

Char.toString : char -> string

print (Char.toString #"a");

Same as str. On my system, appears to be pre-loaded.

Operations on characters

The operators <  <=  =  <>  >=  > can be applied to characters.

Functions that begin with Char. are in the Char library package. Enter load "Char"; to load this library. Enter open Char; to use these functions without typing Char. each time (open will also display a list of the functions thus made available).

Char.contains and Char.notContains use currying: Char.contains "aeiou" returns a function of type char -> bool that will return true if given a vowel as an argument. For example, the following two expressions are equivalent (and both return true):

Char.contains "aeiou" #"e";
(Char.contains "aeiou") #"e";



Char.succ : char -> char

Get the next character in the ASCII sequence; same as chr (ord #"a" + 1).

Char.pred : char -> char

Get the previous character in the ASCII sequence; same as chr (ord #"a" - 1).

Char.toUpper : char -> char

Given a lowercase letter, return the corresponding capital letter. Given any other character, return that same character.

Char.toLower : char -> char

Given a capital letter, return the corresponding lowercase letter. Given any other character, return that same character.

Char.isAlpha : char -> bool

Test if character is alphabetic.

Char.isUpper : char -> bool

Test if character is a capital letter.

Char.isLower : char -> bool

Test if character is a lowercase letter.

Char.isDigit : char -> bool

Test if character is a digit.

Char.isHexDigit : char -> bool

Test if character is a hexadecimal digit. Hex digits A thru F may be uppercase or lowercase.

Char.isAlphaNum : char -> bool

Test if character is alphabetic or numeric.

Char.isAscii : char -> bool

Test if character is a seven-bit character.

Char.isSpace : char -> bool

Test if character is whitespace (blank, tab, newline, etc.)

Char.isPrint : char -> bool

Test if character is printable (including whitespace).

Char.isGraph : char -> bool

Test if character is printable and not whitespace.

Char.contains : string -> char -> bool

Test if the string contains the given character.

Char.notContains : string -> char -> bool

Test if the string does not contain the given character.

Operations on lists

A list is a set of elements, all of the same type, enclosed in brackets and separated by commas. Example: ["hello", "bonjour", "guten Tag"]. The type of this example is string list.

The empty list is represented by [] or by nil.




hd : 'a list -> 'a

hd [3, 5, 7]

The "head" of a list is its first element. Same as car in Scheme.

tl : 'a list -> 'a list

tl [3, 5, 7]

The "tail" of a list is the list with its first element removed. Same as cdr in Scheme.

:: : 'a * 'a list -> 'a list

1 :: [3, 5, 7]
1 :: 3 :: 5 :: 7 :: nil

(Infix, right associative) Inserts an element as the new head of a list. Same as cons in Scheme.

@ : 'a list * 'a list -> 'a list

[1, 3, 5] @ [7, 9]

(Infix, right associative) Append two lists. Same as append in Scheme.

explode : string -> char list

explode inputString

Convert string to list of characters.

implode : char list -> string

implode [#"h", #"i"]

Convert list of characters to string.

Operations on tuples

If T is a tuple of three elements, then #1(T) is the first element, #2(T) is the second element, and #3(T) is the third element. #4(T) would be an error. #n(T) is illegal syntax.


Functions with side effects

The most useful functions with side effects are the I/O functions. Since ML automatically prints the values of expressions entered at the prompt, you can often avoid doing any explicit I/O.

The most commonly used output function is print, which prints only strings. If you want to print something other than a string, it is your responsibility to convert it to a string. For example, if N is an integer, Int.toString (N) is its string representation.




print : string -> unit

print "Hello\n"

print only prints strings.

The function print is automatically available. If you want to use other I/O functions, you should open the TextIO library by saying

open TextIO;

(or you can prepend TextIO. to each of the function names in the following table). TextIO provides the streams stdIn, stdOut, and stdErr; you don't need to explicitly open these streams. An instream is an input stream; an outstream is an output stream.



openIn : string -> instream

The argument is the name of a file to be opened for reading.

endOfStream : instream -> bool

Tests for end of file.

inputN : instream * int -> string

Reads int characters and returns them as a string. Will wait for input from a terminal, or return fewer characters if at the end of a file.

inputLine : instream -> string

Reads up to \n; if no \n, one is added, unless the line is completely empty.

input : instream -> string

Reads the entire file.

input1 : instream -> char option

Result is either NONE or SOME <char>. These values can be used in patterns.

lookahead : instream -> char option

Peeks at next character, if any.

canInput : instream * int -> bool

Test if at least int characters remain.

closeIn : instream -> unit

Close the input stream.



openOut : string -> outstream

The argument is the name of a file to be opened for writing.

openAppend : string -> outstream

The argument is the name of a file to be opened for appending.

output : outstream * string -> unit

Write the string to the file.

flushOut : outstream -> unit

Flush the output buffer, thus ensuring that all data that has been written to the file actually makes it there.

closeOut : outstream -> unit

Close the file.


fun copyFile (inputFileName, outputFileName) =
     val inFile = openIn inputFileName;
     val outFile = openOut outputFileName
     ( output (outFile, input inFile);
       flushOut outFile;
       closeOut outFile

(Note: this function copies the first 60 bytes from the input file to the output file. I assume this is controlled by some setting that I don't know about yet.)



If expressions

The if expression looks like this:

if <boolean expression> then <expression1> else <expression2>

If the <boolean expression> is true, then <expression1> is evaluated and is the value of the expression, otherwise <expression2> is evaluated and is the value of the expression.

Since this is an expression and not a statement, it must have a value; therefore, the else part is required.


Case expressions

The case expression looks like this:

case <expression> of <match>

where a <match> has the form:

   <pattern1> => <expression1>
 => <expression2>
   . . .
 => <expressionN>

First, the initial <expression> is evaluated, then its value is compared against each of the <patterns>. When a matching <patterni> is found, the corresponding <expressioni> is evaluated and becomes the value of the case expression.

The most common patterns are


case (N + 1) of 1 => "a" | 2 => "b" | 3 => "c" | 4 => "d";

case [1] of [] => "empty" | x::xs => "nonempty";

Because every expression must have a value, ML will warn you if it does not think you have a pattern for every possible case.


Sequence of expressions

We can execute a sequence of expressions by separating them with semicolons and enclosing the group in parentheses; the value of the sequence is the value of the last expression in the sequence. This is most useful with expressions evaluated for their side effects.

(print "Hello, "; print name; print "\n");



An "exception" is an error. You can declare new types of exceptions, with or without a parameter, as follows:

exception <name> ;
<name> of <type> ;

To signal that one of these exceptions has occurred, use

raise <name> ;

The way you use exceptions is as follows:

<expression> handle <match>

where <expression> can perhaps raise an exception; if it does, the first rule in <match> that matches the expression will be executed. If you raise an exception and fail to handle it, ML will give you an "uncaught exception" error.



The usual form of a function definition is

fun <name> <parameter> = <expression> ;

A function only has one parameter, but that parameter is frequently a tuple, for example,

fun max (x, y) = if x > y then x else y;

Because a function has only one parameter, the following is legal:

val pair = (3, 5);
max pair;

(The value 5 is returned.)

It doesn't hurt to use a tuple of length 1 in place of a single parameter; the following are equivalent definitions:

fun score  x  = if x < 0 then 0 else x;
fun score (x) = if x < 0 then 0 else x;

Functions that don't use their parameter must still have one; they can be given the unit as the parameter:

fun tab () = print "\t";

Mutually recursive functions

Functions must be defined before they are used. To define mutually recursive functions, use fun...and...and.... The following example (to return every other element of a list) is from Elements of ML Programming by Jeffrey D. Ullman:

   take (L) =
      if L = nil then nil
      else hd(L) :: skip(tl(L))
   skip (L) =
      if L = nil then nil
      else take(tl(L)) ;

Local variables in functions

It is sometimes convenient to declare local variables and functions using For example,

fun circleData (radius) =
      val pi = 3.1415926536;
      val circumference = two * pi * radius;
      fun area radius = pi * radius * radius
      (circumference, area (radius))

Patterns in functions

Functions can use patterns in their arguments. Examples:

fun length (nil) = 0
|   length (x :: xs) = 1 + length xs;

fun member (X, nil) = false
|   member (X, x::xs) =
       if (X = x)
          then true
          else member (X, xs) ;

Patterns can also be used in rules, which have the form

pattern => expression

A match expression consists of one or more rules separated by vertical bars:

pattern1 => expression1 | pattern2 => expression2 | ...

As noted above, the usual form of a function definition is

fun <name> <parameter> = <expression> ;

but an alternate form, using matches, is

val rec <name> = fn <match> ;

(The rec stands for "recursive" and may be omitted for nonrecursive functions.) As an example, here is an alternate definition for length:

val rec length = fn
    nil   => 0
|   x::xs => 1 + length xs;

Anonymous functions

Matches can also be used to define nonrecursive anonymous functions:

(fn x => x + 1) 3

(Result is 4.) Anonymous functions cannot be recursive because, being anonymous, they have no name by which you can call them.

Polymorphic functions

In other languages, "polymorphic" means that you have two or more functions with the same name; in ML it means that a single function can handle more than one type of parameter.

An example of a built-in polymorphic function is the list function hd; it returns the first element of any type of list.

ML functions that you write will be polymorphic if they contain only polymorphic functions and operators. For example, the following function to reverse the two components of a 2-tuple is polymorphic:

fun revPair (x, y) = (y, x);

To write a polymorphic function, you need to avoid arithmetic operators (because ML needs to tell whether the arithmetic is integer or floating point), the inequality comparisons <, <=, >, >=, string concatenation, boolean operators, and type conversion operators. You can use: the list operators hd, tl, ::, @, and brackets and commas, along with the constant nil; parentheses and commas to form tuples, along with #1, #2, and so on; and equality tests = and <>.

Higher-order functions

A higher-order function is one which takes a function (or a function-containing structure) as an argument, or produces a function (or function-containing structure) as its result, or both. For example,

- fun try (a, x) = a x;
> val try = fn : ('a -> 'b) * 'a -> 'b
try (hd, [1,2,3]);
> val it = 1 : int
try (tl, [1,2,3]);
> val it = [2, 3] : int list

Notice in particular the type returned when try is defined. It is a function whose argument is a pair: the first element of the pair is a function of type ('a -> 'b), the second element is a suitable type for that function, 'a, and the result type is the result type of the function, 'b.

To use an infix operator as a function, precede it by the word op. For example, op + (3,5) is the same as 3+5. Hence,

- try (op +, (3, 5));
> val it = 8 : int

Curried functions

A curried function is a higher-order function that takes an argument and produces as result a new function with that argument embedded in the function. For example,

- fun incr x y = x + y;
> val incr = fn : int -> int -> int
incr 5 3;
> val it = 8 : int
(incr 5) 3;
> val it = 8 : int
val incr5 = incr 5;
> val incr5 = fn : int -> int
incr5 3;
> val it = 8 : int

Notice the way the function incr is defined, as if it had two blank-separated arguments. Also notice that when the function is applied, the implicit parenthesization is that of the function and the usual one argument. Finally, notice the way incr5 is defined, as a value rather than as a function.

map is a curried function that takes a function that applies to one thing of type 'a and produces a function that applies to a list 'a. For example,

- round 3.75;
> val it = 4 : int
map round [2.7, 3.1, 3.8, 9.4, 6.5];
> val it = [3, 3, 4, 9, 6] : int list
(map round) [2.7, 3.1, 3.8, 9.4, 6.5];
> val it = [3, 3, 4, 9, 6] : int list

The function List.filter takes a boolean test and returns a function that will extract from a list those elements that pass the test.

- List.filter (fn x => x > 0) [3, 0, 2, ~5, ~8, 4];
> val it = [3, 2, 4] : int list


(To be continued)