This part of the manual is a tutorial introduction to the Objective
Caml language. A good familiarity with programming in a conventional
languages (say, Pascal or C) is assumed, but no prior exposure to
functional languages is required. The present chapter introduces the
core language. Chapter 3 deals with the
object-oriented features, and chapter 2 with the
module system.
For this overview of Caml, we use the interactive system, which
is started by running ocaml from the Unix shell, or by launching the
OCamlwin.exe application under Windows. This tutorial is presented
as the transcript of a session with the interactive system:
lines starting with # represent user input; the system responses are
printed below, without a leading #.
Under the interactive system, the user types Caml phrases, terminated
by ;;, in response to the # prompt, and the system compiles them
on the fly, executes them, and prints the outcome of evaluation.
Phrases are either simple expressions, or let definitions of
identifiers (either values or functions).
#1+2*3;;
- : int = 7
#let pi = 4.0 *. atan 1.0;;
val pi : float = 3.14159265358979312
#let square x = x *. x;;
val square : float -> float = <fun>
#square(sin pi) +. square(cos pi);;
- : float = 1.
The Caml system computes both the value and the type for
each phrase. Even function parameters need no explicit type declaration:
the system infers their types from their usage in the
function. Notice also that integers and floating-point numbers are
distinct types, with distinct operators: + and * operate on
integers, but +. and *. operate on floats.
#1.0 * 2;;
This expression has type float but is here used with type int
Recursive functions are defined with the let rec binding:
#let rec fib n =
if n < 2 then 1 else fib(n-1) + fib(n-2);;
val fib : int -> int = <fun>
#fib 10;;
- : int = 89
In addition to integers and floating-point numbers, Caml offers the
usual basic data types: booleans, characters, and character strings.
#(1 < 2) = false;;
- : bool = false
#'a';;
- : char = 'a'
#"Hello world";;
- : string = "Hello world"
Predefined data structures include tuples, arrays, and lists. General
mechanisms for defining your own data structures are also provided.
They will be covered in more details later; for now, we concentrate on lists.
Lists are either given in extension as a bracketed list of
semicolon-separated elements, or built from the empty list []
(pronounce ``nil'') by adding elements in front using the ::
(``cons'') operator.
#let l = ["is"; "a"; "tale"; "told"; "etc."];;
val l : string list = ["is"; "a"; "tale"; "told"; "etc."]
#"Life" :: l;;
- : string list = ["Life"; "is"; "a"; "tale"; "told"; "etc."]
As with all other Caml data structures, lists do not need to be
explicitly allocated and deallocated from memory: all memory
management is entirely automatic in Caml. Similarly, there is no
explicit handling of pointers: the Caml compiler silently introduces
pointers where necessary.
As with most Caml data structures, inspecting and destructuring lists
is performed by pattern-matching. List patterns have the exact same
shape as list expressions, with identifier representing unspecified
parts of the list. As an example, here is insertion sort on a list:
#let rec sort lst =
match lst with
[] -> []
| head :: tail -> insert head (sort tail)
and insert elt lst =
match lst with
[] -> [elt]
| head :: tail -> if elt <= head then elt :: lst else head :: insert elt tail
;;
val sort : 'a list -> 'a list = <fun>
val insert : 'a -> 'a list -> 'a list = <fun>
#sort l;;
- : string list = ["a"; "etc."; "is"; "tale"; "told"]
The type inferred for sort, 'a list -> 'a list, means that sort
can actually apply to lists of any type, and returns a list of the
same type. The type 'a is a type variable, and stands for any
given type. The reason why sort can apply to lists of any type is
that the comparisons (=, <=, etc.) are polymorphic in Caml:
they operate between any two values of the same type. This makes
sort itself polymorphic over all list types.
#sort [6;2;5;3];;
- : int list = [2; 3; 5; 6]
#sort [3.14; 2.718];;
- : float list = [2.718; 3.14]
The sort function above does not modify its input list: it builds
and returns a new list containing the same elements as the input list,
in ascending order. There is actually no way in Caml to modify
in-place a list once it is built: we say that lists are immutable
data structures. Most Caml data structures are immutable, but a few
(most notably arrays) are mutable, meaning that they can be
modified in-place at any time.
Caml is a functional language: functions in the full mathematical
sense are supported and can be passed around freely just as any other
piece of data. For instance, here is a deriv function that takes any
float function as argument and returns an approximation of its
derivative function:
#let deriv f dx = function x -> (f(x +. dx) -. f(x)) /. dx;;
val deriv : (float -> float) -> float -> float -> float = <fun>
#let sin' = deriv sin 1e-6;;
val sin' : float -> float = <fun>
#sin' pi;;
- : float = -1.00000000013961143
Even function composition is definable:
#let compose f g = function x -> f(g(x));;
val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = <fun>
#let cos2 = compose square cos;;
val cos2 : float -> float = <fun>
Functions that take other functions as arguments are called
``functionals'', or ``higher-order functions''. Functionals are
especially useful to provide iterators or similar generic operations
over a data structure. For instance, the standard Caml library
provides a List.map functional that applies a given function to each
element of a list, and returns the list of the results:
#List.map (function n -> n * 2 + 1) [0;1;2;3;4];;
- : int list = [1; 3; 5; 7; 9]
This functional, along with a number of other list and array
functionals, is predefined because it is often useful, but there is
nothing magic with it: it can easily be defined as follows.
#let rec map f l =
match l with
[] -> []
| hd :: tl -> f hd :: map f tl;;
val map : ('a -> 'b) -> 'a list -> 'b list = <fun>
User-defined data structures include records and variants. Both are
defined with the type declaration. Here, we declare a record type to
represent rational numbers.
#type ratio = {num: int; denum: int};;
type ratio = { num : int; denum : int; }
#let add_ratio r1 r2 =
{num = r1.num * r2.denum + r2.num * r1.denum;
denum = r1.denum * r2.denum};;
val add_ratio : ratio -> ratio -> ratio = <fun>
#add_ratio {num=1; denum=3} {num=2; denum=5};;
- : ratio = {num = 11; denum = 15}
The declaration of a variant type lists all possible shapes for values
of that type. Each case is identified by a name, called a constructor,
which serves both for constructing values of the variant type and
inspecting them by pattern-matching. Constructor names are capitalized
to distinguish them from variable names (which must start with a
lowercase letter). For instance, here is a variant
type for doing mixed arithmetic (integers and floats):
#type number = Int of int | Float of float | Error;;
type number = Int of int | Float of float | Error
This declaration expresses that a value of type number is either an
integer, a floating-point number, or the constant Error representing
the result of an invalid operation (e.g. a division by zero).
Enumerated types are a special case of variant types, where all
alternatives are constants:
#type sign = Positive | Negative;;
type sign = Positive | Negative
#let sign_int n = if n >= 0 then Positive else Negative;;
val sign_int : int -> sign = <fun>
To define arithmetic operations for the number type, we use
pattern-matching on the two numbers involved:
#let add_num n1 n2 =
match (n1, n2) with
(Int i1, Int i2) ->
(* Check for overflow of integer addition *)
if sign_int i1 = sign_int i2 && sign_int(i1 + i2) <> sign_int i1
then Float(float i1 +. float i2)
else Int(i1 + i2)
| (Int i1, Float f2) -> Float(float i1 +. f2)
| (Float f1, Int i2) -> Float(f1 +. float i2)
| (Float f1, Float f2) -> Float(f1 +. f2)
| (Error, _) -> Error
| (_, Error) -> Error;;
val add_num : number -> number -> number = <fun>
#add_num (Int 123) (Float 3.14159);;
- : number = Float 126.14159
The most common usage of variant types is to describe recursive data
structures. Consider for example the type of binary trees:
#type 'a btree = Empty | Node of 'a * 'a btree * 'a btree;;
type 'a btree = Empty | Node of 'a * 'a btree * 'a btree
This definition reads as follow: a binary tree containing values of
type 'a (an arbitrary type) is either empty, or is a node containing
one value of type 'a and two subtrees containing also values of type
'a, that is, two 'a btree.
Operations on binary trees are naturally expressed as recursive functions
following the same structure as the type definition itself. For
instance, here are functions performing lookup and insertion in
ordered binary trees (elements increase from left to right):
#let rec member x btree =
match btree with
Empty -> false
| Node(y, left, right) ->
if x = y then true else
if x < y then member x left else member x right;;
val member : 'a -> 'a btree -> bool = <fun>
#let rec insert x btree =
match btree with
Empty -> Node(x, Empty, Empty)
| Node(y, left, right) ->
if x <= y then Node(y, insert x left, right)
else Node(y, left, insert x right);;
val insert : 'a -> 'a btree -> 'a btree = <fun>
Though all examples so far were written in purely applicative style,
Caml is also equipped with full imperative features. This includes the
usual while and for loops, as well as mutable data structures such
as arrays. Arrays are either given in extension between [| and |]
brackets, or allocated and initialized with the Array.create
function, then filled up later by assignments. For instance, the
function below sums two vectors (represented as float arrays) componentwise.
#let add_vect v1 v2 =
let len = min (Array.length v1) (Array.length v2) in
let res = Array.create len 0.0 in
for i = 0 to len - 1 do
res.(i) <- v1.(i) +. v2.(i)
done;
res;;
val add_vect : float array -> float array -> float array = <fun>
#add_vect [| 1.0; 2.0 |] [| 3.0; 4.0 |];;
- : float array = [|4.; 6.|]
Record fields can also be modified by assignment, provided they are
declared mutable in the definition of the record type:
#type mutable_point = { mutable x: float; mutable y: float };;
type mutable_point = { mutable x : float; mutable y : float; }
#let translate p dx dy =
p.x <- p.x +. dx; p.y <- p.y +. dy;;
val translate : mutable_point -> float -> float -> unit = <fun>
#let mypoint = { x = 0.0; y = 0.0 };;
val mypoint : mutable_point = {x = 0.; y = 0.}
#translate mypoint 1.0 2.0;;
- : unit = ()
#mypoint;;
- : mutable_point = {x = 1.; y = 2.}
Caml has no built-in notion of variable -- identifiers whose current
value can be changed by assignment. (The let binding is not an
assignment, it introduces a new identifier with a new scope.)
However, the standard library provides references, which are mutable
indirection cells (or one-element arrays), with operators ! to fetch
the current contents of the reference and := to assign the contents.
Variables can then be emulated by let-binding a reference. For
instance, here is an in-place insertion sort over arrays:
#let insertion_sort a =
for i = 1 to Array.length a - 1 do
let val_i = a.(i) in
let j = ref i in
while !j > 0 && val_i < a.(!j - 1) do
a.(!j) <- a.(!j - 1);
j := !j - 1
done;
a.(!j) <- val_i
done;;
val insertion_sort : 'a array -> unit = <fun>
References are also useful to write functions that maintain a current
state between two calls to the function. For instance, the following
pseudo-random number generator keeps the last returned number in a
reference:
#let current_rand = ref 0;;
val current_rand : int ref = {contents = 0}
#let random () =
current_rand := !current_rand * 25713 + 1345;
!current_rand;;
val random : unit -> int = <fun>
Again, there is nothing magic with references: they are implemented as
a one-field mutable record, as follows.
#type 'a ref = { mutable contents: 'a };;
type 'a ref = { mutable contents : 'a; }
#let (!) r = r.contents;;
val ( ! ) : 'a ref -> 'a = <fun>
#let (:=) r newval = r.contents <- newval;;
val ( := ) : 'a ref -> 'a -> unit = <fun>
In some special cases, you may need to store a polymorphic function in
a data structure, keeping its polymorphism. Without user-provided
type annotations, this is not allowed, as polymorphism is only
introduced on a global level. However, you can give explicitly
polymorphic types to record fields.
#type idref = { mutable id: 'a. 'a -> 'a };;
type idref = { mutable id : 'a. 'a -> 'a; }
#let r = {id = fun x -> x};;
val r : idref = {id = <fun>}
#let g s = (s.id 1, s.id true);;
val g : idref -> int * bool = <fun>
#r.id <- (fun x -> print_string "called id\n"; x);;
- : unit = ()
#g r;;
called id
called id
- : int * bool = (1, true)
Caml provides exceptions for signalling and handling exceptional
conditions. Exceptions can also be used as a general-purpose non-local
control structure. Exceptions are declared with the exception
construct, and signalled with the raise operator. For instance, the
function below for taking the head of a list uses an exception to
signal the case where an empty list is given.
#exception Empty_list;;
exception Empty_list
#let head l =
match l with
[] -> raise Empty_list
| hd :: tl -> hd;;
val head : 'a list -> 'a = <fun>
#head [1;2];;
- : int = 1
#head [];;
Exception: Empty_list.
Exceptions are used throughout the standard library to signal cases
where the library functions cannot complete normally. For instance,
the List.assoc function, which returns the data associated with a
given key in a list of (key, data) pairs, raises the predefined
exception Not_found when the key does not appear in the list:
#List.assoc 1 [(0, "zero"); (1, "one")];;
- : string = "one"
#List.assoc 2 [(0, "zero"); (1, "one")];;
Exception: Not_found.
Exceptions can be trapped with the try...with construct:
#let name_of_binary_digit digit =
try
List.assoc digit [0, "zero"; 1, "one"]
with Not_found ->
"not a binary digit";;
val name_of_binary_digit : int -> string = <fun>
#name_of_binary_digit 0;;
- : string = "zero"
#name_of_binary_digit (-1);;
- : string = "not a binary digit"
The with part is actually a regular pattern-matching on the
exception value. Thus, several exceptions can be caught by one
try...with construct. Also, finalization can be performed by
trapping all exceptions, performing the finalization, then raising
again the exception:
#let temporarily_set_reference ref newval funct =
let oldval = !ref in
try
ref := newval;
let res = funct () in
ref := oldval;
res
with x ->
ref := oldval;
raise x;;
val temporarily_set_reference : 'a ref -> 'a -> (unit -> 'b) -> 'b = <fun>
| 1.7 |
Symbolic processing of expressions |
|
We finish this introduction with a more complete example
representative of the use of Caml for symbolic processing: formal
manipulations of arithmetic expressions containing variables. The
following variant type describes the expressions we shall manipulate:
#type expression =
Const of float
| Var of string
| Sum of expression * expression (* e1 + e2 *)
| Diff of expression * expression (* e1 - e2 *)
| Prod of expression * expression (* e1 * e2 *)
| Quot of expression * expression (* e1 / e2 *)
;;
type expression =
Const of float
| Var of string
| Sum of expression * expression
| Diff of expression * expression
| Prod of expression * expression
| Quot of expression * expression
We first define a function to evaluate an expression given an
environment that maps variable names to their values. For simplicity,
the environment is represented as an association list.
#exception Unbound_variable of string;;
exception Unbound_variable of string
#let rec eval env exp =
match exp with
Const c -> c
| Var v ->
(try List.assoc v env with Not_found -> raise(Unbound_variable v))
| Sum(f, g) -> eval env f +. eval env g
| Diff(f, g) -> eval env f -. eval env g
| Prod(f, g) -> eval env f *. eval env g
| Quot(f, g) -> eval env f /. eval env g;;
val eval : (string * float) list -> expression -> float = <fun>
#eval [("x", 1.0); ("y", 3.14)] (Prod(Sum(Var "x", Const 2.0), Var "y"));;
- : float = 9.42
Now for a real symbolic processing, we define the derivative of an
expression with respect to a variable dv:
#let rec deriv exp dv =
match exp with
Const c -> Const 0.0
| Var v -> if v = dv then Const 1.0 else Const 0.0
| Sum(f, g) -> Sum(deriv f dv, deriv g dv)
| Diff(f, g) -> Diff(deriv f dv, deriv g dv)
| Prod(f, g) -> Sum(Prod(f, deriv g dv), Prod(deriv f dv, g))
| Quot(f, g) -> Quot(Diff(Prod(deriv f dv, g), Prod(f, deriv g dv)),
Prod(g, g))
;;
val deriv : expression -> string -> expression = <fun>
#deriv (Quot(Const 1.0, Var "x")) "x";;
- : expression =
Quot (Diff (Prod (Const 0., Var "x"), Prod (Const 1., Const 1.)),
Prod (Var "x", Var "x"))
| 1.8 |
Pretty-printing and parsing |
|
As shown in the examples above, the internal representation (also
called abstract syntax) of expressions quickly becomes hard to
read and write as the expressions get larger. We need a printer and a
parser to go back and forth between the abstract syntax and the concrete syntax, which in the case of expressions is the familiar
algebraic notation (e.g. 2*x+1).
For the printing function, we take into account the usual precedence
rules (i.e. * binds tighter than +) to avoid printing unnecessary
parentheses. To this end, we maintain the current operator precedence
and print parentheses around an operator only if its precedence is
less than the current precedence.
#let print_expr exp =
(* Local function definitions *)
let open_paren prec op_prec =
if prec > op_prec then print_string "(" in
let close_paren prec op_prec =
if prec > op_prec then print_string ")" in
let rec print prec exp = (* prec is the current precedence *)
match exp with
Const c -> print_float c
| Var v -> print_string v
| Sum(f, g) ->
open_paren prec 0;
print 0 f; print_string " + "; print 0 g;
close_paren prec 0
| Diff(f, g) ->
open_paren prec 0;
print 0 f; print_string " - "; print 1 g;
close_paren prec 0
| Prod(f, g) ->
open_paren prec 2;
print 2 f; print_string " * "; print 2 g;
close_paren prec 2
| Quot(f, g) ->
open_paren prec 2;
print 2 f; print_string " / "; print 3 g;
close_paren prec 2
in print 0 exp;;
val print_expr : expression -> unit = <fun>
#let e = Sum(Prod(Const 2.0, Var "x"), Const 1.0);;
val e : expression = Sum (Prod (Const 2., Var "x"), Const 1.)
#print_expr e; print_newline();;
2. * x + 1.
- : unit = ()
#print_expr (deriv e "x"); print_newline();;
2. * 1. + 0. * x + 0.
- : unit = ()
Parsing (transforming concrete syntax into abstract syntax) is usually
more delicate. Caml offers several tools to help write parsers:
on the one hand, Caml versions of the lexer generator Lex and the
parser generator Yacc (see chapter 12), which handle
LALR(1) languages using push-down automata; on the other hand, a
predefined type of streams (of characters or tokens) and
pattern-matching over streams, which facilitate the writing of
recursive-descent parsers for LL(1) languages. An example using
ocamllex and ocamlyacc is given in
chapter 12. Here, we will use stream parsers.
The syntactic support for stream parsers is provided by the Camlp4
preprocessor, which can be loaded into the interactive toplevel via
the #load directive below.
##load "camlp4o.cma";;
Camlp4 Parsing version 3.07
#open Genlex;;
let lexer = make_lexer ["("; ")"; "+"; "-"; "*"; "/"];;
val lexer : char Stream.t -> Genlex.token Stream.t = <fun>
For the lexical analysis phase (transformation of the input text into
a stream of tokens), we use a ``generic'' lexer provided in the
standard library module Genlex. The make_lexer function takes a
list of keywords and returns a lexing function that ``tokenizes'' an
input stream of characters. Tokens are either identifiers, keywords,
or literals (integer, floats, characters, strings). Whitespace and
comments are skipped.
#let token_stream = lexer(Stream.of_string "1.0 +x");;
val token_stream : Genlex.token Stream.t = <abstr>
#Stream.next token_stream;;
- : Genlex.token = Float 1.
#Stream.next token_stream;;
- : Genlex.token = Kwd "+"
#Stream.next token_stream;;
- : Genlex.token = Ident "x"
The parser itself operates by pattern-matching on the stream of
tokens. As usual with recursive descent parsers, we use several
intermediate parsing functions to reflect the precedence and
associativity of operators. Pattern-matching over streams is more
powerful than on regular data structures, as it allows recursive calls
to parsing functions inside the patterns, for matching sub-components of
the input stream. See chapter 7 for more details.
In order to use stream parsers at toplevel, we must first load the
camlp4 preprocessor.
##load"camlp4o.cma";;
Camlp4 Parsing version 3.07
Then we are ready to define our parser.
#let rec parse_expr = parser
[< e1 = parse_mult; e = parse_more_adds e1 >] -> e
and parse_more_adds e1 = parser
[< 'Kwd "+"; e2 = parse_mult; e = parse_more_adds (Sum(e1, e2)) >] -> e
| [< 'Kwd "-"; e2 = parse_mult; e = parse_more_adds (Diff(e1, e2)) >] -> e
| [< >] -> e1
and parse_mult = parser
[< e1 = parse_simple; e = parse_more_mults e1 >] -> e
and parse_more_mults e1 = parser
[< 'Kwd "*"; e2 = parse_simple; e = parse_more_mults (Prod(e1, e2)) >] -> e
| [< 'Kwd "/"; e2 = parse_simple; e = parse_more_mults (Quot(e1, e2)) >] -> e
| [< >] -> e1
and parse_simple = parser
[< 'Ident s >] -> Var s
| [< 'Int i >] -> Const(float i)
| [< 'Float f >] -> Const f
| [< 'Kwd "("; e = parse_expr; 'Kwd ")" >] -> e;;
val parse_expr : Genlex.token Stream.t -> expression = <fun>
val parse_more_adds : expression -> Genlex.token Stream.t -> expression =
<fun>
val parse_mult : Genlex.token Stream.t -> expression = <fun>
val parse_more_mults : expression -> Genlex.token Stream.t -> expression =
<fun>
val parse_simple : Genlex.token Stream.t -> expression = <fun>
#let parse_expression = parser [< e = parse_expr; _ = Stream.empty >] -> e;;
val parse_expression : Genlex.token Stream.t -> expression = <fun>
Composing the lexer and parser, we finally obtain a function to read
an expression from a character string:
#let read_expression s = parse_expression(lexer(Stream.of_string s));;
val read_expression : string -> expression = <fun>
#read_expression "2*(x+y)";;
- : expression = Prod (Const 2., Sum (Var "x", Var "y"))
A small puzzle: why do we get different results in the following two
examples?
#read_expression "x - 1";;
- : expression = Diff (Var "x", Const 1.)
#read_expression "x-1";;
Exception: Stream.Error "".
Answer: the generic lexer provided by Genlex recognizes negative
integer literals as one integer token. Hence, x-1 is read as
the token Ident "x" followed by the token Int(-1); this sequence
does not match any of the parser rules. On the other hand,
the second space in x - 1 causes the lexer to return the three
expected tokens: Ident "x", then Kwd "-", then Int(1).
| 1.9 |
Standalone Caml programs |
|
All examples given so far were executed under the interactive system.
Caml code can also be compiled separately and executed
non-interactively using the batch compilers ocamlc or ocamlopt.
The source code must be put in a file with extension .ml. It
consists of a sequence of phrases, which will be evaluated at runtime
in their order of appearance in the source file. Unlike in interactive
mode, types and values are not printed automatically; the program must
call printing functions explicitly to produce some output. Here is a
sample standalone program to print Fibonacci numbers:
(* File fib.ml *)
let rec fib n =
if n < 2 then 1 else fib(n-1) + fib(n-2);;
let main () =
let arg = int_of_string Sys.argv.(1) in
print_int(fib arg);
print_newline();
exit 0;;
main ();;
Sys.argv is an array of strings containing the command-line
parameters. Sys.argv.(1) is thus the first command-line parameter.
The program above is compiled and executed with the following shell
commands:
$ ocamlc -o fib fib.ml
$ ./fib 10
89
$ ./fib 20
10946