Functional programming
using Caml Light
Michel Mauny
January 1995
Contents
1 Introduction 4
I Functional programming 6
2 Functional languages 7
2.1 History of functional languages . . . . . . . . . . . . . . . . 8
2.2 The ML family . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The Miranda family. . . . . . . . . . . . . . . . . . . . . . . 9
3 Basic concepts 10
3.1 Toplevel loop . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Evaluation: from expressions to values . . . . . . . . . . . . 11
3.3 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6 Partial applications. . . . . . . . . . . . . . . . . . . . . . 16
4 Basic types 18
4.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Boolean values. . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Strings and characters. . . . . . . . . . . . . . . . . . . . . 24
4.4 Tuples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5 Patterns and patternmatching . . . . . . . . . . . . . . . . . 25
4.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Lists 31
5.1 Building lists. . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Extracting elements from lists: patternmatching . . . . . . . 32
5.3 Functions over lists. . . . . . . . . . . . . . . . . . . . . . 33
6 Userdefined types 36
6.1 Product types . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.2 Sum types . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
II Caml Light specifics 47
7 Mutable data structures 48
7.1 Userdefined mutable data structures. . . . . . . . . . . . . . 48
1
7.2 The ref type. . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.4 Loops: while and for . . . . . . . . . . . . . . . . . . . . . 52
7.5 Polymorphism and mutable data structures. . . . . . . . . . . . 53
8 Escaping from computations: exceptions 55
8.1 Exceptions. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.2 Raising an exception. . . . . . . . . . . . . . . . . . . . . . 56
8.3 Trapping exceptions . . . . . . . . . . . . . . . . . . . . . . 57
8.4 Polymorphism and exceptions . . . . . . . . . . . . . . . . . . 58
9 Basic input/output 60
9.1 Printable types . . . . . . . . . . . . . . . . . . . . . . . . 60
9.2 Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.3 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.4 Channels on files . . . . . . . . . . . . . . . . . . . . . . . 63
10 Streams and parsers 66
10.1 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
10.2 Stream matching and parsers . . . . . . . . . . . . . . . . . . 68
10.3 Parameterized parsers . . . . . . . . . . . . . . . . . . . . . 70
10.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . 78
11 Standalone programs and separate compilation 79
11.1 Standalone programs . . . . . . . . . . . . . . . . . . . . . . 79
11.2 Programs in several files . . . . . . . . . . . . . . . . . . . 80
11.3 Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . 82
III A complete example 84
12 ASL: A Small Language 85
12.1 ASL abstract syntax trees . . . . . . . . . . . . . . . . . . . 85
12.2 Parsing ASL programs. . . . . . . . . . . . . . . . . . . . . . 86
13 Untyped semantics of ASL programs 92
13.1 Semantic values . . . . . . . . . . . . . . . . . . . . . . . . 92
13.2 Semantic functions. . . . . . . . . . . . . . . . . . . . . . . 93
13.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
14 Encoding recursion 95
14.1 Fixpoint combinators. . . . . . . . . . . . . . . . . . . . . . 95
14.2 Recursion as a primitive construct. . . . . . . . . . . . . . . 97
15 Static typing, polymorphism and type synthesis 98
15.1 The type system . . . . . . . . . . . . . . . . . . . . . . . . 98
15.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 103
15.3 The ASL typesynthesizer. . . . . . . . . . . . . . . . . . . . 105
16 Compiling ASL to an abstract machine code 114
16.1 The Abstract Machine. . . . . . . . . . . . . . . . . . . . . . 114
16.2 Compiling ASL programs into CAM code. . . . . . . . . . . . . . 117
2
16.3 Execution of CAM code . . . . . . . . . . . . . . . . . . . . . 119
17 Answers to exercises 122
18 Conclusions and further reading 134
3
Chapter 1
Introduction
This document is a tutorial introduction to functional programming, and,
more precisely, to the usage of Caml Light. It has been used to teach Caml
Light(1) in different universities and is intended for beginners. It
contains numerous examples and exercises, and absolute beginners should read
it while sitting in front of a Caml Light toplevel loop, testing examples
and variations by themselves.
After generalities about functional programming, some features specific to
Caml Light are described. ML type synthesis and a simple execution model
are presented in a complete example of prototyping a subset of ML.
Part I (chapters 26) may be skipped by users familiar with ML. Users
with experience in functional programming, but unfamiliar with the ML
dialects may skip the very first chapters and start at chapter 6, learning
the Caml Light syntax from the examples. Part I starts with some intuition
about functions and types and gives an overview of ML and other functional
languages (chapter 2). Chapter 3 outlines the interaction with the Caml
Light toplevel loop and its basic objects. Basic types and some of their
associated primitives are presented in chapter 4. Lists (chapter 5) and
userdefined types (chapter 6) are structured data allowing for the
representation of complex objects and their easy creation and
destructuration.
While concepts presented in part I are common (under one form or another)
to many functional languages, part B (chapters 711) is dedicated to
features specific to Caml Light: mutable data structures (chapter 7),
exception handling (chapter 8), input/output (chapter 9) and streams and
parsers (chapter 10) show a more imperative side of the language.
Standalone programs and separate compilation (chapter 11) allow for modular
programming and the creation of standalone applications. Concise examples
of Caml Light features are to be found in this part.
Part C (chapters 1216) is meant for already experienced Caml Light users
willing to know more about how the Caml Light compiler synthesizes the types
of expression and how compilation and evaluation proceeds. Some knowledge
about firstorder unification is assumed. The presentation is rather

1. The ``Caml Strong'' version of these notes is available as an INRIA
technical report [24].
4
informal, and is sometimes terse (specially in the chapter about type
synthesis). We prototype a small and simple functional language (called
ASL): we give the complete prototype implementation, from the ASL parser to
the symbolic execution of code. Lexing and parsing of ASL programs are
presented in chapter 12, providing realistic usages of streams and parsers.
Chapter 13 presents an untyped callbyvalue semantics of ASL programs
through the definition of an ASL interpreter. The encoding of recursion in
untyped ASL is presented in chapter 14, showing the expressive power of the
language. The type synthesis of functional programs is demonstrated in
chapter 15, using destructive unification (on firstorder terms representing
types) as a central tool. Chapter 16 introduces the Categorical Abstract
Machine: a simple execution model for callbyvalue functional programs.
Although the Caml Light execution model is different from the one presented
here, an intuition about the simple compilation of functional languages can
be found in this chapter.
Warning: The programs and remarks (especially contained in parts B and C)
might not be valid in Caml Light versions different from 0.7.
5
Part I
Functional programming
6
Chapter 2
Functional languages
Programming languages are said to be functional when the basic way of
structuring programs is the notion of function and their essential control
structure is function application. For example, the Lisp language [22], and
more precisely its modern successor Scheme [31, 1], has been called
functional because it possesses these two properties.
However, we want the programming notion of function to be as close as
possible to the usual mathematical notion of function. In mathematics,
functions are ``firstclass'' objects: they can be arbitrarily manipulated.
For example, they can be composed, and the composition function is itself a
function.
In mathematics, one would present the successor function in the following
way:
successor: N > N
n > n+1
The functional composition could be presented as:
o: (A=>B)x (C=>A) > (C=>B)
(f,g) > (x > f (g x))
where (A=>B) denotes the space of functions from A to B.
We remark here the importance of:
1. the notion of type; a mathematical function always possesses a domain
and a codomain. They will correspond to the programming notion of type.
2. lexical binding: when we wrote the mathematical definition of
successor, we have assumed that the addition function + had been
previously defined, mapping a pair of natural numbers to a natural
number; the meaning of the successor function is defined using the
meaning of the addition: whatever + denotes in the future, this
successor function will remain the same.
3. the notion of functional abstraction, allowing to express the behavior
of fog as (x > f (g x)), i.e. the function which, when given some x,
7
returns f (g x).
ML dialects (cf. below) respect these notions. But they also allow
nonfunctional programming styles, and, in this sense, they are functional
but not purely functional.
2.1 History of functional languages
Some historical points:
 1930: Alonzo Church developed the lcalculus [6] as an attempt to
provide a basis for mathematics. The lcalculus is a formal theory for
functionality. The three basic constructs of the lcalculus are:
 variable names (e.g. x, y,...);
 application (MN if M and M are terms);
 functional abstraction (lx.M).
Terms of the lcalculus represent functions. The pure lcalculus has
been proved inconsistent as a logical theory. Some type systems have
been added to it in order to remedy this inconsistency.
 1958: Mac Carthy invented Lisp [22] whose programs have some
similarities with terms of the lcalculus. Lisp dialects have been
recently evolving in order to be closer to modern functional languages
(Scheme), but they still do not possess a type system.
 1965: P. Landin proposed the ISWIM [18] language (for ``If You See What
I Mean''), which is the precursor of languages of the ML family.
 1978: J. Backus introduced FP: a language of combinators [3] and a
framework in which it is possible to reason about programs. The main
particularity of FP programs is that they have no variable names.
 1978: R. Milner proposes a language called ML [11], intended to be the
metalanguage of the LCF proof assistant (i.e. the language used to
program the search of proofs). This language is inspired by ISWIM
(close to lcalculus) and possesses an original type system.
 1985: D. Turner proposed the Miranda [36] programming language, which
uses Milner's type system but where programs are submitted to lazy
evaluation.
Currently, the two main families of functional languages are the ML and the
Miranda families.
8
2.2 The ML family
ML languages are based on a sugared(1) version of lcalculus. Their
evaluation regime is callbyvalue (i.e. the argument is evaluated before
being passed to a function), and they use Milner's type system.
The LCF proof system appeared in 1972 at Stanford (Stanford LCF). It has
been further developed at Cambridge (Cambridge LCF) where the ML language
was added to it.
From 1981 to 1986, a version of ML and its compiler was developed in a
collaboration between INRIA and Cambridge by G. Cousineau, G. Huet and L.
Paulson.
In 1981, L. Cardelli implemented a version of ML whose compiler generated
native machine code.
In 1984, a committee of researchers from the universities of Edinburgh and
Cambridge, Bell Laboratories and INRIA, headed by R. Milner worked on a new
extended language called Standard ML [28]. This core language was completed
by a module facility designed by D. MacQueen [23].
Since 1984, the Caml language has been under design in a collaboration
between INRIA and LIENS(2)). Its first release appeared in 1987. The main
implementors of Caml were Ascander Suarez, Pierre Weis and Michel Mauny.
In 1989 appeared Standard ML of NewJersey, developed by Andrew Appel
(Princeton University) and David MacQueen (Bell Laboratories).
Caml Light is a smaller, more portable implementation of the core Caml
language, developed by Xavier Leroy since 1990.
2.3 The Miranda family
All languages in this family use lazy evaluation (i.e. the argument of a
function is evaluated if and when the function needs its valuearguments
are passed unevaluated to functions). They also use Milner's type system.
Languages belonging to the Miranda family find their origin in the SASL
language [34] (1976) developed by D. Turner. SASL and its successors (KRC
[35] 1981, Miranda [36] 1985 and Haskell [15] 1990) use sets of mutually
recursive equations as programs. These equations are written in a script
(collection of declarations) and the user may evaluate expressions using
values defined in this script. LML (Lazy ML) has been developed in Goteborg
(Sweeden); its syntax is close to ML's syntax and it uses a fast execution
model: the Gmachine [16].

1. i.e. with a more userfriendly syntax.
2. Laboratoire d'Informatique de l'Ecole Normale Superieure, 45 Rue d'Ulm,
75005 Paris
9
Chapter 3
Basic concepts
We examine in this chapter some fundamental concepts which we will use and
study in the following chapters. Some of them are specific to the interface
with the Caml language (toplevel, global definitions) while others are
applicable to any functional language.
3.1 Toplevel loop
The first contact with Caml is through its interactive aspect(1). When
running Caml on a computer, we enter a toplevel loop where Caml waits for
input from the user, and then gives a response to what has been entered.
The beginning of a Caml Light session looks like this (assuming $ is the
shell prompt on the host machine):
$camllight
> Caml Light version 0.6
#
On the PC version, the command is called caml.
The ``#'' character is Caml's prompt. When this character appears at the
beginning of a line in an actual Caml Light session, the toplevel loop is
waiting for a new toplevel phrase to be entered.
Throughout this document, the phrases starting by the # character
represent legal input to Caml. Since this document has been preprocessed
by Caml Light and these lines have been effectively given as input to a Caml
Light process, the reader may reproduce by him/herself the session contained
in each chapter (each chapter of the first two parts contains a different
session, the third part is a single session). Lines starting with the ``>''
character are Caml Light error messages. Lines starting by another
character are either Caml responses or (possibly) illegal input. The input

1. Caml Light implementations also possess a batch compiler usable to
compile files and produce standalone applications; this will be discussed in
chapter 11.
10
is printed in typewriter font (like this), and output is written using
slanted typewriter font (like that).
Important: Of course, when developing nontrivial programs, it is
preferable to edit programs in files and then to include the files in the
toplevel, instead of entering the programs directly. Furthermore, when
debugging, it is very useful to trace some functions, to see with what
arguments they are called, and what result they return. See the reference
manual [20] for a description of these facilities.
3.2 Evaluation: from expressions to values
Let us enter an arithmetic expression and see what is Caml's response:
#1+2;;
 : int = 3
The expression ``1+2'' has been entered, followed by ``;;'' which represents
the end of the current toplevel phrase. When encountering ``;;'', Caml
enters the typechecking (more precisely type synthesis) phase, and prints
the inferred type for the expression. Then, it compiles code for the
expression, executes it and, finally, prints the result.
In the previous example, the result of evaluation is printed as ``3'' and
the type is ``int'': the type of integers.
The process of evaluation of a Caml expression can be seen as transforming
this expression until no further transformation is allowed. These
transformations must preserve semantics. For example, if the expression
``1+2'' has the mathematical object 3 as semantics, then the result ``3''
must have the same semantics. The different phases of the Caml evaluation
process are:
 parsing (checking the syntactic legality of input);
 type synthesis;
 compiling;
 loading;
 executing;
 printing the result of execution.
Let us consider another example: the application of the successor function
to 2+3. The expression (function x > x+1) should be read as ``the function
that, given x, returns x+1''. The juxtaposition of this expression to (2+3)
is function application.
#(function x > x+1) (2+3);;
 : int = 6
11
There are several ways to reduce that value before obtaining the result 6.
Here are two of them (the expression being reduced at each step is
underlined):
(function x > x+1) (2+3) (function x > x+1) (2+3)
 
 
(function x > x+1) 5 (2+3) + 1
 
 
5+1 5+1
 
 
6 6
The transformations used by Caml during evaluation cannot be described in
this chapter, since they imply knowledge about compilation of Caml programs
and machine representation of Caml values. However, since the basic control
structure of Caml is function application, it is quite easy to give an idea
of the transformations involved in the Caml evaluation process by using the
kind of rewriting we used in the last example. The evaluation of the
(welltyped) application e1(e2), where e1 and e2 denote arbitrary
expressions, consists in the following steps:
 Evaluate e2, obtaining its value v.
 Evaluate e1 until it becomes a functional value. Because of the
welltyping hypothesis, e1 must represent a function from some type t1
to some t2, and the type of v is t1. We will write (function x > e)
for the result of the evaluation of e1. It denotes the mathematical
object usually written as:
f: t1>t2
x>e (where, of course, e may depend on x)
N.B.: We do not evaluate e before we know the value of x.
 Evaluate e where v has been substituted for all occurrences of x. We
then get the final value of the original expression. The result is of
type t2.
In the previous example, we evaluate:
 2+3 to 5;
 (function x > x+1) to itself (it is already a function body);
 x+1 where 5 is substituted for x, i.e. evaluate 5+1, getting 6 as
result.
It should be noticed that Caml uses callbyvalue: arguments are evaluated
before being passed to functions. The relative evaluation order of the
12
functional expression and of the argument expression is
implementationdependent, and should not be relied upon. The Caml Light
implementation evaluates arguments before functions (righttoleft
evaluation), as shown above; the original Caml implementation evaluates
functions before arguments (lefttoright evaluation).
3.3 Types
Types and their checking/synthesis are crucial to functional programming:
they provide an indication about the correctness of programs.
The universe of Caml values is partitioned into types. A type represents
a collection of values. For example, int is the type of integer numbers,
and float is the type of floatingpoint numbers. Truth values belong to the
bool type. Character strings belong to the string type. Types are divided
into two classes:
 Basic types (int, bool, string, ...).
 Compound types such as functional types. For example, the type of
functions from integers to integers is denoted by int > int. The type
of functions from boolean values to character strings is written
bool > string. The type of pairs whose first component is an integer
and whose second component is a boolean value is written int * bool.
In fact, any combination of the types above (and even more!) is possible.
This could be written as:
BasicType ::= int
 bool
 string
Type ::= BasicType
 Type > Type
 Type * Type
Once a Caml toplevel phrase has been entered in the computer, the Caml
process starts analyzing it. First of all, it performs syntax analysis,
which consists in checking whether the phrase belongs to the language or
not. For example, here is a syntax error(2) (a parenthesis is missing):
#(function x > x+1 (2+3);;
Toplevel input:
>(function x > x+1 (2+3);;
> ^^
Syntax error.

2. Actually, lexical analysis takes place before syntax analysis and
lexical errors may occur, detecting for instance badly formed character
constants.
13
The carets ``^^^'' underline the location where the error was detected.
The second analysis performed is type analysis: the system attempts to
assign a type to each subexpression of the phrase, and to synthesize the
type of the whole phrase. If type analysis fails (i.e. if the system is
unable to assign a sensible type to the phrase), then a type error is
reported and Caml waits for another input, rejecting the current phrase.
Here is a type error (cannot add a number to a boolean):
#(function x > x+1) (2+true);;
Toplevel input:
>(function x > x+1) (2+true);;
> ^^^^
This expression has type bool,
but is used with type int.
The rejection of illtyped phrases is called strong typing. Compilers for
weakly typed languages (C, for example) would instead issue a warning
message and continue their work at the risk of getting a ``Bus error'' or
``Illegal instruction'' message at runtime.
Once a sensible type has been deduced for the expression, then the
evaluation process (compilation, loading and execution) may begin.
Strong typing enforces writing clear and wellstructured programs.
Moreover, it adds security and increases the speed of program development,
since most typos and many conceptual errors are trapped and signaled by the
type analysis. Finally, welltyped programs do not need dynamic type tests
(the addition function does not need to test whether or not its arguments
are numbers), hence static type analysis allows for more efficient machine
code.
3.4 Functions
The most important kind of values in functional programming are functional
values. Mathematically, a function f of type A>B is a rule of
correspondence which associates with each element of type A a unique member
of type B.
If x denotes an element of A, then we will write f(x) for the application
of f to x. Parentheses are often useless (they are used only for grouping
subexpressions), so we could also write (f(x)) as well as (f((x))) or simply
f x. The value of f x is the unique element of B associated with x by the
rule of correspondence for f.
The notation f(x) is the one normally employed in mathematics to denote
functional application. However, we shall be careful not to confuse a
function with its application. We say ``the function f with formal
parameter x'', meaning that f has been defined by:
f:x>e
or, in Caml, that the body of f is something like (function x > ...).
Functions are values as other values. In particular, functions may be
passed as arguments to other functions, and/or returned as result. For
14
example, we could write:
#function f > (function x > (f(x+1)  1));;
 : (int > int) > int > int =
That function takes as parameter a function (let us call it f) and returns
another function which, when given an argument (let us call it x), will
return the predecessor of the value of the application f(x+1).
The type of that function should be read as:
(int > int) > (int > int).
3.5 Definitions
It is important to give names to values. We have already seen some named
values: we called them formal parameters. In the expression
(function x > x+1), the name x is a formal parameter. Its name is
irrelevant: changing it into another one does not change the meaning of the
expression. We could have written that function (function y > y+1).
If now we apply this function to, say, 1+2, we will evaluate the
expression y+1 where y is the value of 1+2. Naming y the value of 1+2 in
y+1 is written as:
#let y=1+2 in y+1;;
 : int = 4
This expression is a legal Caml phrase, and the let construct is indeed
widely used in Caml programs.
The let construct introduces local bindings of values to identifiers.
They are local because the scope of y is restricted to the expression (y+1).
The identifier y kept its previous binding (if any) after the evaluation of
the ``let ...in . ..'' construct. We can check that y has not been
globally defined by trying to evaluate it:
#y;;
Toplevel input:
>y;;
>^
The value identifier y is unbound.
Local bindings using let also introduce sharing of (possibly timeconsuming)
evaluations. When evaluating ``let x=e1 in e2'', e1 gets evaluated only
once. All occurrences of x in e2 access the value of e1 which has been
computed once. For example, the computation of:
let big = sum_of_prime_factors 35461243
in big+(2+big)(4*(big+1));;
will be less expensive than:
(sum_of_prime_factors 35461243)
15
+ (2 + (sum_of_prime_factors 35461243))
 (4 * (sum_of_prime_factors 35461243 + 1));;
The let construct also has a global form for toplevel declarations, as in:
#let successor = function x > x+1;;
successor : int > int =
#let square = function x > x*x;;
square : int > int =
When a value has been declared at toplevel, it is of course available during
the rest of the session.
#square(successor 3);;
 : int = 16
#square;;
 : int > int =
When declaring a functional value, there are some alternative syntaxes
available. For example we could have declared the square function by:
#let square x = x*x;;
square : int > int =
or (closer to the mathematical notation) by:
#let square (x) = x*x;;
square : int > int =
All these definitions are equivalent.
3.6 Partial applications
A partial application is the application of a function to some but not all
of its arguments. Consider, for example, the function f defined by:
#let f x = function y > 2*x+y;;
f : int > int > int =
Now, the expression f(3) denotes the function which when given an argument y
returns the value of 2*3+y. The application f(x) is called a partial
application, since f waits for two successive arguments, and is applied to
only one. The value of f(x) is still a function.
We may thus define an addition function by:
#let addition x = function y > x+y;;
addition : int > int > int =
16
This can also be written:
#let addition x y = x+y;;
addition : int > int > int =
We can then define the successor function by:
#let successor = addition 1;;
successor : int > int =
Now, we may use our successor function:
#successor (successor 1);;
 : int = 3
Exercises
Exercise 3.1 Give examples of functions with the following types:
1. (int > int) > int
2. int > (int > int)
3. (int > int) > (int > int)
Exercise 3.2 We have seen that the names of formal parameters are
meaningless. It is then possible to rename x by y in (function x > x+x).
What should we do in order to rename x in y in
(function x > (function y > x+y))
Hint: rename y by z first. Question: why?
Exercise 3.3 Evaluate the following expressions (rewrite them until no
longer possible):
let x=1+2 in ((function y > y+x) x);;
let x=1+2 in ((function x > x+x) x);;
let f1 = function f2 > (function x > f2 x)
in let g = function x > x+1
in f1 g 2;;
17
Chapter 4
Basic types
We examine in this chapter the Caml basic types.
4.1 Numbers
Caml Light provides two numeric types: integers (type int) and
floatingpoint numbers (type float). Integers are limited to the range
230... 2301. Integer arithmetic is taken modulo 231 ; that is, an
integer operation that overflows does not raise an error, but the result
simply wraps around. Predefined operations (functions) on integers include:
+ addition
 subtraction and unary minus
* multiplication
/ division
mod modulo
Some examples of expressions:
#3 * 4 + 2;;
 : int = 14
#3 * (4 + 2);;
 : int = 18
#3  7  2;;
 : int = 6
There are precedence rules that make * bind tighter than +, and so on. In
doubt, write extra parentheses.
So far, we have not seen the type of these arithmetic operations. They
all expect two integer arguments; so, they are functions taking one integer
and returning a function from integers to integers. The (functional) value
of such infix identifiers may be obtained by taking their prefix version.
#prefix + ;;
18
 : int > int > int =
#prefix * ;;
 : int > int > int =
#prefix mod ;;
 : int > int > int =
As shown by their types, the infix operators +, *, ..., do not work on
floatingpoint values. A separate set of floatingpoint arithmetic
operations is provided:
+. addition
. subtraction and unary minus
*. multiplication
/. division
sqrt square root
exp, log exponentiation and logarithm
sin, cos,.. . usual trigonometric operations
We have two conversion functions to go back and forth between integers and
floatingpoint numbers: int_of_float and float_of_int.
#1 + 2.3;;
Toplevel input:
>1 + 2.3;;
> ^^^
This expression has type float,
but is used with type int.
#float_of_int 1 +. 2.3;;
 : float = 3.3
Let us now give some examples of numerical functions. We start by
defining some very simple functions on numbers:
#let square x = x *. x;;
square : float > float =
#square 2.0;;
 : float = 4.0
#square (2.0 /. 3.0);;
 : float = 0.444444444444
#let sum_of_squares (x,y) = square(x) +. square(y);;
sum_of_squares : float * float > float =
#let half_pi = 3.14159 /. 2.0
#in sum_of_squares(cos(half_pi), sin(half_pi));;
 : float = 1.0
19
We now develop a classical example: the computation of the root of a
function by Newton's method. Newton's method can be stated as follows: if
y is an approximation to a root of a function f, then:
f(y)
y 
f'(y)
is a better approximation, where f'(y) is the derivative of f evaluated at
y. For example, with f(x)=x2 a, we have f'(x) =2x, and therefore:
a
f(y) y2 a y +
y
y =y =
f'(y) 2y 2
We can define a function deriv for approximating the derivative of a
function at a given point by:
#let deriv f x dx = (f(x+.dx) . f(x)) /. dx;;
deriv : (float > float) > float > float > float =
Provided dx is sufficiently small, this gives a reasonable estimate of the
derivative of f at x.
The following function returns the absolute value of its floating point
number argument. It uses the ``if ...then . ..else'' conditional
construct.
#let abs x = if x >. 0.0 then x else . x;;
abs : float > float =
The main function, given below, uses three local functions. The first one,
until, is an example of a recursive function: it calls itself in its body,
and is defined using a let rec construct (rec shows that the definition is
recursive). It takes three arguments: a predicate p on floats, a function
change from floats to floats, and a float x. If p(x) is false, then until
is called with last argument change(x), otherwise, x is the result of the
whole call. We will study recursive functions more closely later. The two
other auxiliary functions, satisfied and improve, take a float as argument
and deliver respectively a boolean value and a float. The function
satisfied asks wether the image of its argument by f is smaller than epsilon
or not, thus testing wether y is close enough to a root of f. The function
improve computes the next approximation using the formula given below. The
three local functions are defined using a cascade of let constructs. The
whole function is:
#let newton f epsilon =
# let rec until p change x =
# if p(x) then x
# else until p change (change(x)) in
# let satisfied y = abs(f y) <. epsilon in
# let improve y = y . (f(y) /. (deriv f y epsilon))
#in until satisfied improve;;
20
newton : (float > float) > float > float > float =
Some examples of equation solving:
#let square_root x epsilon =
# newton (function y > y*.y . x) epsilon x
#and cubic_root x epsilon =
# newton (function y > y*.y*.y . x) epsilon x;;
square_root : float > float > float =
cubic_root : float > float > float =
#square_root 2.0 1e5;;
 : float = 1.41421569553
#cubic_root 8.0 1e5;;
 : float = 2.00000000131
#2.0 *. (newton cos 1e5 1.5);;
 : float = 3.14159265359
4.2 Boolean values
The presence of the conditional construct implies the presence of boolean
values. The type bool is composed of two values true and false.
#true;;
 : bool = true
#false;;
 : bool = false
The functions with results of type bool are often called predicates. Many
predicates are predefined in Caml. Here are some of them:
#prefix <;;
 : 'a > 'a > bool =
#1 < 2;;
 : bool = true
#prefix <.;;
 : float > float > bool =
#3.14159 <. 2.718;;
 : bool = false
There exist also <=, >, >=, and similarly <=., >., >=..
21
4.2.1 Equality
Equality has a special status in functional languages because of functional
values. It is easy to test equality of values of base types (integers,
floatingpoint numbers, booleans, ...):
#1 = 2;;
 : bool = false
#false = (1>2);;
 : bool = true
But it is impossible, in the general case, to decide the equality of
functional values. Hence, equality stops on a runtime error when it
encounters functional values.
#(fun x > x) = (fun x > x);;
Uncaught exception: Invalid_argument "equal: functional value"
Since equality may be used on values of any type, what is its type?
Equality takes two arguments of the same type (whatever type it is) and
returns a boolean value. The type of equality is a polymorphic type, i.e.
may take several possible forms. Indeed, when testing equality of two
numbers, then its type is int > int > bool, and when testing equality of
strings, its type is string > string > bool.
#prefix =;;
 : 'a > 'a > bool =
The type of equality uses type variables, written 'a, `b, etc. Any type can
be substituted to type variables in order to produces specific instances of
types. For example, substituting int for 'a above produces
int > int > bool, which is the type of the equality function used on
integer values.
#1=2;;
 : bool = false
#(1,2) = (2,1);;
 : bool = false
#1 = (1,2);;
Toplevel input:
>1 = (1,2);;
> ^^^
This expression has type int * int,
but is used with type int.
22
4.2.2 Conditional
Conditional expressions are of the form ``if etest then e1 else e2'', where
etest is an expression of type bool and e1, e2 are expressions possessing
the same type. As an example, the logical negation could be written:
#let negate a = if a then false else true;;
negate : bool > bool =
#negate (1=2);;
 : bool = true
4.2.3 Logical operators
The classical logical operators are available in Caml. Disjunction and
conjunction are respectively written or and &:
#true or false;;
 : bool = true
#(1<2) & (2>1);;
 : bool = true
The operators & and or are not functions. They should not be seen as
regular functions, since they evaluate their second argument only if it is
needed. For example, the or operator evaluates its second operand only if
the first one evaluates to false. The behavior of these operators may be
described as follows:
e1 or e2 is equivalent to if e1 then true else e2
e1 & e2 is equivalent to if e1 then e2 else false
Negation is predefined:
#not true;;
 : bool = false
The not identifier receives a special treatment from the parser: the
application ``not f x'' is recognized as ``not (f x)'' while ``f g x'' is
identical to ``(f g) x''. This special treatment explains why the
functional value associated to not can be obtained only using the prefix
keyword:
#prefix not;;
 : bool > bool =
23
4.3 Strings and characters
String constants (type string) are written between " characters
(doublequotes). Singlecharacter constants (type char) are written between
` characters (backquotes). The most used string operation is string
concatenation, denoted by the ^ character.
#"Hello" ^ " World!";;
 : string = "Hello World!"
#prefix ^;;
 : string > string > string =
Characters are ASCII characters:
#`a`;;
 : char = `a`
#`\065`;;
 : char = `A`
and can be built from their ASCII code as in:
#char_of_int 65;;
 : char = `A`
Other operations are available (sub_string, int_of_char, etc). See the Caml
Light reference manual [20] for complete information.
4.4 Tuples
4.4.1 Constructing tuples
It is possible to combine values into tuples (pairs, triples, ...). The
value constructor for tuples is the ``,'' character (the comma). We will
often use parentheses around tuples in order to improve readability, but
they are not strictly necessary.
#1,2;;
 : int * int = 1, 2
#1,2,3,4;;
 : int * int * int * int = 1, 2, 3, 4
#let p = (1+2, 1<2);;
p : int * bool = 3, true
The infix ``*'' identifier is the type constructor for tuples. For
instance, t1t2 corresponds to the mathematical cartesian product of types
t1 and t2.
24
We can build tuples from any values: the tuple value constructor is
generic.
4.4.2 Extracting pair components
Projection functions are used to extract components of tuples. For pairs,
we have:
#fst;;
 : 'a * 'b > 'a =
#snd;;
 : 'a * 'b > 'b =
They have polymorphic types, of course, since they may be applied to any
kind of pair. They are predefined in the Caml system, but could be defined
by the user (in fact, the cartesian product itself could be defined  see
section 6.1, dedicated to userdefined product types):
#let first (x,y) = x
#and second (x,y) = y;;
first : 'a * 'b > 'a =
second : 'a * 'b > 'b =
#first p;;
 : int = 3
#second p;;
 : bool = true
We say that first is a generic value because it works uniformly on several
kinds of arguments (provided they are pairs). We often confuse between
``generic'' and ``polymorphic'', saying that such value is polymorphic
instead of generic.
4.5 Patterns and patternmatching
Patterns and patternmatching play an important role in ML languages. They
appear in all real programs and are strongly related to types (predefined or
userdefined).
A pattern indicates the shape of a value. Patterns are ``values with
holes''. A single variable (formal parameter) is a pattern (with no shape
specified: it matches any value). When a value is matched against a
pattern (this is called patternmatching), the pattern acts as a filter. We
already used patterns and patternmatching in all the functions we wrote:
the function body (function x > ...) does (trivial) patternmatching.
When applied to a value, the formal parameter x gets bound to that value.
The function body (function (x,y) > x+y) does also patternmatching: when
applied to a value (a pair, because of welltyping hypotheses), the x and y
get bound respectively to the first and the second component of that pair.
25
All these patternmatching examples were trivial ones, they did not
involve any test:
 formal parameters such as x do not impose any particular shape to the
value they are supposed to match;
 pair patterns such as (x,y) always match for typing reasons (cartesian
product is a product type).
However, some types do not guarantee such a uniform shape to their values.
Consider the bool type: an element of type bool is either true or false.
If we impose to a value of type bool to have the shape of true, then
patternmatching may fail. Consider the following function:
#let f = function true > false;;
Toplevel input:
>let f = function true > false;;
> ^^^^^^^^^^^^^^^^^^^^^^
Warning: this matching is not exhaustive.
f : bool > bool =
The compiler warns us that the patternmatching may fail (we did not
consider the false case).
Thus, if we apply f to a value that evaluates to true, patternmatching
will succeed (an equality test is performed during execution).
#f (1<2);;
 : bool = false
But, if f's argument evaluates to false, a runtime error is reported:
#f (1>2);;
Uncaught exception: Match_failure ("", 1346, 1368)
Here is a correct definition using patternmatching:
#let negate = function true > false
#  false > true;;
negate : bool > bool =
The patternmatching has now two cases, separated by the ``'' character.
Cases are examined in turn, from top to bottom. An equivalent definition of
negate would be:
#let negate = function true > false
#  x > true;;
negate : bool > bool =
The second case now matches any boolean value (in fact, only false since
true has been caught by the first match case). When the second case is
chosen, the identifier x gets bound to the argument of negate, and could be
26
used in the righthand part of the match case. Since in our example we do
not use the value of the argument in the righthand part of the second match
case, another equivalent definition of negate would be:
#let negate = function true > false
#  _ > true;;
negate : bool > bool =
Where ``_'' acts as a formal paramenter (matches any value), but does not
introduce any binding: it should be read as ``anything else''.
As an example of patternmatching, consider the following function
definition (truth value table of implication):
#let imply = function (true,true) > true
#  (true,false) > false
#  (false,true) > true
#  (false,false) > true;;
imply : bool * bool > bool =
Here is another way of defining imply, by using variables:
#let imply = function (true,x) > x
#  (false,x) > true;;
imply : bool * bool > bool =
Simpler, and simpler again:
#let imply = function (true,x) > x
#  (false,_) > true;;
imply : bool * bool > bool =
#let imply = function (true,false) > false
#  _ > true;;
imply : bool * bool > bool =
Patternmatching is allowed on any type of value (nontrivial
patternmatching is only possible on types with data constructors).
For example:
#let is_zero = function 0 > true  _ > false;;
is_zero : int > bool =
#let is_yes = function "oui" > true
#  "si" > true
#  "ya" > true
#  "yes" > true
#  _ > false;;
is_yes : string > bool =
27
4.6 Functions
The type constructor ``>'' is predefined and cannot be defined in ML's type
system. We shall explore in this section some further aspects of functions
and functional types.
4.6.1 Functional composition
Functional composition is easily definable in Caml. It is of course a
polymorphic function:
#let compose f g = function x > f (g (x));;
compose : ('a > 'b) > ('c > 'a) > 'c > 'b =
The type of compose contains no more constraints than the ones appearing in
the definition: in a sense, it is the most general type compatible with
these constraints.
These constraints are:
 the codomain of g and the domain of f must be the same;
 x must belong to the domain of g;
 compose f g x will belong to the codomain of f.
Let's see compose at work:
#let succ x = x+1;;
succ : int > int =
#compose succ list_length;;
 : '_a list > int =
#(compose succ list_length) [1;2;3];;
 : int = 4
4.6.2 Currying
We can define two versions of the addition function:
#let plus = function (x,y) > x+y;;
plus : int * int > int =
#let add = function x > (function y > x+y);;
add : int > int > int =
These two functions differ only in their way of taking their arguments. The
first one will take an argument belonging to a cartesian product, the second
one will take a number and return a function. The add function is said to
be the curried version of plus (in honor of the logician Haskell Curry).
28
The currying transformation may be written in Caml under the form of a
higherorder function:
#let curry f = function x > (function y > f(x,y));;
curry : ('a * 'b > 'c) > 'a > 'b > 'c =
Its inverse function may be defined by:
#let uncurry f = function (x,y) > f x y;;
uncurry : ('a > 'b > 'c) > 'a * 'b > 'c =
We may check the types of curry and uncurry on particular examples:
#uncurry (prefix +);;
 : int * int > int =
#uncurry (prefix ^);;
 : string * string > string =
#curry plus;;
 : int > int > int =
Exercises
Exercise 4.1 Define functions that compute the surface area and the volume
of wellknown geometric objects (rectangles, circles, spheres, ...).
Exercise 4.2 What would happen in a language submitted to callbyvalue
with recursion if there was no conditional construct (if ... then ... else
...)?
Exercise 4.3 Define the factorial function such that:
factorial n= n*(n 1)* ... *2* 1
Give two versions of factorial: recursive and tail recursive.
Exercise 4.4 Define the fibonacci function that, when given a number n,
returns the nth Fibonacci number, i.e. the nth term un of the sequence
defined by:
u1= 1
u2= 1
un+2= un+1+un
Exercise 4.5 What are the types of the following expressions?
 uncurry compose
29
 compose curry uncurry
 compose uncurry curry
30
Chapter 5
Lists
Lists represent an important data structure, mainly because of their success
in the Lisp language. Lists in ML are homogeneous: a list cannot contain
elements of different types. This may be annoying to new ML users, yet
lists are not as fundamental as in Lisp, since ML provides a facility for
introducing new types allowing the user to define more precisely the data
structures needed by the program (cf. chapter 6).
5.1 Building lists
Lists are empty or non empty sequences of elements. They are built with two
value constructors:
 [], the empty list (read: nil);
 ::, the nonempty list constructor (read: cons). It takes an element e
and a list l, and builds a new list whose first element (head) is e and
rest (tail) is l.
The special syntax [e1; ...; en ] builds the list whose elements are
e1,... ,en. It is equivalent to e1 :: (e2 :: .. .(en :: []). ..).
We may build lists of numbers:
#1::2::[];;
 : int list = [1; 2]
#[3;4;5];;
 : int list = [3; 4; 5]
#let x=2 in [1; 2; x+1; x+2];;
 : int list = [1; 2; 3; 4]
Lists of functions:
#let adds =
# let add x y = x+y
# in [add 1; add 2; add 3];;
31
adds : (int > int) list = [; ; ]
and indeed, lists of anything desired.
We may wonder what are the types of the list (data) constructors. The
empty list is a list of anything (since it has no element), it has thus the
type ``list of anything''. The list constructor :: takes an element and a
list (containing elements with the same type) and returns a new list. Here
again, there is no type constraint on the elements considered.
#[];;
 : 'a list = []
#function head > function tail > head::tail;;
 : 'a > 'a list > 'a list =
We see here that the list type is a recursive type. The :: constructor
receives two arguments; the second argument is itself a list.
5.2 Extracting elements from lists: patternmatching
We know how to build lists; we now show how to test emptiness of lists
(although the equality predicate could be used for that) and extract
elements from lists (e.g. the first one). We have used patternmatching on
pairs, numbers or boolean values. The syntax of patterns also includes list
patterns. (We will see that any data constructor can actually be used in a
pattern.) For lists, at least two cases have to be considered (empty, non
empty):
#let is_null = function [] > true  _ > false;;
is_null : 'a list > bool =
#let head = function x::_ > x
#  _ > raise (Failure "head");;
head : 'a list > 'a =
#let tail = function _::l > l
#  _ > raise (Failure "tail");;
tail : 'a list > 'a list =
The expression raise (Failure "head") will produce a runtime error when
evaluated. In the definition of head above, we have chosen to forbid taking
the head of an empty list. We could have chosen tail [] to evaluate to [],
but we cannot return a value for head [] without changing the type of the
head function.
We say that the types list and bool are sum types, because they are
defined with several alternatives:
 a list is either [] or :: of ...
 a boolean value is either true or false
32
Lists and booleans are typical examples of sum types. Sum types are the
only types whose values need runtime tests in order to be matched by a
nonvariable pattern (i.e. a pattern that is not a single variable).
The cartesian product is a product type (only one alternative). Product
types do not involve runtime tests during patternmatching, because the
type of their values suffices to indicate statically what their structure
is.
5.3 Functions over lists
We will see in this section the definition of some useful functions over
lists. These functions are of general interest, but are not exhaustive.
Some of them are predefined in the Caml Light system. See also [9] or [37]
for other examples of functions over lists.
Computation of the length of a list:
#let rec length = function [] > 0
#  _::l > 1 + length(l);;
length : 'a list > int =
#length [true; false];;
 : int = 2
Concatenating two lists:
#let rec append =
# function [], l2 > l2
#  (x::l1), l2 > x::(append (l1,l2));;
append : 'a list * 'a list > 'a list =
The append function is already defined in Caml, and bound to the infix
identifier @.
#[1;2] @ [3;4];;
 : int list = [1; 2; 3; 4]
Reversing a list:
#let rec rev = function [] > []
#  x::l > (rev l) @ [x];;
rev : 'a list > 'a list =
#rev [1;2;3];;
 : int list = [3; 2; 1]
The map function applies a function on all the elements of a list, and
return the list of the function results. It demonstrates full functionality
(it takes a function as argument), list processing, and polymorphism (note
the sharing of type variables between the arguments of map in its type).
33
#let rec map f =
# function [] > []
#  x::l > (f x)::(map f l);;
map : ('a > 'b) > 'a list > 'b list =
#map (function x > x+1) [1;2;3;4;5];;
 : int list = [2; 3; 4; 5; 6]
#map length [ [1;2;3]; [4;5]; [6]; [] ];;
 : int list = [3; 2; 1; 0]
The following function is a list iterator. It takes a function f, a base
element a and a list [x1;...;xn]. It computes:
itlist f a [x1;. ..; xn]=f (. ..(f (f a x1) x2) ...)xn
For instance, when applied to the curried addition function, to the base
element 0, and to a list of numbers, it computes the sum of all elements in
the list. The expression:
it_list (prefix +) 0 [1;2;3;4;5]
evaluates to ((((0+1)+2)+3)+4)+5
i.e. to 15.
#let rec it_list f a =
# function [] > a
#  x::l > it_list f (f a x) l;;
it_list : ('a > 'b > 'a) > 'a > 'b list > 'a =
#let sigma = it_list prefix + 0;;
sigma : int list > int =
#sigma [1;2;3;4;5];;
 : int = 15
#it_list (prefix *) 1 [1;2;3;4;5];;
 : int = 120
The it_list function is one of the many ways to iterate over a list. For
other list iteration functions, see [9].
Exercises
Exercise 5.1 Define the combine function which, when given a pair of lists,
returns a list of pairs such that:
combine ([x1;...;xn],[y1;...;yn]) = [(x1,y1);...;(xn,yn)]
34
The function generates a runtime error if the argument lists do not have
the same length.
Exercise 5.2 Define a function which, when given a list, returns the list
of all its sublists.
35
Chapter 6
Userdefined types
The user is allowed to define his/her own data types. With this facility,
there is no need to encode the data structures that must be manipulated by a
program into lists (as in Lisp) or into arrays (as in Fortran).
Furthermore, early detection of type errors is enforced, since userdefined
data types reflect precisely the needs of the algorithms.
Types are either:
 product types,
 or sum types.
We have already seen examples of both kinds of types: the bool and list
types are sum types (they contain values with different shapes and are
defined and matched using several alternatives). The cartesian product is
an example of a product type: we know statically the shape of values
belonging to cartesian products.
In this chapter, we will see how to define and use new types in Caml.
6.1 Product types
Product types are finite labeled products of types. They are a
generalization of cartesian product. Elements of product types are called
records.
6.1.1 Defining product types
An example: suppose we want to define a data structure containing
information about individuals. We could define:
#let jean=("Jean",23,"Student","Paris");;
jean : string * int * string * string = "Jean", 23, "Student", "Paris"
and use patternmatching to extract any particular information about the
person jean. The problem with such usage of cartesian product is that a
function name_of returning the name field of a value representing an
36
individual would have the same type as the general first projection for
4tuples (and indeed would be the same function). This type is not precise
enough since it allows for the application of the function to any 4uple,
and not only to values such as jean.
Instead of using cartesian product, we define a person data type:
#type person =
# {Name:string; Age:int; Job:string; City:string};;
Type person defined.
The type person is the product of string, int, string and string. The field
names provide type information and also documentation: it is much easier to
understand data structures such as jean above than arbitrary tuples.
We have labels (i.e. Name, ...) to refer to components of the products.
The order of appearance of the products components is not relevant: labels
are sufficient to uniquely identify the components. The Caml system finds a
canonical order on labels to represent and print record values. The order
is always the order which appeared in the definition of the type.
We may now define the individual jean as:
#let jean = {Job="Student"; City="Paris";
# Name="Jean"; Age=23};;
jean : person = {Name="Jean"; Age=23; Job="Student"; City="Paris"}
This example illustrates the fact that order of labels is not relevant.
6.1.2 Extracting products components
The canonical way of extracting product components is patternmatching.
Patternmatching provides a way to mention the shape of values and to give
(local) names to components of values. In the following example, we name n
the numerical value contained in the field Age of the argument, and we
choose to forget values contained in other fields (using the _ character).
#let age_of = function
# {Age=n; Name=_; Job=_; City=_} > n;;
age_of : person > int =
#age_of jean;;
 : int = 23
It is also possible to access the value of a single field, with the . (dot)
operator:
#jean.Age;;
 : int = 23
#jean.Job;;
 : string = "Student"
Labels always refer to the most recent type definition: when two record
37
types possess some common labels, then these labels always refer to the most
recently defined type. When using modules (see section 11.2) this problem
arises for types defined in the same module. For types defined in different
modules, the full name of labels (i.e. with the name of the modules
prepended) disambiguates such situations.
6.1.3 Parameterized product types
It is important to be able to define parameterized types in order to define
generic data structures. The list type is parameterized, and this is the
reason why we may build lists of any kind of values. If we want to define
the cartesian product as a Caml type, we need type parameters because we
want to be able to build cartesian product of any pair of types.
#type ('a,'b) pair = {Fst:'a; Snd:'b};;
Type pair defined.
#let first x = x.Fst and second x = x.Snd;;
first : ('a, 'b) pair > 'a =
second : ('a, 'b) pair > 'b =
#let p={Snd=true; Fst=1+2};;
p : (int, bool) pair = {Fst=3; Snd=true}
#first(p);;
 : int = 3
Warning: the pair type is similar to the Caml cartesian product *, but it
is a different type.
#fst p;;
Toplevel input:
>fst p;;
> ^
This expression has type (int, bool) pair,
but is used with type 'a * 'b.
Actually, any two type definitions produce different types. If we enter
again the previous definition:
#type ('a,'b) pair = {Fst:'a; Snd:'b};;
Type pair defined.
we cannot any longer extract the Fst component of x:
#p.Fst;;
Toplevel input:
>p.Fst;;
>^
This expression has type (int, bool) pair,
but is used with type ('a, 'b) pair.
38
since the label Fst refers to the latter type pair that we defined, while
p's type is the former pair.
6.2 Sum types
A sum type is the finite labeled disjoint union of several types. A sum
type definition defines a type as being the union of some other types.
6.2.1 Defining sum types
Example: we want to have a type called identification whose values can be:
 either strings (name of an individual),
 or integers (encoding of social security number as a pair of integers).
We then need a type containing both int * int and character strings. We
also want to preserve static typechecking, we thus want to be able to
distinguish pairs from character strings even if they are injected in order
to form a single type.
Here is what we would do:
#type identification = Name of string
#  SS of int * int;;
Type identification defined.
The type identification is the labeled disjoint union of string and
int * int. The labels Name and SS are injections. They respectively inject
int * int and string into a single type identification.
Let us use these injections in order to build new values:
#let id1 = Name "Jean";;
id1 : identification = Name "Jean"
#let id2 = SS (1670728,280305);;
id2 : identification = SS (1670728, 280305)
Values id1 and id2 belong to the same type. They may for example be put
into lists as in:
#[id1;id2];;
 : identification list = [Name "Jean"; SS (1670728, 280305)]
Injections may possess one argument (as in the example above), or none.
The latter case corresponds(1) to enumerated types, as in Pascal. An
example of enumerated type is:

1. In Caml Light, there is no implicit order on values of sum types.
39
#type suit = Heart
#  Diamond
#  Club
#  Spade;;
Type suit defined.
#Club;;
 : suit = Club
The type suit contains only 4 distinct elements. Let us continue this
example by defining a type for cards.
#type card = Ace of suit
#  King of suit
#  Queen of suit
#  Jack of suit
#  Plain of suit * int;;
Type card defined.
The type card is the disjoint union of:
 suit under the injection Ace,
 suit under the injection King,
 suit under the injection Queen,
 suit under the injection Jack,
 the product of int and suit under the injection Plain.
Let us build a list of cards:
#let figures_of c = [Ace c; King c; Queen c; Jack c]
#and small_cards_of c =
# map (function n > Plain(c,n)) [7;8;9;10];;
figures_of : suit > card list =
small_cards_of : suit > card list =
#figures_of Heart;;
 : card list = [Ace Heart; King Heart; Queen Heart; Jack Heart]
#small_cards_of Spade;;
 : card list =
[Plain (Spade, 7); Plain (Spade, 8); Plain (Spade, 9); Plain (Spade, 10)]
6.2.2 Extracting sum components
Of course, patternmatching is used to extract sum components. In case of
sum types, patternmatching uses dynamic tests for this extraction. The
40
next example defines a function color returning the name of the color of the
suit argument:
#let color = function Diamond > "red"
#  Heart > "red"
#  _ > "black";;
color : suit > string =
Let us count the values of cards (assume we are playing ``belote''):
#let count trump = function
# Ace _ > 11
#  King _ > 4
#  Queen _ > 3
#  Jack c > if c = trump then 20 else 2
#  Plain (c,10) > 10
#  Plain (c,9) > if c = trump then 14 else 0
#  _ > 0;;
count : suit > card > int =
6.2.3 Recursive types
Some types possess a naturally recursive structure, lists, for example. It
is also the case for treelike structures, since trees have subtrees that
are trees themselves.
Let us define a type for abstract syntax trees of a simple arithmetic
language(2). An arithmetic expression will be either a numeric constant, or
a variable, or the addition, multiplication, difference, or division of two
expressions.
#type arith_expr = Const of int
#  Var of string
#  Plus of args
#  Mult of args
#  Minus of args
#  Div of args
#and args = {Arg1:arith_expr; Arg2:arith_expr};;
Type arith_expr defined.
Type args defined.
The two types arith_expr and args are simultaneously defined, and arith_expr
is recursive since its definition refers to args which itself refers to
arith_expr. As an example, one could represent the abstract syntax tree of
the arithmetic expression ``x+(3*y)'' as the Caml value:

2. Syntax trees are said to be abstract because they are pieces of abstract
syntax contrasting with concrete syntax which is the ``string'' form of
programs: analyzing (parsing) concrete syntax usually produces abstract
syntax.
41
#Plus {Arg1=Var "x";
# Arg2=Mult{Arg1=Const 3; Arg2=Var "y"}};;
 : arith_expr = Plus {Arg1=Var "x"; Arg2=Mult {Arg1=Const 3; Arg2=Var "y"}}
The recursive definition of types may lead to types such that it is hard
or impossible to build values of these types. For example:
#type stupid = {Next:stupid};;
Type stupid defined.
Elements of this type are infinite data structures. Essentially, the only
way to construct one is:
#let rec stupid_value = {Next=stupid_value};;
stupid_value : stupid =
{Next=
{Next=
{Next=
{Next=
{Next=
{Next=
{Next=
{Next=
{Next={Next={Next={Next={Next={Next={Next={Next=.}}}}}}}}}}}}}}
}}
Recursive type definitions should be wellfounded (i.e. possess a
nonrecursive case, or base case) in order to work well with callbyvalue.
6.2.4 Parameterized sum types
Sum types may also be parameterized. Here is the definition of a type
isomorphic to the list type:
#type 'a sequence = Empty
#  Sequence of 'a * 'a sequence;;
Type sequence defined.
A more sophisticated example is the type of generic binary trees:
#type ('a,'b) btree = Leaf of 'b
#  Btree of ('a,'b) node
#and ('a,'b) node = {Op:'a;
# Son1: ('a,'b) btree;
# Son2: ('a,'b) btree};;
Type btree defined.
Type node defined.
A binary tree is either a leaf (holding values of type 'b) or a node
composed of an operator (of type 'a) and two sons, both of them being binary
trees.
42
Binary trees can also be used to represent abstract trees for arithmetic
expressions (with only binary operators and only one kind of leaves). The
abstract syntax tree t of ``1+2'' could be defined as:
#let t = Btree {Op="+"; Son1=Leaf 1; Son2=Leaf 2};;
t : (string, int) btree = Btree {Op="+"; Son1=Leaf 1; Son2=Leaf 2}
Finally, it is time to notice that patternmatching is not restricted to
function bodies, i.e. constructs such as:
function P1 > E1
 .. .
 Pn > En
but there is also a construct dedicated to patternmatching of actual
values:
match E with P1 > E1
 .. .
 Pn > En
which matches the value of the expression E against each of the patterns
Pi, selecting the first one that matches, and giving control to the
corresponding expression. For example, we can match the tree t previously
defined by:
#match t with Btree{Op=x; Son1=_; Son2=_} > x
#  Leaf l > "No operator";;
 : string = "+"
6.2.5 Data constructors and functions
One may ask: ``What is the difference between a sum data constructor and a
function?''. At first sight, they look very similar. We assimilate
constant data constructors (such as Heart) to constants. Similarly, in Caml
Light, sum data constructors with arguments also possess a functional type:
#Ace;;
 : suit > card =
However, a data constructor possesses particular properties that a general
function does not possess, and it is interesting to understand these
differences. From the mathematical point of view, a sum data constructor is
known to be an injection while a Caml function is a general function without
further information. A mathematical injection f:A>B admits an inverse
function f1 from its image f(A)# B to A.
From the examples above, if we consider the King constructor, then:
#let king c = King c;;
king : suit > card =
43
king is the general function associated to the King constructor, and:
#function King c > c;;
Toplevel input:
>function King c > c;;
>^^^^^^^^^^^^^^^^^^^^
Warning: this matching is not exhaustive.
 : card > suit =
is the inverse function for king. It is a partial function, since
patternmatching may fail.
6.2.6 Degenerate cases: when sums meet products
What is the status of a sum type with a single case such as:
#type counter1 = Counter of int;;
Type counter1 defined.
Of course, the type counter1 is isomorphic to int. The injection
function x > Counter x is a total function from int to counter1. It is
thus a bijection.
Another way to define a type isomorphic to int would be:
#type counter2 = {Counter: int};;
Type counter2 defined.
The types counter1 and counter2 are isomorphic to int. They are at the same
time sum and product types. Their patternmatching does not perform any
runtime test.
The possibility of defining arbitrary complex data types permits the easy
manipulation of abstract syntax trees in Caml (such as the arith_expr type
above). These abstract syntax trees are supposed to represent programs of a
language (e.g. a language of arithmetic expressions). These kind of
languages which are defined in Caml are called objectlanguages and Caml is
said to be their metalanguage.
6.3 Summary
 New types can be introduced in Caml.
 Types may be parameterized by type variables. The syntax of type
parameters is:
::

 ( [, ]* )
44
 Types can be recursive.
 Product types:
 Mathematical product of several types.
 The construct is:
type =
{: ; ...}
where the s may contain type variables appearing in .
 Sum types:
 Mathematical disjoint union of several types.
 The construct is:
type =
[of ]  ...
where the s may contain type variables appearing in .
Exercises
Exercise 6.1 Define a function taking as argument a binary tree and
returning a pair of lists: the first one contains all operators of the
tree, the second one contains all its leaves.
Exercise 6.2 Define a function map_btree analogous to the map function on
lists. The function map_btree should take as arguments two functions f and
g, and a binary tree. It should return a new binary tree whose leaves are
the result of applying f to the leaves of the tree argument, and whose
operators are the results of applying the g function to the operators of the
argument.
Exercise 6.3 We can associate to the list type definition an canonical
iterator in the following way. We give a functional interpretation to the
data constructors of the list type.
45
We change the list constructors [] and :: respectively into a constant a
and an operator + (used as a prefix identifier), and abstract with respect
to these two operators, obtaining the list iterator satisfying:
listit + a [] = a
listit + a (x1::.. .::xn::[]) = x1 + (.. .+ (xn + a)... )
Its Caml definition would be:
#let rec list_it f a =
# function [] > a
#  x::l > f x (list_it f a l);;
list_it : ('a > 'b > 'b) > 'b > 'a list > 'b =
As an example, the application of it_list to the functional composition and
to its neutral element (the identity function), computes the composition of
lists of functions (try it!).
Define, using the same method, a canonical iterator over binary trees.
46
Part II
Caml Light specifics
47
Chapter 7
Mutable data structures
The definition of a sum or product type may be annotated to allow physical
(destructive) update on data structures of that type. This is the main
feature of the imperative programming style. Writing values into memory
locations is the fundamental mechanism of imperative languages such as
Pascal. The Lisp language, while mostly functional, also provides the
dangerous functions rplaca and rplacd to physically modify lists. Mutable
structures are required to implement many efficient algorithms. They are
also very convenient to represent the current state of a state machine.
7.1 Userdefined mutable data structures
Assume we want to define a type person as in the previous chapter. Then, it
seems natural to allow a person to change his/her age, job and the city that
person lives in, but not his/her name. We can do this by annotating some
labels in the type definition of person by the mutable keyword:
#type person =
# {Name: string; mutable Age: int;
# mutable Job: string; mutable City: string};;
Type person defined.
We can build values of type person in the very same way as before:
#let jean =
# {Name="Jean"; Age=23; Job="Student"; City="Paris"};;
jean : person = {Name="Jean"; Age=23; Job="Student"; City="Paris"}
But now, the value jean may be physically modified in the fields specified
to be mutable in the definition (and only in these fields).
We can modify the field Field of an expression in order to assign
it the value of by using the following construct:
.Field <
For example; if we want jean to become one year older, we would write:
48
#jean.Age < jean.Age + 1;;
 : unit = ()
Now, the value jean has been modified into:
#jean;;
 : person = {Name="Jean"; Age=24; Job="Student"; City="Paris"}
We may try to change the Name of jean, but we won't succeed: the typecheker
will not allow us to do that.
#jean.Name < "Paul";;
Toplevel input:
>jean.Name < "Paul";;
>^^^^^^^^^^^^^^^^^^^
The label Name is not mutable.
It is of course possible to use such constructs in functions as in:
#let get_older ({Age=n; _} as p) = p.Age < n + 1;;
get_older : person > unit =
In that example, we named n the current Age of the argument, but we also
named p the argument. This is an alias pattern: it saves us the bother of
writing:
#let get_older p =
# match p with {Age=n} > p.Age < n + 1;;
get_older : person > unit =
Notice that in the two previous expressions, we did not specify all fields
of the record p. Other examples would be:
#let move p new_city = p.City < new_city
#and change_job p j = p.Job < j;;
move : person > string > unit =
change_job : person > string > unit =
#change_job jean "Teacher"; move jean "Cannes";
#get_older jean; jean;;
 : person = {Name="Jean"; Age=25; Job="Teacher"; City="Cannes"}
We used the ``;'' character between the different changes we imposed to
jean. This is the sequencing of evaluations: it permits to evaluate
successively several expressions, discarding the result of each (except the
last one). This construct becomes useful in the presence of sideeffects
such as physical modifications and input/output, since we want to explicitly
specify the order in which they are performed.
49
7.2 The ref type
The ref type is the predefined type of mutable indirection cells. It is
present in the Caml system for reasons of compatibility with earlier
versions of Caml. The ref type could be defined as follows (we don't use
the ref name in the following definition because we want to preserve the
original ref type):
#type 'a reference = {mutable Ref: 'a};;
Type reference defined.
Example of building a value of type ref:
#let r = ref (1+2);;
r : int ref = ref 3
The ref identifier is syntactically presented as a sum data constructor.
The definition of r should be read as ``let r be a reference to the value of
1+2''. The value of r is nothing but a memory location whose contents can
be overwritten.
We consult a reference (i.e. read its memory location) with the ``!''
symbol:
#!r + 1;;
 : int = 4
We modify values of type ref with the := infix function:
#r:=!r+1;;
 : unit = ()
#r;;
 : int ref = ref 4
Some primitives are attached to the ref type, for example:
#incr;;
 : int ref > unit =
#decr;;
 : int ref > unit =
which increments (resp. decrements) references on integers.
7.3 Arrays
Arrays are modifiable data structures. They belong to the parameterized
type 'a vect. Array expressions are bracketed by [ and ], and elements
are separated by semicolons:
50
#let a = [ 10; 20; 30 ];;
a : int vect = [10; 20; 30]
The length of an array is returned by with the function vect_length:
#vect_length a;;
 : int = 3
7.3.1 Accessing array elements
Accesses to array elements can be done using the following syntax:
#a.(0);;
 : int = 10
or, more generally: e1.(e2), where e1 evaluates to an array and e2 to an
integer. Alternatively, the function vect_item is provided:
#vect_item;;
 : 'a vect > int > 'a =
The first element of an array is at index 0. Arrays are useful because
accessing an element is done in constant time: an array is a contiguous
fragment of memory, while accessing list elements takes linear time.
7.3.2 Modifying array elements
Modification of an array element is done with the construct:
e1.(e2) < e3
where e3 has the same type as the elements of the array e1. The expression
e2 computes the index at which the modification will occur.
As for accessing, a function for modifying array elements is also
provided:
#vect_assign;;
 : 'a vect > int > 'a > unit =
For example:
#a.(0) < (a.(0)1);;
 : unit = ()
#a;;
 : int vect = [9; 20; 30]
#vect_assign a 0 ((vect_item a 0)  1);;
 : unit = ()
51
#a;;
 : int vect = [8; 20; 30]
7.4 Loops: while and for
Imperative programming (i.e. using sideeffects such as physical
modification of data structures) traditionally makes use of sequences and
explicit loops. Sequencing evaluation in Caml Light is done by using the
semicolon ``;''. Evaluating expression e1, discarding the value returned,
and then evaluating e2 is written:
e1 ; e2
If e1 and e2 perform sideeffects, this construct ensures that they will be
performed in the specified order (from left to right). In order to
emphasize sequential sideeffects, instead of using parentheses around
sequences, one can use begin and end, as in:
#let x = ref 1 in
# begin
# x := !x + 1;
# x := !x * !x
# end;;
 : unit = ()
The keywords begin and end are equivalent to opening and closing
parentheses. The program above could be written as:
#let x = ref 1 in
# (x := !x + 1; x := !x * !x);;
 : unit = ()
Explicit loops are not strictly necessary per se: a recursive function
could perform the same work. However, the usage of an explicit loop locally
emphasizes a more imperative style. Two loops are provided:
 while: while e1 do e2 done evaluates e1 which must return a boolean
expression, if e1 return true, then e2 (which is usually a sequence) is
evaluated, then e1 is evaluated again and so on until e1 returns false.
 for: two variants, increasing and decreasing
 for v=e1 to e2 do e3 done
 for v=e1 downto e2 do e3 done
52
where v is an identifier. Expressions e1 and e2 are the bounds of the
loop: they must evaluate to integers. In the case of the increasing
loop, the expressions e1 and e2 are evaluated producing values n1 and
n2 : if n1 is strictly greater than n2, then nothing is done.
Otherwise, e3 is evaluated (n2n1 )+1 times, with the variable v bound
successively to n1, n1+1, . .., n2.
The behavior of the decreasing loop is similar: if n1 is strictly
smaller than n2, then nothing is done. Otherwise, e3 is evaluated
(n1n2) +1 times with v bound to successive values decreasing from n1
to n2.
Both loops return the value () of type unit.
#for i=0 to (vect_length a)  1 do a.(i) < i done;;
 : unit = ()
#a;;
 : int vect = [0; 1; 2]
7.5 Polymorphism and mutable data structures
There are some restrictions concerning polymorphism and mutable data
structures. One cannot enclose polymorphic objects inside mutable data
structures.
#let r = ref [];;
r : '_a list ref = ref []
The reason is that once the type of r, ('a list) ref, has been computed, it
cannot be changed. But the value of r can be changed: we could write:
r:=[1;2];;
and now, r would be a reference on a list of numbers while its type would
still be ('a list) ref, allowing us to write:
r:= true::!r;;
making r a reference on [true; 1; 2], which is an illegal Caml object.
Thus the Caml typechecker imposes that modifiable data structures
appearing at toplevel must possess monomorphic types (i.e. not
polymorphic).
Exercises
Exercise 7.1 Give a mutable data type defining the Lisp type of lists and
define the four functions car, cdr, rplaca and rplacd.
53
Exercise 7.2 Define a stamp function, of type unit > int, that returns a
fresh integer each time it is called. That is, the first call returns 1;
the second call returns 2; and so on.
Exercise 7.3 Define a quick_sort function on arrays of floating point
numbers following the quicksort algorithm [13]. Information about the
quicksort algorithm can be found in [33], for example.
54
Chapter 8
Escaping from computations: exceptions
In some situations, it is necessary to abort computations. If we are trying
to compute the integer division of an integer n by 0, then we must escape
from that embarrassing situation without returning any result.
Another example of the usage of such an escape mechanism appears when we
want to define the head function on lists:
#let head = function
# x::L > x
#  [] > raise (Failure "head: empty list");;
Toplevel input:
> x::L > x
> ^
Warning: the variable L starts with an upper case letter in this pattern.
head : 'a list > 'a =
We cannot give a regular value to the expression head [] without losing the
polymorphism of head. We thus choose to escape: we raise an exception.
8.1 Exceptions
An exception is a Caml value of the builtin type exn, very similar to a sum
type. An exception:
 has a name (Failure in our example),
 and holds zero or one value ("head: empty list" of type string in the
example).
Some exceptions are predefined, like Failure. New exceptions can be
defined with the following construct:
exception [of ]
Example:
#exception Found of int;;
55
Exception Found defined.
The exception Found has been declared, and it carries integer values. When
we apply it to an integer, we get an exception value (of type exn):
#Found 5;;
 : exn = Found 5
8.2 Raising an exception
Raising an exception is done by applying the primitive function raise to a
value of type exn:
#raise;;
 : exn > 'a =
#raise (Found 5);;
Uncaught exception: Found 5
Here is a more realistic example:
#let find_index p =
# let rec find n =
# function [] > raise (Failure "not found")
#  x::L > if p(x) then raise (Found n)
# else find (n+1) L
# in find 1;;
Toplevel input:
>  x::L > if p(x) then raise (Found n)
> ^
Warning: the variable L starts with an upper case letter in this pattern.
find_index : ('a > bool) > 'a list > 'b =
The find_index function always fails. It raises:
 Found n, if there is an element x of the list such that p(x), in this
case n is the index of x in the list,
 the Failure exception if no such x has been found.
Raising exceptions is more than an error mechanism: it is a programmable
control structure. In the find_index example, there was no error when
raising the Found exception: we only wanted to quickly escape from the
computation, since we found what we were looking for. This is why it must
be possible to trap exceptions: we want to trap possible errors, but we
also want to get our result in the case of the find_index function.
56
8.3 Trapping exceptions
Trapping exceptions is achieved by the following construct:
try with
This construct evaluates . If no exception is raised during
the evaluation, then the result of the try construct is the result of
. If an exception is raised during this evaluation, then the
raised exception is matched against the . If a case matches,
then control is passed to it. If no case matches, then the exception is
propagated outside of the try construct, looking for the enclosing try.
Example:
#let find_index p L =
# let rec find n =
# function [] > raise (Failure "not found")
#  x::L > if p(x) then raise (Found n)
# else find (n+1) L
# in
# try find 1 L with Found n > n;;
Toplevel input:
>let find_index p L =
> ^
Warning: the variable L starts with an upper case letter in this pattern.
Toplevel input:
>  x::L > if p(x) then raise (Found n)
> ^
Warning: the variable L starts with an upper case letter in this pattern.
find_index : ('a > bool) > 'a list > int =
#find_index (function n > (n mod 2) = 0) [1;3;5;7;9;10];;
 : int = 6
#find_index (function n > (n mod 2) = 0) [1;3;5;7;9];;
Uncaught exception: Failure "not found"
The part of the try construct is a regular pattern matching on
values of type exn. It is thus possible to trap any exception by using the
_ symbol. As an example, the following function traps any exception raised
during the application of its two arguments. Warning: the _ will also trap
interrupts from the keyboard such as controlC!
#let catch_all f arg default =
# try f(arg) with _ > default;;
catch_all : ('a > 'b) > 'a > 'b > 'b =
It is even possible to catch all exceptions, do something special (close or
remove opened files, for example), and raise again that exception, to
propagate it upwards.
57
#let show_exceptions f arg =
# try f(arg) with x > print_string "Exception raised!\n"; raise x;;
show_exceptions : ('a > 'b) > 'a > 'b =
In the example above, we print a message to the standard output channel (the
terminal), before raising again the trapped exception.
#catch_all (function x > raise (Failure "foo")) 1 0;;
 : int = 0
#catch_all (show_exceptions (function x > raise (Failure "foo"))) 1 0;;
Exception raised!
 : int = 0
8.4 Polymorphism and exceptions
Exceptions must not be polymorphic for a reason similar to the one for
references (although it is a bit harder to give an example).
#exception Exc of 'a list;;
Toplevel input:
>exception Exc of 'a list;;
> ^^
The type variable a is unbound.
One reason is that the excn type is not a parameterized type, but one deeper
reason is that if the exception Exc is declared to be polymorphic, then a
function may raise Exc [1;2]. There might be no mention of that fact in the
type inferred for the function. Then, another function may trap that
exception, obtaining the value [1;2] whose real type is int list. But the
only type known by the typechecker is 'a list: the try form should refer to
the Exc data constructor, which is known to be polymorphic. It may then be
possible to build an illtyped Caml value [true; 1; 2], since the
typechecker does not possess any further type information than 'a list.
The problem is thus the absence of static connection from exceptions that
are raised and the occurrences where they are trapped. Another example
would be the one of a function raising Exc with an integer or a boolean
value, depending on some condition. Then, in that case, when trying to trap
these exceptions, we cannot decide wether they will hold integers or boolean
values.
Exercises
Exercise 8.1 Define the function find_succeed which given a function f and
a list L returns the first element of L on which the application of f
succeeds.
Exercise 8.2 Define the function map_succeed which given a function f and a
list L returns the list of the results of successful applications of f to
58
elements of L.
59
Chapter 9
Basic input/output
We describe in this chapter the Caml Light input/output model and some of
its primitive operations. More complete information about IO can be found
in the Caml Light manual [20].
Caml Light has an imperative input/output model: an IO operation should
be considered as a sideeffect, and is thus dependent on the order of
evaluation. IOs are performed onto channels with types in_channel and
out_channel. These types are abstract, i.e. their representation is not
accessible.
Three channels are predefined:
#std_in;;
 : in_channel =
#std_out;;
 : out_channel =
#std_err;;
 : out_channel =
They are the ``standard'' IO channels: std_in is usually connected to the
keyboard, and printing onto std_out and std_err usually appears on the
screen.
9.1 Printable types
It is not possible to print and read every value. Functions, for example,
are typically not readable, unless a suitable string representation is
designed and reading such a representation is followed by an interpretation
computing the desired function.
We call printable type a type for which there are input/output primitives
implemented in Caml Light. The main printable types are:
 characters: type char;
 strings: type string;
60
 integers: type int;
 floating point numbers: type float.
We know all these types from the previous chapters. Strings and characters
support a notation for escaping to ASCII codes or to denote special
characters such as newline:
#`A`;;
 : char = `A`
#`\065`;;
 : char = `A`
#`\\`;;
 : char = `\\`
#`\n`;;
 : char = `\n`
#"string with\na newline inside";;
 : string = "string with\na newline inside"
The ``\'' character is used as an escape and is useful for nonprintable or
special characters.
Of course, character constants can be used as constant patterns:
#function `a` > 0  _ > 1;;
 : char > int =
On types such as char that have a finite number of constant elements, it may
be useful to use orpatterns, gathering constants in the same matching rule:
#let is_vowel = function
# `a`  `e`  `i`  `o`  `u`  `y` > true
# _ > false;;
is_vowel : char > bool =
The first rule is chosen if the argument matches one of the cases. Since
there is a total ordering on characters, the syntax of character patterns is
enriched with a ``..'' notation:
#let is_lower_case_letter = function
# `a`..`z` > true
# _ > false;;
is_lower_case_letter : char > bool =
Of course, orpatterns and this notation can be mixed, as in:
#let is_letter = function
# `a`..`z`  `A`..`Z` > true
61
# _ > false;;
is_letter : char > bool =
In the next sections, we give the most commonly used IO primitives on
these printable types. A complete listing of predefined IO operations is
given in [20].
9.2 Output
Printing on standard output is performed by the following functions:
#print_char;;
 : char > unit =
#print_string;;
 : string > unit =
#print_int;;
 : int > unit =
#print_float;;
 : float > unit =
Printing is buffered, i.e. the effect of a call to a printing function may
not be seen immediately: flushing explicitly the output buffer is sometimes
required, unless a printing function flushes it implicitly. Flushing is
done with the flush function:
#flush;;
 : out_channel > unit =
#print_string "Hello!"; flush std_out;;
Hello! : unit = ()
The print_newline function prints a newline character and flushes the
standard output:
#print_newline;;
 : unit > unit =
Flushing is required when writing standalone applications, in which the
application may terminate without all printing being done. Standalone
applications should terminate by a call to the exit function (from the io
module), which flushes all pending output on std_out and std_err.
Printing on the standard error channel std_err is done with the following
functions:
#prerr_char;;
 : char > unit =
62
#prerr_string;;
 : string > unit =
#prerr_int;;
 : int > unit =
#prerr_float;;
 : float > unit =
The following function prints its string argument followed by a newline
character to std_err and then flushes std_err.
#prerr_endline;;
 : string > unit =
9.3 Input
These input primitives flush the standard output and read from the standard
input:
#read_line;;
 : unit > string =
#read_int;;
 : unit > int =
#read_float;;
 : unit > float =
Because of their names and types, these functions do not need further
explanation.
9.4 Channels on files
When programs have to read from or print to files, it is necessary to open
and close channels on these files.
9.4.1 Opening and closing channels
Opening and closing is performed with the following functions:
#open_in;;
 : string > in_channel =
#open_out;;
 : string > out_channel =
#close_in;;
 : in_channel > unit =
63
#close_out;;
 : out_channel > unit =
The open_in function checks the existence of its filename argument, and
returns a new input channel on that file; open_out creates a new file (or
truncates it to zero length if it exists) and returns an output channel on
that file. Both functions fail if permissions are not sufficient for
reading or writing.
Warning:
 Closing functions close their channel argument. Since their behavior is
unspecified on already closed channels, anything can happen in this
case!
 Closing one of the standard IO channels (std_in, std_out, std_err) have
unpredictable effects!
9.4.2 Reading or writing from/to specified channels
Some of the functions on standard input/output have corresponding functions
working on channels:
#output_char;;
 : out_channel > char > unit =
#output_string;;
 : out_channel > string > unit =
#input_char;;
 : in_channel > char =
#input_line;;
 : in_channel > string =
9.4.3 Failures
The exception End_of_file is raised when an input operation cannot complete
because the end of the file has been reached.
#End_of_file;;
 : exn = End_of_file
The exception sys__Sys_error (Sys_error from the module sys) is raised
when some manipulation of files is forbidden by the operating system:
#open_in "abracadabra";;
Uncaught exception: sys__Sys_error "abracadabra: No such file or directory"
The functions that we have seen in this chapter are sufficient for our
needs. Many more exist (useful mainly when working with files) and are
64
described in [20].
Exercises
Exercise 9.1 Define a function copy_file taking two filenames (of type
string) as arguments, and copying the contents of the first file on the
second one. Error messages must be printed on std_err.
Exercise 9.2 Define a function wc taking a filename as argument and
printing on the standard output the number of characters and lines appearing
in the file. Error messages must be printed on std_err.
Note: it is good practice to develop a program in defining small functions.
A single function doing the whole work is usually harder to debug and to
read. With small functions, one can trace them and see the arguments they
are called on and the result they produce.
65
Chapter 10
Streams and parsers
In the next part of these course notes, we will implement a small functional
language. Parsing valid programs of this language requires writing a
lexical analyzer and a parser for the language. For the purpose of writing
easily such programs, Caml Light provides a special data structure:
streams. Their main usage is to be interfaced to input channels or strings
and to be matched against stream patterns.
10.1 Streams
Streams belong to an abstract data type: their actual representation
remains hidden from the user. However, it is still possible to build
streams either ``by hand'' or by using some predefined functions.
10.1.1 The stream type
The type stream is a parameterized type. One can build streams of integers,
of characters or of any other type. Streams receive a special syntax,
looking like the one for lists. The empty stream is written:
#[< >];;
 : '_a stream =
A non empty stream possesses elements that are written preceded by the ``'''
(quote) character.
#[< '0; '1; '2 >];;
 : int stream =
Elements that are not preceded by ``''' are substreams that are expanded in
the enclosing stream:
#[< '0; [<'1;'2>]; '3 >];;
 : int stream =
#let s = [< '"abc" >] in [< s; '"def" >];;
66
 : string stream =
Thus, stream concatenation can be defined as:
#let stream_concat s t = [< s; t >];;
stream_concat : 'a stream > 'a stream > 'a stream =
Building streams in this way can be useful while testing a parsing function
or defining a lexical analyzer (taking as argument a stream of characters
and returning a stream of tokens). Stream concatenation does not copy
substreams: they are simply put in the same stream. Since (as we will see
later) stream matching has a destructive effect on streams (streams are
physically ``eaten'' by stream matching), parsing [< t; t >] will in fact
parse t only once: the first occurrence of t will be consumed, and the
second occurrence will be empty before its parsing will be performed.
Interfacing streams with an input channel can be done with the function:
#stream_of_channel;;
 : in_channel > char stream =
returning a stream of characters which are read from the channel argument.
The end of stream will coincide with the end of the file associated to the
channel.
In the same way, one can build the character stream associated to a
character string using:
#stream_of_string;;
 : string > char stream =
#let s = stream_of_string "abc";;
s : char stream =
10.1.2 Streams are lazily evaluated
Stream expressions are submitted to lazy evaluation, i.e. they are
effectively build only when required. This is useful in that it allows for
the easy manipulation of ``interactive'' streams like the stream built from
the standard input. If this was not the case, i.e. if streams were
immediately completely computed, a program evaluating
``stream_of_channel std_in'' would read everything up to an endoffile on
standard input before giving control to the rest of the program.
Furthermore, lazy evaluation of streams allows for the manipulation of
infinite streams. As an example, we can build the infinite stream of
integers, using side effects to show precisely when computations occur:
#let rec ints_from n =
# [< '(print_int n; print_char ` `; flush std_out; n);
# ints_from (n+1) >];;
ints_from : int > int stream =
67
#let ints = ints_from 0;;
ints : int stream =
We notice that no printing occurred and that the program terminates: this
shows that none of the elements have been evaluated and that the infinite
stream has not been built. We will see in the next section that these
sideeffects will occur on demand, i.e. when tests will be needed by a
matching function on streams.
10.2 Stream matching and parsers
The syntax for building streams can be used for patternmatching over them.
However, stream matching is more complex than the usual pattern matching.
10.2.1 Stream matching is destructive
Let us start with a simple example:
#let next = function [< 'x >] > x;;
next : 'a stream > 'a =
The next function returns the first element of its stream argument, and
fails if the stream is empty:
#let s = [< '0; '1; '2 >];;
s : int stream =
#next s;;
 : int = 0
#next s;;
 : int = 1
#next s;;
 : int = 2
#next s;;
Uncaught exception: Parse_failure
We can see from the previous examples that the stream pattern [< 'x >]
matches an initial segment of the stream. Such a pattern must be read as
``the stream whose first element matches x''. Furthermore, once stream
matching has succeeded, the stream argument has been physically modified and
does not contain any longer the part that has been recognized by the next
function.
If we come back to the infinite stream of integers, we can see that the
calls to next provoke the evaluation of the necessary part of the stream:
#next ints; next ints; next ints;;
0 1 2  : int = 2
68
Thus, successive calls to next remove the first elements of the stream until
it becomes empty. Then, next fails when applied to the empty stream, since,
in the definition of next, there is no stream pattern that matches an
initial segment of the empty stream.
It is of course possible to specify several stream patterns as in:
#let next = function
# [< 'x >] > x
# [< >] > raise (Failure "empty");;
next : 'a stream > 'a =
Cases are tried in turn, from top to bottom.
Stream pattern components are not restricted to quoted patterns (intended
to match stream elements), but can be also function calls (corresponding to
nonterminals, in the grammar terminology). Functions appearing as stream
pattern components are intended to match substreams of the stream argument:
they are called on the actual stream argument, and they are followed by a
pattern which should match the result of this call. For example, if we
define a parser recognizing a non empty sequence of characters `a`:
#let seq_a =
# let rec seq = function
# [< '`a`; seq l >] > `a`::l
#  [< >] > []
# in function [< '`a`; seq l >] > `a`::l;;
seq_a : char stream > char list =
we used the recursively defined function seq inside the stream pattern of
the first rule. This definition should be read as:
 if the stream is not empty and if its first element matches `a`, apply
seq to the rest of the stream, let l be the result of this call and
return `a`::l,
 otherwise, fail (raise Parse_failure);
and seq should be read in the same way (except that, since it recognizes
possibly empty sequences of `a`, it never fails).
Less operationally, we can read it as: ``a nonempty sequence of `a`
starts with an `a`, and is followed by a possibly empty sequence of `a`.
Another example is the recognition of a nonempty sequence of `a` followed
by a `b`, or a `b` alone:
#let seq_a_b = function
# [< seq_a l; '`b` >] > l@[`b`]
# [< '`b` >] > [`b`];;
seq_a_b : char stream > char list =
Here, operationally, once an `a` has been recognized, the first matching
rule is chosen. Any further mismatch (either from seq_a or from the last
`b`) will raise a Parse_error exception, and the whole parsing will fail.
69
On the other hand, if the first character is not an `a`, seq_a will raise
Parse_failure, and the second rule ([< '`b` >] > ...) will be tried.
This behavior is typical of predictive parsers. Predictive parsing is
recursivedescent parsing with one lookahead token. In other words, a
predictive parser is a set of (possibly mutually recursive) procedures,
which are selected according to the shape of (at most) the first token.
10.2.2 Sequential binding in stream patterns
Bindings in stream patterns occur sequentially, in contrast with bindings in
regular patterns, which can be thought as occurring in parallel. Stream
matching is guaranteed to be performed from left to right. For example,
computing the sum of the elements of an integer stream could be defined as:
#let rec stream_sum n = function
# [< '0; (stream_sum n) p >] > p
# [< 'x; (stream_sum (n+x)) p >] > p
# [< >] > n;;
stream_sum : int > int stream > int =
#stream_sum 0 [< '0; '1; '2; '3; '4 >];;
 : int = 10
The stream_sum function uses its first argument as an accumulator holding
the sum computed so far. The call (stream_sum (n+x)) uses x which was bound
in the stream pattern component occurring at the left of the call.
Warning: streams patterns are legal only in the function and match
constructs. The let and other forms are restricted to usual patterns.
Furthermore, a stream pattern cannot appear inside another pattern.
10.3 Parameterized parsers
Since a parser is a function like any other function, it can be
parameterized or be used as a parameter. Parameters used only in the
righthand side of streammatching rules simulate inherited attributes of
attribute grammars. Parameters used as parsers in stream patterns allow for
the implementation of higherorder parsers. We will use the next example to
motivate the introduction of parameterized parsers.
10.3.1 Example: a parser for arithmetic expressions
Before building a parser for arithmetic expressions, we need a lexical
analyzer able to recognize arithmetic operations and integer constants. Let
us first define a type for tokens:
#type token =
# PLUS  MINUS  TIMES  DIV  LPAR  RPAR
# INT of int;;
Type token defined.
70
Skipping blank spaces is performed by the spaces function defined as:
#let rec spaces = function
# [< '` ``\t``\n`; spaces _ >] > ()
# [< >] > ();;
spaces : char stream > unit =
The conversion of a digit (character) into its integer value is done by:
#let int_of_digit = function
# `0`..`9` as c > (int_of_char c)  (int_of_char `0`)
# _ > raise (Failure "not a digit");;
int_of_digit : char > int =
The ``as'' keyword allows for naming a pattern: in this case, the variable
c will be bound to the actual digit matched by `0`..`9`. Pattern built with
as are called alias patterns.
For the recognition of integers, we already feel the need for a
parameterized parser. Integer recognition is done by the integer analyzer
defined below. It is parameterized by a numeric value representing the
value of the first digits of the number:
#let rec integer n = function
# [< ' `0`..`9` as c; (integer (10*n + int_of_digit c)) r >] > r
# [< >] > n;;
integer : int > char stream > int =
#integer 0 (stream_of_string "12345");;
 : int = 12345
We are now ready to write the lexical analyzer, taking a stream of
characters, and returning a stream of tokens. Returning a token stream
which will be explored by the parser is a simple, reasonably efficient and
intuitive way of composing a lexical analyzer and a parser.
#let rec lexer s = match s with
# [< '`(`; spaces _ >] > [< 'LPAR; lexer s >]
# [< '`)`; spaces _ >] > [< 'RPAR; lexer s >]
# [< '`+`; spaces _ >] > [< 'PLUS; lexer s >]
# [< '``; spaces _ >] > [< 'MINUS; lexer s >]
# [< '`*`; spaces _ >] > [< 'TIMES; lexer s >]
# [< '`/`; spaces _ >] > [< 'DIV; lexer s >]
# [< '`0`..`9` as c; (integer (int_of_digit c)) n; spaces _ >]
# > [< 'INT n; lexer s >];;
lexer : char stream > token stream =
We assume there is no leading space in the input.
Now, let us examine the language that we want to recognize. We shall have
integers, infix arithmetic operations and parenthesized expressions. The
BNF form of the grammar is:
71
Expr ::= Expr + Expr
 Expr  Expr
 Expr * Expr
 Expr / Expr
 ( Expr )
 INT
The values computed by the parser will be abstract syntax trees (by contrast
with concrete syntax, which is the input string or stream). Such trees
belong to the following type:
#type atree =
# Int of int
# Plus of atree * atree
# Minus of atree * atree
# Mult of atree * atree
# Div of atree * atree;;
Type atree defined.
The Expr grammar is ambiguous. To make it unambiguous, we will adopt the
usual precedences for arithmetic operators and assume that all operators
associate to the left. Now, to use stream matching for parsing, we must
take into account the fact that matching rules are chosen according to the
behavior of the first component of each matching rule. This is predictive
parsing, and, using wellknown techniques, it is easy to rewrite the grammar
above in such a way that writing the corresponding predictive parser becomes
trivial. These techniques are described in [2], and consist in adding a
nonterminal for each precedence level, and removing leftrecursion. We
obtain:
Expr ::= Mult
 Mult + Expr
 Mult  Expr
Mult ::= Atom
 Atom * Mult
 Atom / Mult
Atom ::= INT
 ( Expr )
While removing leftrecursion, we forgot about left associativity of
operators. This is not a problem, as long as we build correct abstract
trees.
Since stream matching bases its choices on the first component of stream
patterns, we cannot see the grammar above as a parser. We need a further
transformation, factoring common prefixes of grammar rules (leftfactor).
We obtain:
Expr ::= Mult RestExpr
72
RestExpr ::= + Mult RestExpr
  Mult RestExpr
 (* nothing *)
Mult ::= Atom RestMult
RestMult ::= * Atom RestMult
 / Atom RestMult
 (* nothing *)
Atom ::= INT
 ( Expr )
Now, we can see this grammar as a parser (note that the order of rules
becomes important, and empty productions must appear last). The shape of
the parser is:
let rec expr =
let rec restexpr ? = function
[< 'PLUS; mult ?; restexpr ? >] > ?
 [< 'MINUS; mult ?; restexpr ? >] > ?
 [< >] > ?
in function [< mult e1; restexpr ? >] > ?
and mult =
let rec restmult ? = function
[< 'TIMES; atom ?; restmult ? >] > ?
 [< 'DIV; atom ?; restmult ? >] > ?
 [< >] > ?
in function [< atom e1; restmult ? >] > ?
and atom = function
[< 'INT n >] > Int n
 [< 'LPAR; expr e; 'RPAR >] > e
We used question marks where parameters, bindings and results still have to
appear. Let us consider the expr function: clearly, as soon as e1 is
recognized, we must be ready to build the leftmost subtree of the result.
This leftmost subtree is either restricted to e1 itself, in case restexpr
does not encounter any operator, or it is the tree representing the addition
(or subtraction) of e1 and the expression immediately following the additive
operator. Therefore, restexpr must be called with e1 as an intermediate
result, and accumulate subtrees built from its intermediate result, the tree
constructor corresponding to the operator and the last expression
encountered. The body of expr becomes:
let rec expr =
let rec restexpr e1 = function
[< 'PLUS; mult e2; restexpr (Plus (e1,e2)) e >] > e
 [< 'MINUS; mult e2; restexpr (Minus (e1,e2)) e >] > e
 [< >] > e1
73
in function [< mult e1; (restexpr e1) e2 >] > e2
Now, expr recognizes a product e1 (by mult), and applies (restexpr e1) to
the rest of the stream. According to the additive operator encountered (if
any), this function will apply mult which will return some e2. Then the
process continues with Plus(e1,e2) as intermediate result. In the end, a
correctly balanced tree will be produced (using the last rule of restexpr).
With the same considerations on mult and restmult, we can complete the
parser, obtaining:
#let rec expr =
# let rec restexpr e1 = function
# [< 'PLUS; mult e2; (restexpr (Plus (e1,e2))) e >] > e
#  [< 'MINUS; mult e2; (restexpr (Minus (e1,e2))) e >] > e
#  [< >] > e1
#in function [< mult e1; (restexpr e1) e2 >] > e2
#
#and mult =
# let rec restmult e1 = function
# [< 'TIMES; atom e2; (restmult (Mult (e1,e2))) e >] > e
#  [< 'DIV; atom e2; (restmult (Div (e1,e2))) e >] > e
#  [< >] > e1
#in function [< atom e1; (restmult e1) e2 >] > e2
#
#and atom = function
# [< 'INT n >] > Int n
# [< 'LPAR; expr e; 'RPAR >] > e;;
expr : token stream > atree =
mult : token stream > atree =
atom : token stream > atree =
And we can now try our parser:
#expr (lexer (stream_of_string "(1+2+3*4)567"));;
 : atree = Minus (Plus (Plus (Int 1, Int 2), Mult (Int 3, Int 4)), Int 567)
10.3.2 Parameters simulating inherited attributes
In the previous example, the parsers restexpr and restmult take an abstract
syntax tree e1 as argument and pass it down to the result through recursive
calls such as (restexpr (Plus(e1,e2))). If we see such parsers as
nonterminals (RestExpr from the grammar above) this parameter acts as an
inherited attribute of the nonterminal. Synthesized attributes are
simulated by the right hand sides of stream matching rules.
10.3.3 Higherorder parsers
In the definition of expr, we may notice that the parsers expr and mult on
the one hand and restexpr and restmult on the other hand have exactly the
same structure. To emphasize this similarity, if we define parsers for
74
additive (resp. multiplicative) operators by:
#let addop = function
# [< 'PLUS >] > (function (x,y) > Plus(x,y))
# [< 'MINUS >] > (function (x,y) > Minus(x,y))
#and multop = function
# [< 'TIMES >] > (function (x,y) > Mult(x,y))
# [< 'DIV >] > (function (x,y) > Div(x,y));;
addop : token stream > atree * atree > atree =
multop : token stream > atree * atree > atree =
we can rewrite the expr parser as:
#let rec expr =
# let rec restexpr e1 = function
# [< addop f; mult e2; (restexpr (f (e1,e2))) e >] > e
#  [< >] > e1
#in function [< mult e1; (restexpr e1) e2 >] > e2
#
#and mult =
# let rec restmult e1 = function
# [< multop f; atom e2; (restmult (f (e1,e2))) e >] > e
#  [< >] > e1
#in function [< atom e1; (restmult e1) e2 >] > e2
#
#and atom = function
# [< 'INT n >] > Int n
# [< 'LPAR; expr e; 'RPAR >] > e;;
expr : token stream > atree =
mult : token stream > atree =
atom : token stream > atree =
Now, we take advantage of these similarities in order to define a general
parser for leftassociative operators. Its name is left_assoc and is
parameterized by a parser for operators and a parser for expressions:
#let rec left_assoc op term =
# let rec rest e1 = function
# [< op f; term e2; (rest (f (e1,e2))) e >] > e
#  [< >] > e1
# in function [< term e1; (rest e1) e2 >] > e2;;
left_assoc :
('a stream > 'b * 'b > 'b) > ('a stream > 'b) > 'a stream 
> 'b =
Now, we can redefine expr as:
#let rec expr str = left_assoc addop mult str
#and mult str = left_assoc multop atom str
#and atom = function
# [< 'INT n >] > Int n
75
# [< 'LPAR; expr e; 'RPAR >] > e;;
expr : token stream > atree =
mult : token stream > atree =
atom : token stream > atree =
And we can now try our definitive parser:
#expr (lexer (stream_of_string "(1+2+3*4)567"));;
 : atree = Minus (Plus (Plus (Int 1, Int 2), Mult (Int 3, Int 4)), Int 567)
Parameterized parsers are useful for defining general parsers such as
left_assoc that can be used with different instances. Another example of a
useful general parser is the star parser defined as:
#let rec star p = function
# [< p x; (star p) l >] > x::l
# [< >] > [];;
star : ('a stream > 'b) > 'a stream > 'b list =
The star parser iterates zero or more times its argument p and returns the
list of results. We still have to be careful in using these general parsers
because of the predictive nature of parsing. For example, star p will never
successfully terminate if p has a rule for the empty stream pattern: this
rule will make the second rule of star useless!
10.3.4 Example: parsing a non contextfree language
As an example of parsing with parameterized parsers, we shall build a parser
for the language {wCw  w#(AB)*}, which is known to be non contextfree.
First, let us define a type for this alphabet:
#type token = A  B  C;;
Type token defined.
Given an input of the form wCw, the idea for a parser recognizing it is:
 first, to recognize the sequence w with a parser wd (for word
definition) returning information in order to build a parser recognizing
only w;
 then to recognize C;
 and to use the parser built at the first step to recognize the sequence
w.
The definition of wd is as follows:
#let rec wd = function
# [< 'A; wd l >] > (function [< 'A >] > "a")::l
# [< 'B; wd l >] > (function [< 'B >] > "b")::l
# [< >] > [];;
76
wd : token stream > (token stream > string) list =
The wu function (for word usage) builds a parser sequencing a list of
parsers:
#let rec wu = function
# p::pl > (function [< p x; (wu pl) l >] > x^l)
# [] > (function [< >] > "");;
wu : ('a stream > string) list > 'a stream > string =
The wu function builds, from a list of parsers pi, for i=1..n, a single
parser
function [] > [x1;.. .;xn]
which is the sequencing of parsers pi. The main parser w is:
#let w = function [< wd l; 'C; (wu l) r >] > r;;
w : token stream > string =
#w [< 'A; 'B; 'B; 'C; 'A; 'B; 'B >];;
 : string = "abb"
#w [< 'C >];;
 : string = ""
In the previous parser, we used an intermediate list of parsers in order
to build the second parser. We can redefine wd without using such a list:
#let w =
# let rec wd wr = function
# [< 'A; (wd (function [< wr r; 'A >] > r^"a")) p >] > p
#  [< 'B; (wd (function [< wr r; 'B >] > r^"b")) p >] > p
#  [< >] > wr
# in function [< (wd (function [< >] > "")) p; 'C; p str >] > str;;
w : token stream > string =
#w [< 'A; 'B; 'B; 'C; 'A; 'B; 'B >];;
 : string = "abb"
#w [< 'C >];;
 : string = ""
Here, wd is made local to w, and takes as parameter wr (for word recognizer)
whose initial value is the parser with an empty stream pattern. This
parameter accumulates intermediate results, and is delivered at the end of
parsing the initial sequence w. After checking for the presence of C, it is
used to parse the second sequence w.
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10.4 Further reading
A summary of the constructs over streams and of primitives over streams is
given in [20].
An alternative to parsing with streams and stream matching are the camllex
and camlyacc programs.
A detailed presentation of streams and stream matching following
``predictive parsing'' semantics can be found in [25], where alternative
semantics are given with some possible implementations.
Exercises
Exercise 10.1 Define a parser for the language of prefix arithmetic
expressions generated by the grammar:
Expr ::= INT
 + Expr Expr
  Expr Expr
 * Expr Expr
 / Expr Expr
Use the lexical analyzer for arithmetic expressions given above. The result
of the parser must be the integer resulting from the evaluation of the
arithmetic expression, i.e. its type must be:
token > int
Exercise 10.2 Enrich the type token above with a constructor IDENT of
string for identifiers, and enrich the lexical analyzer for it to recognize
identifiers built from alphabetic letters (upper or lowercase). Length of
identifiers may be limited.
78
Chapter 11
Standalone programs and separate compilation
So far, we have used Caml Light in an interactive way. It is also possible
to program in Caml Light in a batchoriented way: writing source code in a
file, having it compiled into an executable program, and executing the
program outside of the Caml Light environment. Interactive use is great for
learning the language and quickly testing new functions. Batch use is more
convenient to develop larger programs, that should be usable without
knowledge of Caml Light.
Note for Macintosh users: batch compilation is not available in the
standalone Caml Light application. It requires the MPW environment (see the
Caml Light manual).
11.1 Standalone programs
Standalone programs are composed of a sequence of phrases, contained in one
or several text files. Phrases are the same as at toplevel: expressions,
value declarations, type declarations, exception declarations, and
directives. When executing the standalone program produced by the
compiler, all phrases are executed in order. The values of expressions and
declared global variables are not printed, however. A standalone program
has to perform input and output explicitly.
Here is a sample program, that prints the number of characters and the
number of lines of its standard input, like the wc Unix utility.
let chars = ref 0;;
let lines = ref 0;;
try
while true do
let c = input_char std_in in
chars := !chars + 1;
if c = `\n` then lines := !lines + 1 else ()
done
with End_of_file >
print_int !chars; print_string " characters, ";
print_int !lines; print_string " lines.\n";
exit 0
;;
79
The input_char function reads the next character from an input channel
(here, std_in, the channel connected to standard input). It raises
exception End_of_file when reaching the end of the file. The exit function
aborts the process. Its argument is the exit status of the process.
Calling exit is absolutely necessary to ensure proper flushing of the output
channels.
Assume this program is in file count.ml. To compile it, simply run the
camlc command from the command interpreter:
camlc o count count.ml
The compiler produces an executable file count. You can now run count with
the help of the "camlrun" command:
camlrun count < count.ml
This should display something like:
306 characters, 13 lines.
Under Unix, the count file can actually be executed directly, just like any
other Unix command, as in:
./count < count.ml
This also works under MSDOS, provided the executable file is given
extension .exe. That is, if we compile count.ml as follows:
camlc o count.exe count.ml
we can run count.exe directly, as in:
count.exe < count.ml
See the reference manual for more information on camlc.
11.2 Programs in several files
It is possible to split one program into several source files, separately
compiled. This way, local changes do not imply a full recompilation of the
program. Let us illustrate that on the previous example. We split it in
two modules: one that implements integer counters; another that performs
the actual counting. Here is the first one, counter.ml:
(* counter.ml *)
type counter = { mutable val: int };;
let new init = { val = init };;
let incr c = c.val < c.val + 1;;
let read c = c.val;;
80
Here is the source for the main program, in file main.ml.
(* main.ml *)
let chars = counter__new 0;;
let lines = counter__new 0;;
try
while true do
let c = input_char std_in in
counter__incr chars;
if c = `\n` then counter__incr lines else ()
done
with End_of_file >
print_int (counter__read chars); print_string " characters, ";
print_int (counter__read lines); print_string " lines.\n";
exit 0
;;
Notice that references to identifiers defined in module counter.ml are
prefixed with the name of the module, counter, and by __ (the ``long dash''
symbol: two underscore characters). If we had simply entered new 0, for
instance, the compiler would have assumed new is an identifier declared in
the current module, and issued an ``undefined identifier'' error.
Compiling this program requires two compilation steps, plus one final
linking step.
camlc c counter.ml
camlc c main.ml
camlc o main counter.zo main.zo
Running the program is done as before:
camlrun main < counter.ml
The c option to camlc means ``compile only''; that is, the compiler should
not attempt to produce a standalone executable program from the given file,
but simply an object code file (files counter.zo, main.zo). The final
linking phases takes the two .zo files and produces the executable main.
Object files must be linked in the right order: for each global identifier,
the module defining it must come before the modules that use it.
Prefixing all external identifiers by the name of their defining module is
sometimes tedious. Therefore, the Caml Light compiler provides a mechanism
to omit the module__ part in external identifiers. The system maintains a
list of ``default'' modules, called the currently opened modules, and
whenever it encounters an identifier without the module__ part, it searches
through the opened modules, to find one that defines this identifier.
Searched modules always include the module being compiled (searched first),
and some library modules of general use. In addition, two directives are
provided to add and to remove modules from the list of opened modules:
 #open "module";; to add module in front of the list;
81
 #close "module";; to remove module from the list.
For instance, we can rewrite the main.ml file above as:
#open "counter";;
let chars = new 0;;
let lines = new 0;;
try
while true do
let c = input_char std_in in
incr chars;
if c = `\n` then incr lines
done
with End_of_file >
print_int (read chars);
print_string " characters, ";
print_int (read lines);
print_string " lines.\n";
exit 0
;;
After the #open "counter" directive, the identifier new automatically
resolves to counters__new.
If two modules, say mod1 and mod2, define both a global value f, then f in
a client module client resolves to mod1__f if mod1 is opened but not mod2,
or if mod1 has been opened more recently than mod2. Otherwise, it resolves
to mod2__f. For instance, the two system modules int and float both define
the infix identifier +. Both modules int and float are opened by default,
but int comes first. Hence, x + y is understood as the integer addition,
since + is resolved to int__+. But after the directive #open "float";;,
module float comes first, and the identifier + is resolved to float__+.
11.3 Abstraction
Some globals defined in a module are not intended to be used outside of this
module. Then, it is good programming style not to export them outside of
the module, so that the compiler can check they are not used in another
module. Also, one may wish to export a data type abstractly, that is,
without publicizing the structure of the type. This ensures that other
modules cannot build or inspect objects of that type without going through
one of the functions on that type exported in the defining module. This
helps in writing clean, wellstructured programs.
The way to do that in Caml Light is to write an explicit interface, or
output signature, specifying those identifiers that are visible from the
outside. All other identifiers will remain local to the module. For global
values, their types must be given by hand. The interface is contained in a
file whose name is the module name, with extension .mli.
Here is for instance an interface for the counter module, that abstracts
the type counter:
82
(* counter.mli *)
type counter;; (* an abstract type *)
value new : int > counter
and incr : counter > unit
and read : counter > int;;
Interfaces must be compiled separately. However, once the interface for
module A has been compiled, any module B that uses A can be immediately
compiled, even if the implementation of A is not yet compiled or even not
yet written. Consider:
camlc c counter.mli
camlc c main.ml
camlc c counter.ml
camlc o main counter.zo main.zo
The implementation main.ml could be compiled before counter.ml. The only
requirement for compiling main.ml is the existence of counter.zi, the
compiled interface of the counter module.
Exercises
Exercise 11.1 Complete the count command: it should be able to operate on
several files, given on the command line. Hint: sys__command_line is an
array of strings, containing the commandline arguments to the process.
83
Part III
A complete example
84
Chapter 12
ASL: A Small Language
We present in this chapter a simple language: ASL (A Small Language). This
language is basically the lcalculus (the purely functional kernel of Caml)
enriched with a conditional construct. The conditional must be a special
construct, because our language will be submitted to callbyvalue: thus,
the conditional cannot be a function.
ASL programs are built up from numbers, variables, functional expressions
(labstractions), applications and conditionals. An ASL program consists of
a global declaration of an identifier getting bound to the value of an
expression. The primitive functions that are available are equality between
numbers and arithmetic binary operations. The concrete syntax of ASL
expressions can be described (ambiguously) as:
Expr ::= INT
 IDENT
 "if" Expr "then" Expr "else" Expr "fi"
 "(" Expr ")"
 "\" IDENT "." Expr
and the syntax of declarations is given as:
Decl ::= "let" IDENT "be" Expr ";"
 Expr ";"
Arithmetic binary operations will be written in prefix position and will
belong to the class IDENT. The \ symbol will play the role of the Caml
keyword function.
We start by defining the abstract syntax of ASL expressions and of ASL
toplevel phrases. Then we define a parser in order to produce abstract
syntax trees from the concrete syntax of ASL programs.
12.1 ASL abstract syntax trees
We encode variable names by numbers. These numbers represent the binding
depth of variables. For instance, the function of x returning x (the ASL
85
identity function) will be represented as:
Abs("x", Var 1)
And the ASL application function which would be written in Caml:
(function f > (function x > f(x)))
would be represented as:
Abs("f", Abs("x", App(Var 2, Var 1)))
and should be viewed as the tree:
Abs
"f" / \ Abs
"x" / \ App
Var 2/\ Var 1
Var n should be read as ``an occurrence of the variable bound by the nth
abstraction node encountered when going toward the root of the abstract
syntax tree''. In our example, when going from Var 2 to the root, the 2nd
abstraction node we encounter introduces the "f" variable.
The numbers encoding variables in abstract syntax trees of functional
expressions are called ``De Bruijn(1) numbers''. The characters that we
attach to abstraction nodes simply serve as documentation: they will not be
used by any of the semantic analyses that we will perform on the trees. The
type of ASL abstract syntax trees is defined by:
#type asl = Const of int
#  Var of int
#  Cond of asl * asl * asl
#  App of asl * asl
#  Abs of string * asl
#
#and top_asl = Decl of string * asl;;
Type asl defined.
Type top_asl defined.
12.2 Parsing ASL programs
Now we come to the problem of defining a concrete syntax for ASL programs
and declarations.

1. They have been proposed by N.G. De Bruijn in [10] in order to facilitate
the mechanical treatment of lcalculus terms.
86
The choice of the concrete aspect of the programs is simply a matter of
taste. The one we choose here is close to the syntax of lcalculus (except
that we will use the backslash character because there is no ``l'' on our
keyboards). We will use the curried versions of equality and arithmetic
functions. We will also use a prefix notation (a la Lisp) for their
application. We will write ``+ (+ 1 2) 3'' instead of ``(1+2)+3''. The
``if e1 then e2 else e3'' construct will be written ``if e1 then e2 else e3
fi'', and will return the then part when e1 is different from 0 (0 acts
thus as falsity in ASL conditionals).
12.2.1 Lexical analysis
The concrete aspect of ASL programs will be either declarations of the form:
let identifier be expression;
or:
expression;
which will be understood as:
let it be expression;
The tokens produced by the lexical analyzer will represent the keywords let,
be, if and else, the \ binder, the dot, parentheses, integers, identifiers,
arithmetic operations and terminating semicolons. We reuse here most of the
code that we developed in chapter 10 or in the answers to its exercises.
Skipping blank spaces:
#let rec spaces = function
# [< '` ``\t``\n`; spaces _ >] > ()
# [< >] > ();;
spaces : char stream > unit =
The type of tokens is given by:
#type token = LET  BE  LAMBDA  DOT  LPAR  RPAR
#  IF  THEN  ELSE  FI  SEMIC
#  INT of int  IDENT of string;;
Type token defined.
Integers:
#let int_of_digit = function
# `0`..`9` as c > (int_of_char c)  (int_of_char `0`)
# _ > raise (Failure "not a digit");;
int_of_digit : char > int =
#let rec integer n = function
87
# [< ' `0`..`9` as c; (integer (10*n + int_of_digit c)) r >] > r
# [< >] > INT n;;
integer : int > char stream > token =
We restrict ASL identifiers to be composed of lowercase letters, the eight
first being significative. An explanation about the ident function can be
found in the chapter dedicated to the answers to exercises (chapter 17).
The function given here is slightly different and tests its result in order
to see wether it is a keyword (let, be, ...) or not:
#let ident_buf = make_string 8 ` `;;
ident_buf : string = " "
#let rec ident len = function
# [< ' `a`..`z` as c;
# (if len >= 8 then ident len
# else begin
# set_nth_char ident_buf len c;
# ident (succ len)
# end) s >] > s
# [< >] > (match sub_string ident_buf 0 len
# with "let" > LET
#  "be" > BE
#  "if" > IF
#  "then" > THEN
#  "else" > ELSE
#  "fi" > FI
#  s > IDENT s);;
ident : int > char stream > token =
A reasonable lexical analyzer would use a hash table to recognize keywords
faster.
Primitive operations are recognized by the following function, which also
detects illegal operators and ends of input:
#let oper = function
# [< '`+````*``/``=` as c >] > IDENT(make_string 1 c)
# [< 'c >] > prerr_string "Illegal character: ";
# prerr_endline (char_for_read c);
# raise (Failure "ASL parsing")
# [< >] > prerr_endline "Unexpected end of input";
# raise (Failure "ASL parsing");;
oper : char stream > token =
The lexical analyzer has the same structure as the one given in chapter 10
except that leading blanks are skipped.
#let rec lexer str = spaces str;
#match str with
# [< '`(`; spaces _ >] > [< 'LPAR; lexer str >]
# [< '`)`; spaces _ >] > [< 'RPAR; lexer str >]
88
# [< '`\\`; spaces _ >] > [< 'LAMBDA; lexer str >]
# [< '`.`; spaces _ >] > [< 'DOT; lexer str >]
# [< '`;`; spaces _ >] > [< 'SEMIC; lexer str >]
# [< '`0`..`9` as c;
# (integer (int_of_digit c)) tok;
# spaces _ >] > [< 'tok; lexer str >]
# [< '`a`..`z` as c;
# (set_nth_char ident_buf 0 c; ident 1) tok;
# spaces _ >] > [< 'tok; lexer str >]
# [< oper tok; spaces _ >] > [< 'tok; lexer str >]
#;;
lexer : char stream > token stream =
The lexical analyzer returns a stream of tokens that the parser will receive
as argument.
12.2.2 Parsing
The final output of our parser will be abstract syntax trees of type asl or
top_asl. This implies that we will detect unbound identifiers at
parsetime. In this case, we will raise the Unbound exception defined as:
#exception Unbound of string;;
Exception Unbound defined.
We also need a function which will compute the binding depths of variables.
That function simply looks for the position of the first occurrence of a
variable name in a list. It will raise Unbound if there is no such
occurrence.
#let binding_depth s rho =
# let rec bind n = function
# [] > raise (Unbound s)
#  t::l > if s = t then Var n else bind (n+1) l
# in bind 1 rho
#;;
binding_depth : string > string list > asl =
We also need a global environment, containing names of already bound
identifiers. The global environment contains predefined names for the
equality and arithmetic functions. We represent the global environment as a
reference since each ASL declaration will augment it with a new name.
#let init_env = ["+";"";"*";"/";"="];;
init_env : string list = ["+"; ""; "*"; "/"; "="]
#let global_env = ref init_env;;
global_env : string list ref = ref ["+"; ""; "*"; "/"; "="]
We now give a parsing function for ASL programs. Blanks at the beginning of
the string are skipped.
89
#let rec top = function
# [< 'LET; 'IDENT id; 'BE; expression e; 'SEMIC >] > Decl(id,e)
#  [< expression e; 'SEMIC >] > Decl("it",e)
#
#and expression = function
# [< (expr !global_env) e >] > e
#
#and expr rho =
# let rec rest e1 = function
# [< (atom rho) e2; (rest (App(e1,e2))) e >] > e
#  [< >] > e1
# in function
# [< 'LAMBDA; 'IDENT id; 'DOT; (expr (id::rho)) e >] > Abs(id,e)
#  [< (atom rho) e1; (rest e1) e2 >] > e2
#
#and atom rho = function
# [< 'IDENT id >] >
# (try binding_depth id rho with Unbound s >
# print_string "Unbound ASL identifier: ";
# print_string s; print_newline();
# raise (Failure "ASL parsing"))
#  [< 'INT n >] > Const n
#  [< 'IF; (expr rho) e1; 'THEN; (expr rho) e2;
# 'ELSE; (expr rho) e3; 'FI >] > Cond(e1,e2,e3)
#  [< 'LPAR; (expr rho) e; 'RPAR >] > e;;
top : token stream > top_asl =
expression : token stream > asl =
expr : string list > token stream > asl =
atom : string list > token stream > asl =
The complete parser that we will use reads a string, converts it into a
stream, and produces the token stream that is parsed:
#let parse_top s = top(lexer(stream_of_string s));;
parse_top : string > top_asl =
Let us try our grammar (we do not augment the global environment at each
declaration: this will be performed after the semantic treatment of ASL
programs). We need to write double \ inside strings, since \ is the string
escape character.
#parse_top "let f be \\x.x;";;
 : top_asl = Decl ("f", Abs ("x", Var 1))
#parse_top "let x be + 1 ((\\x.x) 2);";;
 : top_asl =
Decl ("x", App (App (Var 1, Const 1), App (Abs ("x", Var 1), Const 2)))
Unbound identifiers and undefined operators are correctly detected:
#parse_top "let y be g 3;";;
90
Unbound ASL identifier: g
Uncaught exception: Failure "ASL parsing"
#parse_top "f (if 0 then + else  fi) 2 3;";;
Unbound ASL identifier: f
Uncaught exception: Failure "ASL parsing"
#parse_top "^ x y;";;
Illegal character: ^
Uncaught exception: Failure "ASL parsing"
91
Chapter 13
Untyped semantics of ASL programs
In this section, we give a semantic treatment of ASL programs. We will use
dynamic typechecking, i.e. we will test the type correctness of programs
during their interpretation.
13.1 Semantic values
We need a type for ASL semantic values (representing results of
computations). A semantic value will be either an integer, or a Caml
functional value from ASL values to ASL values.
#type semval = Constval of int
#  Funval of (semval > semval);;
Type semval defined.
We now define two exceptions. The first one will be used when we encounter
an illtyped program and will represent runtime type errors. The other one
is helpful for debugging: it will be raised when our interpreter (semantic
function) goes into an illegal situation.
The following two exceptions will be raised in case of runtime ASL type
error, and in case of bug of our semantic treatment:
#exception Illtyped;;
Exception Illtyped defined.
#exception SemantBug of string;;
Exception SemantBug defined.
We must give a semantic value to our basic functions (equality and
arithmetic operations). The next function transforms a Caml function into
an ASL value.
#let init_semantics caml_fun =
# Funval
# (function Constval n >
# Funval(function Constval m > Constval(caml_fun n m)
#  _ > raise Illtyped)
92
#  _ > raise Illtyped);;
init_semantics : (int > int > int) > semval =
Now, associate a Caml Light function to each ASL predefined function:
#let caml_function = function
# "+" > prefix +
#  "" > prefix 
#  "*" > prefix *
#  "/" > prefix /
#  "=" > (fun n m > if n=m then 1 else 0)
#  s > raise (SemantBug "Unknown primitive");;
caml_function : string > int > int > int =
In the same way as, for parsing, we needed a global environment from which
the binding depth of identifiers was computed, we need a semantic
environment from which the interpreter will fetch the value represented by
identifiers. The global semantic environment will be a reference on the
list of predefined ASL values.
#let init_sem = map (fun x > init_semantics(caml_function x))
# init_env;;
init_sem : semval list =
[Funval ; Funval ; Funval ; Funval ; Funval ]
#let global_sem = ref init_sem;;
global_sem : semval list ref =
ref [Funval ; Funval ; Funval ; Funval ; Funval ]
13.2 Semantic functions
The semantic function is the interpreter itself. There is one for
expressions and one for declarations. The one for expressions computes the
value of an ASL expression from an environment rho. The environment will
contain the values of globally defined ASL values or of temporary ASL
values. It is organized as a list, and the numbers representing variable
occurrences will be used as indices into the environment.
#let rec nth n = function
# [] > raise (Failure "nth")
#  x::l > if n=1 then x else nth (n1) l;;
nth : int > 'a list > 'a =
#let rec semant rho =
# let rec sem = function
# Const n > Constval n
#  Var(n) > nth n rho
#  Cond(e1,e2,e3) >
# (match sem e1 with Constval 0 > sem e3
93
#  Constval n > sem e2
#  _ > raise Illtyped)
#  Abs(_,e') > Funval(fun x > semant (x::rho) e')
#  App(e1,e2) > (match sem e1
# with Funval(f) > f (sem e2)
#  _ > raise Illtyped)
# in sem
#;;
semant : semval list > asl > semval =
The main function must be able to treat an ASL declaration, evaluate it, and
update the global environments (global_env and global_sem).
#let semantics = function Decl(s,e) >
# let result = semant !global_sem e
# in global_env := s::!global_env;
# global_sem := result::!global_sem;
# print_string "ASL Value of ";
# print_string s;
# print_string " is ";
# (match result with
# Constval n > print_int n
#  Funval f > print_string "");
# print_newline();;
semantics : top_asl > unit =
13.3 Examples
#semantics (parse_top "let f be \\x. + x 1;");;
ASL Value of f is
 : unit = ()
#semantics (parse_top "let i be \\x. x;");;
ASL Value of i is
 : unit = ()
#semantics (parse_top "let x be i (f 2);");;
ASL Value of x is 3
 : unit = ()
#semantics (parse_top "let y be if x then (\\x.x) else 2 fi 0;");;
ASL Value of y is 0
 : unit = ()
94
Chapter 14
Encoding recursion
14.1 Fixpoint combinators
We have seen that we do not have recursion in ASL. However, it is possible
to encode recursion by defining a fixpoint combinator. A fixpoint
combinator is a function F such that:
F M is equivalent to M (F M) modulo the evaluation rules.
for any expression M. A consequence of the equivalence given above is that
fixpoint combinators can encode recursion. Let us note M=N if expressions
M and N are equivalent modulo the evaluation rules. Then, consider ffact to
be the functional obtained from the body of the factorial function by
abstracting (i.e. using as a parameter) the fact identifier, and fix an
arbitrary fixpoint combinator. We have:
ffact is \fact.(\n. if = n 0 then 1 else * n (fact ( n 1)) fi)
Now, let us consider the expression E=(fix ffact) 3. Using our intuition
about the evaluation rules, and the definition of a fixpoint combinator, we
obtain:
E= ffact (fix ffact) 3
Replacing ffact by its definition, we obtain:
E= (\fact.(\n. if = n 0 then 1 else * n (fact ( n 1)) fi)) (fix
ffact) 3
We can now pass the two arguments to the first abstraction, instantiating
fact and n respectively to fix ffact and 3:
E= if = 3 0 then 1 else * 3 (fix ffact ( 3 1)) fi
We can now reduce the conditional into its else branch:
E= * 3 (fix ffact ( 3 1))
Continuing this way, we eventually compute:
E= * 3 (* 2 (* 1 1)) = 6
This is the expected behavior of the factorial function. Given an
appropriate fixpoint combinator fix, we could define the factorial function
as fix ffact, where ffact is the expression above.
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Unfortunately, when using callbyvalue, any application of a fixpoint
combinator F such that:
F M evaluates to M (F M)
leads to nontermination of the evaluation (because evaluation of (F M)
leads to evaluating (M (F M)), and thus (F M) again).
We will use the Z fixpoint combinator defined by:
Z=lf.((lx. f (ly. (x x) y))(lx. f (ly. (x x) y)))
The fixpoint combinator Z has the particularity of being usable under
callbyvalue evaluation regime (in order to check that fact, it is
necessary to know the evaluation rules of lcalculus). Since the name z
looks more like an ordinary parameter name, we will call fix the ASL
expression corresponding to the Z fixpoint combinator.
#semantics (parse_top
# "let fix be \\f.((\\x.f(\\y.(x x) y))(\\x.f(\\y.(x x) y)));");;
ASL Value of fix is
 : unit = ()
We are now able to define the ASL factorial function:
#semantics (parse_top
# "let fact be fix (\\f.(\\n. if = n 0 then 1
# else * n (f ( n 1)) fi));");;
ASL Value of fact is
 : unit = ()
#semantics (parse_top "fact 8;");;
ASL Value of it is 40320
 : unit = ()
and the ASL Fibonacci function:
#semantics (parse_top
# "let fib be fix (\\f.(\\n. if = n 1 then 1
# else if = n 2 then 1
# else + (f ( n 1)) (f (
n 2)) fi fi));");;
ASL Value of fib is
 : unit = ()
#semantics (parse_top "fib 9;");;
ASL Value of it is 34
 : unit = ()
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14.2 Recursion as a primitive construct
Of course, in a more realistic prototype, we would extend the concrete and
abstract syntaxes of ASL in order to support recursion as a primitive
construct. We do not do it here because we want to keep ASL simple. This
is an interesting non trivial exercise!
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Chapter 15
Static typing, polymorphism and type synthesis
We now want to perform static typechecking of ASL programs, that is, to
complete typechecking before evaluation, making runtime type tests
unnecessary during evaluation of ASL programs.
Furthermore, we want to have polymorphism (i.e. allow the identity
function, for example, to be applicable to any kind of data).
Type synthesis may be seen as a game. When learning a game, we must:
 learn the rules (what is allowed, and what is forbidden);
 learn a winning strategy.
In type synthesis, the rules of the game are called a type system, and the
winning strategy is the typechecking algorithm.
In the following sections, we give the ASL type system, the algorithm and
an implementation of that algorithm. Most of this presentation is borrowed
from [7].
15.1 The type system
We study in this section a type system for the ASL language. Then, we
present an algorithm performing the type synthesis of ASL programs, and its
Caml Light implementation. Because of subtle aspects of the notation used
(inference rules), and since some important mathematical notions, such as
unification of firstorder terms, are not presented here, this chapter may
seem obscure at first reading.
The type system we will consider for ASL has been first given by Milner
[27] for a subset of the ML language (in fact, a superset of lcalculus). A
type is either:
 the type Number;
 or a type variable (a, b, ...);
 or t1>t2, where t1 and t2 are types.
In a type, a type variable is an unknown, i.e. a type that we are
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computing. We will use t, t', t1, ..., as metavariables(1) representing
types. This notation is important: we shall use other greek letters to
denote other notions in the following sections.
Example (a>Number)>b>b is a type.
O
A type scheme, is a type where some variables are distinguished as being
generic. We can represent type schemes by:
for all a1,.. .,an.t where t is a type.
Example
for all a.(a>Number)>b>b and (a>Number)>b>b are type schemes.
O We will use s, s', s1, ..., as metavariables representing type schemes.
# #
We may also write type schemes as for all a.t. In this case, a represent a
(possibly empty) set of generic type variables. When the set of generic
variables is empty, we write for all .t or simply t.
We will write FV(s) for the set of unknowns occurring in the type scheme
s. Unknowns are also called free variables (they are not bound by a
for all quantifier).
We also write BV(s) (bound type variables of s) for the set of type
variables occurring in s which are not free (i.e. the set of variables
universally quantified). Bound type variables are also said to be generic.
Example If s denotes the type scheme for all a.(a>Number)>b>b, then we
have:
FV(s)= {b}
and
BV(s)= {a}
O A substitution instance s' of a type scheme s is the type scheme S(s)
where S is a substitution of types for free type variables appearing in s.
When applying a substitution to a type scheme, a renaming of some bound type
variables may become necessary, in order to avoid the capture of a free type
variable by a quantifier.
Example
 If s denotes for all b.(b>a)>a and s' is
for all b.(b>(g>g))>(g>g), then s' is a substitution instance of s

1. A metavariable should not be confused with a variable or a type
variable.
99
because s'= S(s) where S={a<(g>g)}, i.e. S substitutes the type g>g
for the variable a.
 If s denotes for all b.(b>a)>a and s' is
for all d.(d>(b>b))>(b>b), then s' is a substitution instance of s
because s'= S(s) where S={a<(b>b)}. In this case, the renaming of b
into d was necessary: we did not want the variable b introduced by S to
be captured by the universal quantification for all b.
O The type scheme s'=for all b1 ... bm.t' is said to be a generic instance
of s=for all a1 ... an.t if there exists a substitution S such that:
 the domain of S is included in {a1,... ,an};
 t'= S(t);
 no bi occurs free in s.
In other words, a generic instance of a type scheme is obtained by giving
more precise values to some generic variables, and (possibly) quantifying
some of the new type variables introduced.
Example If s=for all b.(b>a)>a, then s' =for all g.((g>g)>a)>a is a
generic instance of s. We changed b into (g>g), and we universally
quantified on the newly introduced type variable g.
O
We express this type system by means of inference rules. An inference
rule is written as a fraction:
 the numerator is called the premisses;
 the denominator is called the conclusion.
An inference rule:
P1 ... Pn

C
may be read in two ways:
 ``If P1, ...and Pn , then C''.
 ``In order to prove C, it is sufficient to prove P1, ...and Pn ''.
An inference rule may have no premise: such a rule will be called an axiom.
100
A complete proof will be represented by a proof tree of the following form:
Pm ... ... Pk
1 l
 
.
.
.
... 
P1 ... P1
1 n

C
where the leaves of the tree (Pm, ..., Pk) are instances of axioms.
1 l
In the premisses and the conclusions appear judgements having the form:
G  e :s
Such a judgement should be read as ``under the typing environment G, the
expression e has type scheme s''. Typing environments are sets of typing
hypotheses of the form x:s where x is an identifier name and s is a type
scheme: typing environments give types to the variables occurring free
(i.e. unbound) in the expression.
When typing lcalculus terms, the typing environment is managed as a stack
(because identifiers possess local scopes). We represent that fact in the
presentation of the type system by removing the typing hypothesis concerning
an identifier name x (if such a typing hypothesis exists) before adding a
new typing hypothesis concerning x.
We write GG(x) for the set ot typing hypotheses obtained from G by
removing the typing hypothesis concerning x (if it exists).
Any numeric constant is of type Number:
 (NUM)
G  Const n :Number
We obtain type schemes for variables from the typing environment G:
 (TAUT)
G U {x: s}  Var x: s
It is possible to instantiate type schemes. The ``GenInstance'' relation
represents generic instantiation.
G  e :s s' =GenInstance(s)
 (INST)
G  e:s'
It is possible to generalize type schemes with respect to variables that do
not occur free in the set of hypotheses:
G  e:s a not in FV(G)
 (GEN)
G  e:for all a.s
101
Typing a conditional:
G  e1 :Number G  e2 :t G  e3 :t
 (IF)
G  (if e1 then e2 else e3 fi): t
Typing an application:
G  e1 :t>t' G  e2 :t
 (APP)
G  (e1 e2) :t'
Typing an abstraction:
(G G(x)) U {x:t}  e :t'
 (ABS)
G  (lx .e):t>t'
The special rule below is the one that introduces polymorphism: this
corresponds to the ML let construct.
G  e :s (G G(x)) U {x: s}  e': t
 (LET)
G  (lx .e') e :t
This type system has been proven to be semantically sound, i.e. the
semantic value of a welltyped expression (an expression admitting a type)
cannot be an error value due to a type error. This is usually expressed as:
Welltyped programs cannot go wrong.
This fact implies that a clever compiler may produce code without any
dynamic type test for a welltyped expression.
Example Let us check, using the set of rules above, that the following is
true:
0  let f =lx.x in f f :b>b
In order to do so, we will use the equivalence between the let construct and
an application of an immediate abstraction (i.e. an expression having the
following shape: (lv.M)N. The (LET) rule will be crucial: without it, we
could not check the judgement above.
 (TAU (TAU (TAUT)
{x: a}  x: a G  f :for all a.a>a G  f: for all a.a>a
 (ABS (INS (INST)
0  (lx.x) :a>a G  f :(b>b)>(b>b) G  f: b>b
 (GE(APP)
0  (lx.x) :for all a.a>a G ={f :for all a.a>a}  f f :b>b
 (LET)
0  let f =lx.x in f f :b>b
O
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This type system does not tell us how to find the best type for an
expression. But what is the best type for an expression? It must be such
that any other possible type for that expression is more specific; in other
words, the best type is the most general.
15.2 The algorithm
How do we find the most general type for an expression of our language? The
problem with the set of rules above, is that we could instantiate and
generalize types at any time, introducing type schemes, while the most
important rules (application and abstraction) used only types.
Let us write a new set of inference rules that we will read as an
algorithm (close to a Prolog program):
Any numeric constant is of type Number:
 (NUM)
G  Const n :Number
The types of identifiers are obtained by taking generic instances of type
schemes appearing in the typing environment. These generic instances will
be types and not type schemes: this restriction appears in the rule below,
where the type t is expected to be a generic instance of the type scheme s.
As it is presented (belonging to a deduction system), the following rule
will have to anticipate the effect of the equality constraints between types
in the other rules (multiple occurrences of a type metavariable), when
choosing the instance t.
t =GenInstance(s)
 (INST)
G U {x: s}  Var x: t
When we read this set of inference rules as an algorithm, the (INST) rule
will be implemented by:
1. taking as t the ``most general generic instance'' of s that is a type
(the rule requires t to be a type and not a type scheme),
2. making t more specific by unification [32] when encountering equality
constraints.
Typing a conditional requires only the test part to be of type Number, and
both alternatives to be of the same type t. This is an example of equality
constraint between the types of two expressions.
G  e1 :Number G  e2 :t G  e3 :t
 (COND)
G  (if e1 then e2 else e3 fi): t
Typing an application produces also equality constraints that are to be
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solved by unification:
G  e1 :t>t' G  e2 :t
 (APP)
G  (e1 e2) :t'
Typing an abstraction ``pushes'' a typing hypothesis for the abstracted
identifier: unification will make it more precise during the typing of the
abstraction body:
(G G(x)) U {x: for all .t}  e: t'
 (ABS)
G  (lx .e):t>t'
Typing a let construct involves a generalization step: we generalize as
much as possible.
G  e: t' {a ,.. .,a } =FV(t') FV(G) (G G(x)) U {x: for all a ... a .t'}  e': t
1 n 1 n
 (LET)
G  (lx .e') e: t
This set of inference rules represents an algorithm because there is
exactly one conclusion for each syntactic ASL construct (giving priority to
the (LET) rule over the regular application rule). This set of rules may be
read as a Prolog program.
This algorithm has been proven to be:
 syntactically sound: when the algorithm succeeds on an expression e and
returns a type t, then e:t.
 complete: if an expression e possesses a type t, then the algorithm
will find a type t' such that t is an instance of t'. The returned
type t' is thus the most general type of e.
Example
We compute the type that we simply checked in our last example. What is
drawn below is the result of the type synthesis: in fact, we run our
algorithm with type variables representing unknowns, modifying the previous
applications of the (INST) rule when necessary (i.e. when encountering an
equality constraint). This is valid, since it can be proved that the
correction of the whole deduction tree is preserved by substitution of types
for type variables. In a real implementation of the algorithm, the data
structures representing types will be submitted to a unification mechanism.
 (INS (INS(INST)
{x :a}  x :a G  f: (b>b)>(b>b) G  f:b>b
 (AB(APP)
0  (lx.x): a>a G= {f: for all a.a>a}  f f: b>b
 (LET)
0  let f =lx.x in f f :b>b
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Once again, this expression is not typable without the use of the (LET)
rule: an error would occur because of the type equality constraints between
all occurrences of a variable bound by a ``l''. In an effective
implementation, a unification error would occur.
O
We may notice from the example above that the algorithm is
syntaxdirected: since, for a given expression, a type deduction for that
expression uses exactly one rule per subexpression, the deduction possesses
the same structure as the expression. We can thus reconstruct the ASL
expression from its type deduction tree. From the deduction tree above, if
we write upper rules as being ``arguments'' of the ones below and if we
annotate the applications of the (INST) and (ABS) rules by the name of the
subject variable, we obtain:
LETf(ABSx(INSTx), APP(INSTf, INSTf))
This is an illustration of the ``typesaspropositions and
programsasproofs'' paradigm, also known as the ``CurryHoward
isomorphism'' (cf. [14]). In this example, we can see the type of the
considered expression as a proposition and the expression itself as the
proof, and, indeed, we recognize the expression as the deduction tree.
15.3 The ASL typesynthesizer
We now implement the set of inference rules given above.
We need:
 a Caml representation of ASL types and type schemes,
 a management of type environments,
 a unification procedure,
 a typing algorithm.
15.3.1 Representation of ASL types and type schemes
We first need to define a Caml type for our ASL type data structure:
#type asl_type = Unknown
#  Number
#  TypeVar of vartype
#  Arrow of asl_type * asl_type
#and vartype = {Index:int; mutable Value:asl_type}
#and asl_type_scheme = Forall of int list * asl_type ;;
Type asl_type defined.
Type vartype defined.
Type asl_type_scheme defined.
The Unknown ASL type is not really a type: it is the initial value of fresh
105
ASL type variables. We will consider as abnormal a situation where Unknown
appears in place of a regular ASL type. In such situations, we will raise
the following exception:
#exception TypingBug of string;;
Exception TypingBug defined.
Type variables are allocated by the new_vartype function, and their global
counter (a local reference) is reset by reset_vartypes.
#let new_vartype, reset_vartypes =
#(* Generating and resetting unknowns *)
# let counter = ref 0
# in (function () > counter := !counter + 1;
# {Index = !counter; Value = Unknown}),
# (function () > counter := 0);;
new_vartype : unit > vartype =
reset_vartypes : unit > unit =
15.3.2 Destructive unification of ASL types
We will need to ``shorten'' type variables: since they are indirections to
ASL types, we need to follow these indirections in order to obtain the type
that they represent. For the sake of efficiency, we take advantage of this
operation to replace multiple indirections by single indirections
(shortening).
#let rec shorten t =
# match t with
# TypeVar {Index=_; Value=Unknown} > t
#  TypeVar ({Index=_;
# Value=TypeVar {Index=_;
# Value=Unknown} as tv}) > tv
#  TypeVar ({Index=_; Value=TypeVar tv1} as tv2)
# > tv2.Value < tv1.Value; shorten t
#  TypeVar {Index=_; Value=t'} > t'
#  Unknown > raise (TypingBug "shorten")
#  t' > t';;
shorten : asl_type > asl_type =
An ASL type error will be represented by the following exception:
#exception TypeClash of asl_type * asl_type;;
Exception TypeClash defined.
We will need unification on ASL types with occurcheck. The following
function implements occurcheck:
#let occurs {Index=n;Value=_} =
# let rec occrec = function
106
# TypeVar{Index=m;Value=_} > (n=m)
#  Number > false
#  Arrow(t1,t2) > (occrec t1) or (occrec t2)
#  Unknown > raise (TypingBug "occurs")
# in occrec
#;;
occurs : vartype > asl_type > bool =
The unification function: implements destructive unification. Instead of
returning the most general unifier, it returns the unificand of two types
(their most general common instance). The two arguments are physically
modified in order to represent the same type. The unification function will
detect type clashes.
#let rec unify (tau1,tau2) =
# match (shorten tau1, shorten tau2)
# with (* type variable n and type variable m *)
# (TypeVar({Index=n; Value=Unknown} as tv1) as t1),
# (TypeVar({Index=m; Value=Unknown} as tv2) as t2)
# > if n <> m then tv1.Value < t2
#  (* type t1 and type variable *)
# t1, (TypeVar ({Index=_;Value=Unknown} as tv) as t2)
# > if not(occurs tv t1) then tv.Value < t1
# else raise (TypeClash (t1,t2))
#  (* type variable and type t2 *)
# (TypeVar ({Index=_;Value=Unknown} as tv) as t1), t2
# > if not(occurs tv t2) then tv.Value < t2
# else raise (TypeClash (t1,t2))
#  Number, Number > ()
#  Arrow(t1,t2), (Arrow(t'1,t'2) as t)
# > unify(t1,t'1); unify(t2,t'2)
#  (t1,t2) > raise (TypeClash (t1,t2));;
unify : asl_type * asl_type > unit =
15.3.3 Representation of typing environments
We use asl_type_scheme list as typing environments, and we will use the
encoding of variables as indices into the environment.
The initial environment is a list of types (Number > (Number > Number)),
which are the types of the ASL primitive functions.
#let init_typing_env =
# map (function s >
# Forall([],Arrow(Number,
# Arrow(Number,Number))))
# init_env;;
The global typing environment is initialized to the initial typing
environment, and will be updated with the type of each ASL declaration,
after they are typechecked.
107
#let global_typing_env = ref init_typing_env;;
15.3.4 From types to type schemes: generalization
In order to implement generalization, we will need some functions collecting
types variables occurring in ASL types.
The following function computes the list of type variables of its
argument.
#let vars_of_type tau =
# let rec vars vs = function
# Number > vs
#  TypeVar {Index=n; Value=Unknown}
# > if mem n vs then vs else n::vs
#  TypeVar {Index=_; Value= t} > vars vs t
#  Arrow(t1,t2) > vars (vars vs t1) t2
#  Unknown > raise (TypingBug "vars_of_type")
# in vars [] tau
#;;
vars_of_type : asl_type > int list =
The unknowns_of_type(bv,t) application returns the list of variables
occurring in t that do not appear in bv. The subtract function returns the
difference of two lists.
#let unknowns_of_type (bv,t) =
# subtract (vars_of_type t) bv;;
unknowns_of_type : int list * asl_type > int list =
We need to compute the list of unknowns of a type environment for the
generalization process (unknowns belonging to that list cannot become
generic). The set of unknowns of a type environment is the union of the
unknowns of each type. The flat function flattens a list of lists.
#let flat = it_list (prefix @) [];;
flat : '_a list list > '_a list =
#let unknowns_of_type_env env =
# flat (map (function Forall(gv,t) > unknowns_of_type (gv,t)) env);;
unknowns_of_type_env : asl_type_scheme list > int list =
The generalization of a type is relative to a typing environment. The
make_set function eliminates duplicates in its list argument.
#let rec make_set = function
# [] > []
#  x::l > if mem x l then make_set l else x :: make_set l;;
make_set : 'a list > 'a list =
#let generalise_type (gamma, tau) =
108
# let genvars =
# make_set (subtract (vars_of_type tau)
# (unknowns_of_type_env gamma))
# in Forall(genvars, tau)
#;;
generalise_type : asl_type_scheme list * asl_type > asl_type_scheme =
15.3.5 From type schemes to types: generic instantiation
The following function returns a generic instance of its type scheme
argument. A generic instance is obtained by replacing all generic type
variables by new unknowns:
#let gen_instance (Forall(gv,tau)) =
# (* We associate a new unknown to each generic variable *)
# let unknowns = map (function n > n, TypeVar(new_vartype())) gv in
# let rec ginstance = function
# (TypeVar {Index=n; Value=Unknown} as t) >
# (try assoc n unknowns
# with Not_found > t)
#  TypeVar {Index=_; Value= t} > ginstance t
#  Number > Number
#  Arrow(t1,t2) > Arrow(ginstance t1, ginstance t2)
#  Unknown > raise (TypingBug "gen_instance")
# in ginstance tau
#;;
gen_instance : asl_type_scheme > asl_type =
15.3.6 The ASL type synthesizer
The type synthesizer is the asl_typing_expr function. Each of its match
cases corresponds to an inference rule given above.
#let rec asl_typing_expr gamma =
# let rec type_rec = function
# Const _ > Number
#  Var n >
# let sigma =
# try nth n gamma
# with Failure _ > raise (TypingBug "Unbound")
# in gen_instance sigma
#  Cond (e1,e2,e3) >
# unify(Number, type_rec e1);
# let t2 = type_rec e2 and t3 = type_rec e3
# in unify(t2, t3); t3
#  App((Abs(x,e2) as f), e1) > (* LET case *)
# let t1 = type_rec e1 in
# let sigma = generalise_type (gamma,t1)
# in asl_typing_expr (sigma::gamma) e2
109
#  App(e1,e2) >
# let u = TypeVar(new_vartype())
# in unify(type_rec e1,Arrow(type_rec e2,u)); u
#  Abs(x,e) >
# let u = TypeVar(new_vartype()) in
# let s = Forall([],u)
# in Arrow(u,asl_typing_expr (s::gamma) e)
# in type_rec;;
asl_typing_expr : asl_type_scheme list > asl > asl_type =
15.3.7 Typing, trapping type clashes and printing ASL types
Now, we define some auxiliary functions in order to build a ``goodlooking''
type synthesizer. First of all, a printing routine for ASL type schemes is
defined (using a function tvar_name which computes a decent name for type
variables).
#let tvar_name n =
# (* Computes a name "'a", ... for type variables, *)
# (* given an integer n representing the position *)
# (* of the type variable in the list of generic *)
# (* type variables *)
# let rec name_of n =
# let q,r = (n / 26), (n mod 26) in
# let s = make_string 1 (char_of_int (96+r)) in
# if q=0 then s else (name_of q)^s
# in "'"^(name_of n)
#;;
tvar_name : int > string =
Then a printing function for type schemes.
#let print_type_scheme (Forall(gv,t)) =
# (* Prints a type scheme. *)
# (* Fails when it encounters an unknown *)
# let names = let rec names_of = function
# (n,[]) > []
#  (n,(v1::Lv)) > (tvar_name n)::(names_of (n+1, Lv))
# in names_of (1,gv) in
# let tvar_names = combine (rev gv, names) in
# let rec print_rec = function
# TypeVar{Index=n; Value=Unknown} >
# let name = try assoc n tvar_names
# with Not_found >
# raise (TypingBug "Non generic variable")
# in print_string name
#  TypeVar{Index=_;Value=t} > print_rec t
#  Number > print_string "Number"
#  Arrow(t1,t2) >
# print_string "("; print_rec t1;
110
# print_string " > "; print_rec t2;
# print_string ")"
#  Unknown > raise (TypingBug "print_type_scheme")
# in print_rec t
#;;
Toplevel input:
>  (n,(v1::Lv)) > (tvar_name n)::(names_of (n+1, Lv))
> ^^
Warning: the variable Lv starts with an upper case letter in this pattern.
print_type_scheme : asl_type_scheme > unit =
Now, the main function which resets the type variables counter, calls the
type synthesizer, traps ASL type clashes and prints the resulting types. At
the end, the global environments are updated.
#let typing (Decl(s,e)) =
# reset_vartypes();
# let tau = (* TYPING *)
# try asl_typing_expr !global_typing_env e
# with TypeClash(t1,t2) > (* A typing error *)
# let vars=vars_of_type(t1)@vars_of_type(t2) in
# print_string "ASL Type clash between ";
# print_type_scheme (Forall(vars,t1));
# print_string " and ";
# print_type_scheme (Forall(vars,t2));
# print_newline();
# raise (Failure "ASL typing") in
# let sigma = generalise_type (!global_typing_env,tau) in
# (* UPDATING ENVIRONMENTS *)
# global_env := s::!global_env;
# global_typing_env := sigma::!global_typing_env;
# reset_vartypes ();
# (* PRINTING RESULTING TYPE *)
# print_string "ASL Type of ";
# print_string s;
# print_string " is ";
# print_type_scheme sigma; print_newline();;
typing : top_asl > unit =
15.3.8 Typing ASL programs
We reinitialize the parsing environment:
#global_env:=init_env; ();;
 : unit = ()
Now, let us run some examples through the ASL type checker:
#typing (parse_top "let x be 1;");;
ASL Type of x is Number
111
 : unit = ()
#typing (parse_top "+ 2 ((\\x.x) 3);");;
ASL Type of it is Number
 : unit = ()
#typing (parse_top "if + 0 1 then 1 else 0 fi;");;
ASL Type of it is Number
 : unit = ()
#typing (parse_top "let id be \\x.x;");;
ASL Type of id is ('a > 'a)
 : unit = ()
#typing (parse_top "+ (id 1) (id id 2);");;
ASL Type of it is Number
 : unit = ()
#typing (parse_top "let f be (\\x.x x) (\\x.x);");;
ASL Type of f is ('a > 'a)
 : unit = ()
#typing (parse_top "+ (\\x.x) 1;");;
ASL Type clash between Number and ('a > 'a)
Uncaught exception: Failure "ASL typing"
15.3.9 Typing and recursion
The Z fixpoint combinator does not have a type in Milner's type system:
#typing (parse_top
# "let fix be \\f.((\\x.f(\\z.(x x)z)) (\\x.f(\\z.(x x)z)));");;
ASL Type clash between 'a and ('a > 'b)
Uncaught exception: Failure "ASL typing"
This is because we try to apply x to itself, and the type of x is not
polymorphic. In fact, no fixpoint combinator is typable in ASL. This is why
we need a special primitive or syntactic construct in order to express
recursivity.
If we want to assign types to recursive programs, we have to predefine the
Z fixpoint combinator. Its type scheme should be for all a.((a>a)>a),
because we take fixpoints of functions.
#global_env := "fix"::init_env;
#global_typing_env:=
# (Forall([1],
# Arrow(Arrow(TypeVar{Index=1;Value=Unknown},
# TypeVar{Index=1;Value=Unknown}),
# TypeVar{Index=1;Value=Unknown})))
# ::init_typing_env;
112
#();;
 : unit = ()
We can now define our favorite functions as:
#typing (parse_top
# "let fact be fix (\\f.(\\n. if = n 0 then 1
# else * n (f ( n 1))
# fi));");;
ASL Type of fact is (Number > Number)
 : unit = ()
#typing (parse_top "fact 8;");;
ASL Type of it is Number
 : unit = ()
#typing (parse_top
# "let fib be fix (\\f.(\\n. if = n 1 then 1
# else if = n 2 then 1
# else + (f( n 1)) (f( n 2))
# fi
# fi));");;
ASL Type of fib is (Number > Number)
 : unit = ()
#typing (parse_top "fib 9;");;
ASL Type of it is Number
 : unit = ()
113
Chapter 16
Compiling ASL to an abstract machine code
In order to fully take advantage of the static typing of ASL programs, we
have to:
 either write a new interpreter without type tests (difficult, because we
used patternmatching in order to realize type tests);
 or design an untyped machine and produce code for it.
We choose here the second solution: it will permit us to give some
intuition about the compiling process of functional languages, and to
describe a typical execution model for (strict) functional languages. The
machine that we will use is a simplified version the Categorical Abstract
Machine (CAM, for short).
We will call CAM our abstract machine, despite its differences with the
original CAM. For more informations on the CAM, see [8, 26].
16.1 The Abstract Machine
The execution model is a stack machine (i.e. a machine using a stack). In
this section, we define in Caml the states of the CAM and its instructions.
A state is composed of:
 a register (holding values and environments),
 a program counter, represented here as a list of instructions whose
first element is the current instruction being executed,
 and a stack represented as a list of code addresses (instruction lists),
values and environments.
The first Caml type that we need is the type for CAM instructions. We will
study later the effect of each instruction.
#type instruction =
# Quote of int (* Integer constants *)
# Plus  Minus (* Arithmetic operations *)
114
# Divide  Equal  Times
# Nth of int (* Variable accesses *)
# Branch of instruction list * instruction list
# (* Conditional execution *)
# Push (* Pushes onto the stack *)
# Swap (* Exch. top of stack and register *)
# Clos of instruction list (* Builds a closure with the current environ
ment *)
# Apply (* Function application *)
#;;
Type instruction defined.
We need a new type for semantic values since instruction lists have now
replaced abstract syntax trees. The semantic values are merged in a type
object. The type object behaves as data in a computer memory: we need
higherlevel information (such as type information) in order to interpret
them. Furthermore, some data do not correspond to anything (for example an
environment composed of environments represents neither an ASL value nor an
intermediate data in a legal computation process).
#type object = Constant of int
#  Closure of object * object
#  Address of instruction list
#  Environment of object list
#;;
Type object defined.
The type state is a product type with mutable components.
#type state = {mutable Reg: object;
# mutable PC: instruction list;
# mutable Stack: object list}
#;;
Type state defined.
Now, we give the operational semantics of CAM instructions. The effect of
an instruction is to change the state configuration. This is what we
describe now with the step function. Code executions will be arbitrary
iterations of this function.
#exception CAMbug of string;;
Exception CAMbug defined.
#exception CAM_End of object;;
Exception CAM_End defined.
#let step state = match state with
# {Reg=_; PC=Quote(n)::code; Stack=s} >
# state.Reg < Constant(n); state.PC < code
#
# {Reg=Constant(m); PC=Plus::code; Stack=Constant(n)::s} >
115
# state.Reg < Constant(n+m); state.Stack < s;
# state.PC < code
#
# {Reg=Constant(m); PC=Minus::code; Stack=Constant(n)::s} >
# state.Reg < Constant(nm); state.Stack < s;
# state.PC < code
#
# {Reg=Constant(m); PC=Times::code; Stack=Constant(n)::s} >
# state.Reg < Constant(n*m); state.Stack < s;
# state.PC < code
#
# {Reg=Constant(m); PC=Divide::code; Stack=Constant(n)::s} >
# state.Reg < Constant(n/m); state.Stack < s;
# state.PC < code
#
# {Reg=Constant(m); PC=Equal::code; Stack=Constant(n)::s} >
# state.Reg < Constant(if n=m then 1 else 0);
# state.Stack < s; state.PC < code
#
# {Reg=Constant(m); PC=Branch(code1,code2)::code; Stack=r::s} >
# state.Reg < r;
# state.Stack < Address(code)::s;
# state.PC < (if m=0 then code2 else code1)
#
# {Reg=r; PC=Push::code; Stack=s} >
# state.Stack < r::s; state.PC < code
#
# {Reg=r1; PC=Swap::code; Stack=r2::s} >
# state.Reg < r2; state.Stack < r1::s;
# state.PC < code
#
# {Reg=r; PC=Clos(code1)::code; Stack=s} >
# state.Reg < Closure(Address(code1),r);
# state.PC < code
#
# {Reg=_; PC=[]; Stack=Address(code)::s} >
# state.Stack < s; state.PC < code
#
# {Reg=v; PC=Apply::code;
# Stack=Closure(Address(code1),Environment(e))::s} >
# state.Reg < Environment(v::e);
# state.Stack < Address(code)::s;
# state.PC < code1
#
# {Reg=v; PC=[]; Stack=[]} >
# raise (CAM_End v)
# {Reg=_; PC=(PlusMinusTimesDivideEqual)::code; Stack=_::_} >
# raise (CAMbug "IllTyped")
#
# {Reg=Environment(e); PC=Nth(n)::code; Stack=_} >
# state.Reg < (try nth n e
116
# with Failure _ > raise (CAMbug "IllTyped"));
# state.PC < code
# _ > raise (CAMbug "Wrong configuration")
#;;
step : state > unit =
We may notice that the empty code sequence denotes a (possibly final) return
instruction.
We could argue that patternmatching in the Camlstep function implements a
kind of dynamic typechecking. In fact, in a concrete (lowlevel)
implementation of the machine (expansion of the CAM instructions in assembly
code, for example), these tests would not appear. They are useless since we
trust the typechecker and the compiler. So, any execution error in a real
implementation comes from a bug in one of the above processes and would lead
to memory errors or illegal instructions (usually detected by the computer's
operating system).
16.2 Compiling ASL programs into CAM code
We give in this section a compiling function taking the abstract syntax tree
of an ASL expression and producing CAM code. The compilation scheme is very
simple:
 the code of a constant is Quote;
 a variable is compiled as an access to the appropriate component of the
current environment (Nth);
 the code of a conditional expression will save the current environment
(Push), evaluate the condition part, and, according to the boolean value
obtained, select the appropriate code to execute (Branch);
 the code of an application will also save the environment on the stack
(Push), execute the function part of the application, then exchange the
functional value and the saved environment (Swap), evaluate the argument
and, finally, apply the functional value (which is at the top of the
stack) to the argument held in the register with the Apply instruction;
 the code of an abstraction simply consists in building a closure
representing the functional value: the closure is composed of the code
of the function and the current environment.
Here is the compiling function:
#let rec code_of = function
# Const(n) > [Quote(n)]
# Var n > [Nth(n)]
# Cond(e_test,e_t,e_f) >
# Push::(code_of e_test)
# @[Branch(code_of e_t, code_of e_f)]
117
# App(e1,e2) > Push::(code_of e1)
# @[Swap]@(code_of e2)
# @[Apply]
# Abs(_,e) > [Clos(code_of e)];;
code_of : asl > instruction list =
A global environment is needed in order to maintain already defined values.
Any CAM execution will start in a state whose register part contains this
global environment.
#let init_CAM_env =
# let basic_instruction = function
# "+" > Plus
#  "" > Minus
#  "*" > Times
#  "/" > Divide
#  "=" > Equal
#  s > raise (CAMbug "Unknown primitive")
# in map (function s >
# Closure(Address[Clos(Push::Nth(2)
# ::Swap::Nth(1)
# ::[basic_instruction s])],
# Environment []))
# init_env;;
init_CAM_env : object list =
[Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Plus]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Minus]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Times]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Divide]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Equal]], Environment [])]
#let global_CAM_env = ref init_CAM_env;;
global_CAM_env : object list ref =
ref
[Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Plus]], Environ
ment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Minus]], Environ
ment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Times]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Divide]], Environment []);
Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Equal]], Environment [])]
As an example, here is the code for some ASL expressions.
#code_of (expression(lexer(stream_of_string "1;")));;
 : instruction list = [Quote 1]
#code_of (expression(lexer(stream_of_string "+ 1 2;")));;
 : instruction list =
[Push; Push; Nth 6; Swap; Quote 1; Apply; Swap; Quote 2; Apply]
118
#code_of (expression(lexer(stream_of_string "(\\x.x) ((\\x.x) 0);")));;
 : instruction list =
[Push; Clos [Nth 1]; Swap; Push; Clos [Nth 1]; Swap; Quote 0; Apply; Apply]
#code_of (expression(lexer(stream_of_string
# "+ 1 (if 0 then 2 else 3 fi);")));;
 : instruction list =
[Push; Push; Nth 6; Swap; Quote 1; Apply; Swap; Push; Quote 0;
Branch ([Quote 2], [Quote 3]); Apply]
16.3 Execution of CAM code
The main function for executing compiled ASL manages the global environment
until execution has succeeded.
#let run (Decl(s,e)) =
# (* TYPING *)
# reset_vartypes();
# let tau =
# try asl_typing_expr !global_typing_env e
# with TypeClash(t1,t2) >
# let vars=vars_of_type(t1) @ vars_of_type(t2) in
# print_string "ASL Type clash between ";
# print_type_scheme (Forall(vars,t1));
# print_string " and ";
# print_type_scheme (Forall(vars,t2));
# raise (Failure "ASL typing")
#  Unbound s > raise (TypingBug ("Unbound: "^s)) in
# let sigma = generalise_type (!global_typing_env,tau) in
# (* PRINTING TYPE INFORMATION *)
# print_string "ASL Type of ";
# print_string s; print_string " is ";
# print_type_scheme sigma; print_newline();
# (* COMPILING *)
# let code = code_of e in
# let state = {Reg=Environment(!global_CAM_env); PC=code; Stack=[]} in
# (* EXECUTING *)
# let result = try while true do step state done; state.Reg
# with CAM_End v > v in
# (* UPDATING ENVIRONMENTS *)
# global_env := s::!global_env;
# global_typing_env := sigma::!global_typing_env;
# global_CAM_env := result::!global_CAM_env;
# (* PRINTING RESULT *)
# (match result
# with Constant(n) > print_int n
#  Closure(_,_) > print_string ""
#  _ > raise (CAMbug "Wrong state configuration"));
# print_newline();;
119
run : top_asl > unit =
Now, let us run some examples:
#(* Reinitializing environments *)
#global_env:=init_env;
#global_typing_env:=init_typing_env;
#global_CAM_env:=init_CAM_env;
#();;
 : unit = ()
#run (parse_top "1;");;
ASL Type of it is Number
1
 : unit = ()
#run (parse_top "+ 1 2;");;
ASL Type of it is Number
3
 : unit = ()
#run (parse_top "(\\f.(\\x.f x)) (\\x. + x 1) 3;");;
ASL Type of it is Number
4
 : unit = ()
We may now introduce the Z fixpoint combinator as a predefined function fix.
#begin
# global_env:="fix"::init_env;
# global_typing_env:=
# (Forall([1],
# Arrow(Arrow(TypeVar{Index=1;Value=Unknown},
# TypeVar{Index=1;Value=Unknown}),
# TypeVar{Index=1;Value=Unknown})))
# ::init_typing_env;
# global_CAM_env:=
# (match code_of (expression(lexer(stream_of_string
# "\\f.((\\x.f(\\z.(x x)z)) (\\x.f(\\z.(x x)z)));")))
# with [Clos(C)] > Closure(Address(C), Environment [])
#  _ > raise (CAMbug "Wrong code for fix"))
# ::init_CAM_env
#end;;
Toplevel input:
> with [Clos(C)] > Closure(Address(C), Environment [])
> ^
Warning: the variable C starts with an upper case letter in this pattern.
 : unit = ()
#run (parse_top
120
# "let fact be fix (\\f.(\\n. if = n 0 then 1
# else * n (f ( n 1))
# fi));");;
ASL Type of fact is (Number > Number)
 : unit = ()
#run (parse_top
# "let fib be fix (\\f.(\\n. if = n 1 then 1
# else if = n 2 then 1
# else + (f( n 1)) (f( n 2))
# fi
# fi));");;
ASL Type of fib is (Number > Number)
 : unit = ()
#run (parse_top "fact 8;");;
ASL Type of it is Number
40320
 : unit = ()
#run (parse_top "fib 9;");;
ASL Type of it is Number
34
 : unit = ()
It is of course possible (and desirable) to introduce recursion by using a
specific syntactic construct, special instructions and a dedicated case to
the compiling function. See [26] for efficient compilation of recursion,
data structures etc.
Exercise 16.1 Interesting exercises for which we won't give solutions
consist in enriching according to your taste the ASL language. Also,
building a standalone ASL interpreter is a good exercise in modular
programming.
121
Chapter 17
Answers to exercises
We give in this chapter one possible solution for each exercise contained in
this document. Exercises are referred to by their number and the page where
they have been proposed: for example, ``2.1, p. 15'' refers to the first
exercise in chapter 2; this exercise is located on page 15.
3.1, p. 17
The following (anonymous) functions have the required types:
#function f > (f 2)+1;;
 : (int > int) > int =
#function m > (function n > n+m+1);;
 : int > int > int =
#(function f > (function m > f(m+1) / 2));;
 : (int > int) > int > int =
3.2, p. 17
We must first rename y to z, obtaining:
(function x > (function z > x+z))
and finally:
(function y > (function z > y+z))
Without renaming, we would have obtained:
(function y > (function y > y+y))
which does not denote the same function.
122
3.3, p. 17
We write successively the reduction steps for each expressions, and then we
use Caml in order to check the result.
 let x=1+2 in ((function y > y+x) x);;
(function y > y+(1+2)) (1+2);;
(function y > y+(1+2)) 3;;
3+(1+2);;
3+3;;
6;;
Caml says:
#let x=1+2 in ((function y > y+x) x);;
 : int = 6
 let x=1+2 in ((function x > x+x) x);;
(function x > x+x) (1+2);;
3+3;;
6;;
Caml says:
#let x=1+2 in ((function x > x+x) x);;
 : int = 6
 let f1 = function f2 > (function x > f2 x)
in let g = function x > x+1
in f1 g 2;;
let g = function x > x+1
in function f2 > (function x > f2 x) g 2;;
(function f2 > (function x > f2 x)) (function x > x+1) 2;;
(function x > (function x > x+1) x) 2;;
(function x > x+1) 2;;
2+1;;
3;;
Caml says:
#let f1 = function f2 > (function x > f2 x)
#in let g = function x > x+1
# in f1 g 2;;
 : int = 3
123
4.1, p. 29
To compute the surface area of a rectangle and the volume of a sphere:
#let surface_rect len wid = len * wid;;
surface_rect : int > int > int =
#let pi = 4.0 *. atan 1.0;;
pi : float = 3.14159265359
#let volume_sphere r = 4.0 /. 3.0 *. pi *. (power r 3.);;
volume_sphere : float > float =
4.2, p. 29
In a callbyvalue language without conditional construct (and without sum
types), all programs involving a recursive definition never terminate.
4.3, p. 29
#let rec factorial n = if n=1 then 1 else n*(factorial(n1));;
factorial : int > int =
#factorial 5;;
 : int = 120
#let tail_recursive_factorial n =
# let rec fact n m = if n=1 then m else fact (n1) (n*m)
# in fact n 1;;
tail_recursive_factorial : int > int =
#tail_recursive_factorial 5;;
 : int = 120
4.4, p. 29
#let rec fibonacci n =
# if n=1 then 1
# else if n=2 then 1
# else fibonacci(n1) + fibonacci(n2);;
fibonacci : int > int =
#fibonacci 20;;
 : int = 6765
124
4.5, p. 29
#let compose f g = function x > f (g (x));;
compose : ('a > 'b) > ('c > 'a) > 'c > 'b =
#let curry f = function x > (function y > f(x,y));;
curry : ('a * 'b > 'c) > 'a > 'b > 'c =
#let uncurry f = function (x,y) > f x y;;
uncurry : ('a > 'b > 'c) > 'a * 'b > 'c =
#uncurry compose;;
 : ('_a > '_b) * ('_c > '_a) > '_c > '_b =
#compose curry uncurry;;
 : ('_a > '_b > '_c) > '_a > '_b > '_c =
#compose uncurry curry;;
 : ('_a * '_b > '_c) > '_a * '_b > '_c =
5.1, p. 34
#let rec combine =
# function [],[] > []
#  (x1::l1),(x2::l2) > (x1,x2)::(combine(l1,l2))
#  _ > raise (Failure "combine: lists of different length");;
combine : 'a list * 'b list > ('a * 'b) list =
#combine ([1;2;3],["a";"b";"c"]);;
 : (int * string) list = [1, "a"; 2, "b"; 3, "c"]
#combine ([1;2;3],["a";"b"]);;
Uncaught exception: Failure "combine: lists of different length"
5.2, p. 35
#let rec sublists =
# function [] > [[]]
#  x::l > let sl = sublists l
# in sl @ (map (fun l > x::l) sl);;
sublists : 'a list > 'a list list =
#sublists [];;
 : '_a list list = [[]]
#sublists [1;2;3];;
 : int list list = [[]; [3]; [2]; [2; 3]; [1]; [1; 3]; [1; 2]; [1; 2; 3]]
#sublists ["a"];;
125
 : string list list = [[]; ["a"]]
6.1, p. 45
#type ('a,'b) btree = Leaf of 'b
#  Btree of ('a,'b) node
#and ('a,'b) node = {Op:'a;
# Son1: ('a,'b) btree;
# Son2: ('a,'b) btree};;
Type btree defined.
Type node defined.
#let rec nodes_and_leaves =
# function Leaf x > ([],[x])
#  Btree {Op=x; Son1=s1; Son2=s2} >
# let (nodes1,leaves1) = nodes_and_leaves s1
# and (nodes2,leaves2) = nodes_and_leaves s2
# in (x::nodes1@nodes2, leaves1@leaves2);;
nodes_and_leaves : ('a, 'b) btree > 'a list * 'b list =
#nodes_and_leaves (Btree {Op="+"; Son1=Leaf 1; Son2=Leaf 2});;
 : string list * int list = ["+"], [1; 2]
6.2, p. 45
#let rec map_btree f g = function
# Leaf x > Leaf (f x)
#  Btree {Op=op; Son1=s1; Son2=s2}
# > Btree {Op=g op; Son1=map_btree f g s1;
# Son2=map_btree f g s2};;
map_btree : ('a > 'b) > ('c > 'd) > ('c, 'a) btree > ('d, 'b) btree =