Functional programming using Caml Light Michel Mauny January 1995 CCoonntteennttss 11 IInnttrroodduuccttiioonn 44 II FFuunnccttiioonnaall pprrooggrraammmmiinngg 66 22 FFuunnccttiioonnaall llaanngguuaaggeess 77 2.1 History of functional languages . . . . . . . . . . . . . . . . 8 2.2 The ML family . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Miranda family. . . . . . . . . . . . . . . . . . . . . . . 9 33 BBaassiicc ccoonncceeppttss 1100 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 44 BBaassiicc ttyyppeess 1188 4.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Boolean values. . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Strings and characters. . . . . . . . . . . . . . . . . . . . . 24 4.4 Tuples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5 Patterns and pattern-matching . . . . . . . . . . . . . . . . . 25 4.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 55 LLiissttss 3311 5.1 Building lists. . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Extracting elements from lists: pattern-matching . . . . . . . 32 5.3 Functions over lists. . . . . . . . . . . . . . . . . . . . . . 33 66 UUsseerr--ddeeffiinneedd ttyyppeess 3366 6.1 Product types . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 Sum types . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 IIII CCaammll LLiigghhtt ssppeecciiffiiccss 4477 77 MMuuttaabbllee ddaattaa ssttrruuccttuurreess 4488 7.1 User-defined 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 88 EEssccaappiinngg ffrroomm ccoommppuuttaattiioonnss:: eexxcceeppttiioonnss 5555 8.1 Exceptions. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2 Raising an exception. . . . . . . . . . . . . . . . . . . . . . 56 8.3 Trapping exceptions . . . . . . . . . . . . . . . . . . . . . . 57 8.4 Polymorphism and exceptions . . . . . . . . . . . . . . . . . . 58 99 BBaassiicc iinnppuutt//oouuttppuutt 6600 9.1 Printable types . . . . . . . . . . . . . . . . . . . . . . . . 60 9.2 Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9.3 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 9.4 Channels on files . . . . . . . . . . . . . . . . . . . . . . . 63 1100 SSttrreeaammss aanndd ppaarrsseerrss 6666 10.1 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 10.2 Stream matching and parsers . . . . . . . . . . . . . . . . . . 68 10.3 Parameterized parsers . . . . . . . . . . . . . . . . . . . . . 70 10.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . 78 1111 SSttaannddaalloonnee pprrooggrraammss aanndd sseeppaarraattee ccoommppiillaattiioonn 7799 11.1 Standalone programs . . . . . . . . . . . . . . . . . . . . . . 79 11.2 Programs in several files . . . . . . . . . . . . . . . . . . . 80 11.3 Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . 82 IIIIII AA ccoommpplleettee eexxaammppllee 8844 1122 AASSLL:: AA SSmmaallll LLaanngguuaaggee 8855 12.1 ASL abstract syntax trees . . . . . . . . . . . . . . . . . . . 85 12.2 Parsing ASL programs. . . . . . . . . . . . . . . . . . . . . . 86 1133 UUnnttyyppeedd sseemmaannttiiccss ooff AASSLL pprrooggrraammss 9922 13.1 Semantic values . . . . . . . . . . . . . . . . . . . . . . . . 92 13.2 Semantic functions. . . . . . . . . . . . . . . . . . . . . . . 93 13.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1144 EEnnccooddiinngg rreeccuurrssiioonn 9955 14.1 Fixpoint combinators. . . . . . . . . . . . . . . . . . . . . . 95 14.2 Recursion as a primitive construct. . . . . . . . . . . . . . . 97 1155 SSttaattiicc ttyyppiinngg,, ppoollyymmoorrpphhiissmm aanndd ttyyppee ssyynntthheessiiss 9988 15.1 The type system . . . . . . . . . . . . . . . . . . . . . . . . 98 15.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 103 15.3 The ASL type-synthesizer. . . . . . . . . . . . . . . . . . . . 105 1166 CCoommppiilliinngg AASSLL ttoo aann aabbssttrraacctt mmaacchhiinnee ccooddee 111144 16.1 The Abstract Machine. . . . . . . . . . . . . . . . . . . . . . 114 16.2 Compiling ASL programs into CAM code. . . . . . . . . . . . . . 117 2 16.3 Execution of CAM code . . . . . . . . . . . . . . . . . . . . . 119 1177 AAnnsswweerrss ttoo eexxeerrcciisseess 112222 1188 CCoonncclluussiioonnss aanndd ffuurrtthheerr rreeaaddiinngg 113344 3 CChhaapptteerr 11 IInnttrroodduuccttiioonn 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 2--6) 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 user-defined 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 7--11) 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 12--16) 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 first-order unification is assumed. The presentation is rather ------------------------------ 1. The ``Caml Strong'' version of these notes is available as an INRIA technical report [26]. 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 call-by-value 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 first-order terms representing types) as a central tool. Chapter 16 introduces the Categorical Abstract Machine: a simple execution model for call-by-value 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. WWaarrnniinngg:: The programs and remarks (especially contained in parts B and C) might not be valid in Caml Light versions different from 0.7. 5 PPaarrtt II FFuunnccttiioonnaall pprrooggrraammmmiinngg 6 CChhaapptteerr 22 FFuunnccttiioonnaall llaanngguuaaggeess Programming languages are said to be f_u_n_c_t_i_o_n_a_l_ when the basic way of structuring programs is the notion of f_u_n_c_t_i_o_n_ and their essential control structure is f_u_n_c_t_i_o_n_ a_p_p_l_i_c_a_t_i_o_n_. 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 ``first-class'' 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 s_u_c_c_e_s_s_o_r_ function in the following way: s_u_c_c_e_s_s_o_r_: NN -> NN 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 t_y_p_e_; a mathematical function always possesses a d_o_m_a_i_n_ and a c_o_d_o_m_a_i_n_. They will correspond to the programming notion of type. 2. lexical binding: when we wrote the mathematical definition of s_u_c_c_e_s_s_o_r_, 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 s_u_c_c_e_s_s_o_r_ function is defined using the m_e_a_n_i_n_g_ of the addition: whatever + denotes in the future, this s_u_c_c_e_s_s_o_r_ function will remain the same. 3. the notion of f_u_n_c_t_i_o_n_a_l_ a_b_s_t_r_a_c_t_i_o_n_, 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 non-functional programming styles, and, in this sense, they are functional but not p_u_r_e_l_y_ f_u_n_c_t_i_o_n_a_l_. 22..11 HHiissttoorryy ooff ffuunnccttiioonnaall llaanngguuaaggeess Some historical points: - 1930: Alonzo Church developed the l-calculus [6] as an attempt to provide a basis for mathematics. The l-calculus is a formal theory for functionality. The three basic constructs of the l-calculus are: - variable names (e.g. x, y,...); - application (MN if M and M are terms); - functional abstraction (lx.M). Terms of the l-calculus represent functions. The pure l-calculus has been proved inconsistent as a logical theory. Some t_y_p_e_ s_y_s_t_e_m_s_ 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 l-calculus. 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 m_e_t_a_l_a_n_g_u_a_g_e_ of the LCF proof assistant (i.e. the language used to program the search of proofs). This language is inspired by ISWIM (close to l-calculus) 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 l_a_z_y_ e_v_a_l_u_a_t_i_o_n_. Currently, the two main families of functional languages are the ML and the Miranda families. 8 22..22 TThhee MMLL ffaammiillyy ML languages are based on a sugared(1) version of l-calculus. Their evaluation regime is c_a_l_l_-_b_y_-_v_a_l_u_e_ (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 New-Jersey, 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. 22..33 TThhee MMiirraannddaa ffaammiillyy All languages in this family use l_a_z_y_ e_v_a_l_u_a_t_i_o_n_ (i.e. the argument of a function is evaluated if and when the function needs its value---arguments 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 s_e_t_s_ o_f_ m_u_t_u_a_l_l_y_ r_e_c_u_r_s_i_v_e_ e_q_u_a_t_i_o_n_s_ as programs. These equations are written in a s_c_r_i_p_t_ (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 G-machine [16]. ------------------------------ 1. i.e. with a more user-friendly syntax. 2. Laboratoire d'Informatique de l'Ecole Normale Superieure, 45 Rue d'Ulm, 75005 Paris 9 CChhaapptteerr 33 BBaassiicc ccoonncceeppttss 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. 33..11 TToopplleevveell lloooopp The first contact with Caml is through its interactive aspect(1). When running Caml on a computer, we enter a t_o_p_l_e_v_e_l_ l_o_o_p_ 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 pre-processed 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). IImmppoorrttaanntt:: Of course, when developing non-trivial 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 t_r_a_c_e_ some functions, to see with what arguments they are called, and what result they return. See the reference manual [21] for a description of these facilities. 33..22 EEvvaalluuaattiioonn:: ffrroomm eexxpprreessssiioonnss ttoo vvaalluueess 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 type-checking (more precisely t_y_p_e_ s_y_n_t_h_e_s_i_s_) 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 f_u_n_c_t_i_o_n_ a_p_p_l_i_c_a_t_i_o_n_. #(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 (well-typed) 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 well-typing 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 call-by-value: arguments are evaluated before being passed to functions. The relative evaluation order of the 12 functional expression and of the argument expression is implementation-dependent, and should not be relied upon. The Caml Light implementation evaluates arguments before functions (right-to-left evaluation), as shown above; the original Caml implementation evaluates functions before arguments (left-to-right evaluation). 33..33 TTyyppeess 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 t_y_p_e_s_. A type represents a collection of values. For example, int is the type of integer numbers, and float is the type of floating-point 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 s_y_n_t_a_x_ a_n_a_l_y_s_i_s_, 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 l_e_x_i_c_a_l_ e_r_r_o_r_s_ 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 t_y_p_e_ a_n_a_l_y_s_i_s_: 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 ill-typed phrases is called s_t_r_o_n_g_ t_y_p_i_n_g_. 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 run-time. 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 well-structured 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, well-typed 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. 33..44 FFuunnccttiioonnss 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). 33..55 DDeeffiinniittiioonnss It is important to give names to values. We have already seen some named values: we called them f_o_r_m_a_l_ p_a_r_a_m_e_t_e_r_s_. 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 l_o_c_a_l_ b_i_n_d_i_n_g_s_ o_f_ v_a_l_u_e_s_ t_o_ i_d_e_n_t_i_f_i_e_r_s_. They are l_o_c_a_l_ 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 s_h_a_r_i_n_g_ of (possibly time-consuming) evaluations. When evaluating ``let x=e1 in e2'', e1 gets evaluated only once. All occurrences of x in e2 access the v_a_l_u_e_ 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. 33..66 PPaarrttiiaall aapppplliiccaattiioonnss A p_a_r_t_i_a_l_ a_p_p_l_i_c_a_t_i_o_n_ 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 p_a_r_t_i_a_l_ a_p_p_l_i_c_a_t_i_o_n_, 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 EExxeerrcciisseess EExxeerrcciissee 33..11 Give examples of functions with the following types: 1. (int -> int) -> int 2. int -> (int -> int) 3. (int -> int) -> (int -> int) EExxeerrcciissee 33..22 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? EExxeerrcciissee 33..33 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 CChhaapptteerr 44 BBaassiicc ttyyppeess We examine in this chapter the Caml basic types. 44..11 NNuummbbeerrss Caml Light provides two numeric types: integers (type int) and floating-point numbers (type float). Integers are limited to the range -230... 230-1. 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 p_r_e_f_i_x_ 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 floating-point values. A separate set of floating-point 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 floating-point 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 r_e_c_u_r_s_i_v_e_ 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 1e-5;; - : float = 1.41421569553 #cubic_root 8.0 1e-5;; - : float = 2.00000000131 #2.0 *. (newton cos 1e-5 1.5);; - : float = 3.14159265359 44..22 BBoooolleeaann vvaalluueess 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 p_r_e_d_i_c_a_t_e_s_. 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 44..22..11 EEqquuaalliittyy Equality has a special status in functional languages because of functional values. It is easy to test equality of values of base types (integers, floating-point 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 run-time 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 p_o_l_y_m_o_r_p_h_i_c_ t_y_p_e_, 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 t_y_p_e_ v_a_r_i_a_b_l_e_s_, written 'a, `b, etc. Any type can be substituted to type variables in order to produces specific i_n_s_t_a_n_c_e_s_ 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 44..22..22 CCoonnddiittiioonnaall 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 44..22..33 LLooggiiccaall ooppeerraattoorrss 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 44..33 SSttrriinnggss aanndd cchhaarraacctteerrss String constants (type string) are written between " characters (double-quotes). Single-character 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 [21] for complete information. 44..44 TTuupplleess 44..44..11 CCoonnssttrruuccttiinngg ttuupplleess It is possible to combine values into tuples (pairs, triples, ...). The v_a_l_u_e_ c_o_n_s_t_r_u_c_t_o_r_ 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 t_y_p_e_ c_o_n_s_t_r_u_c_t_o_r_ 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 g_e_n_e_r_i_c_. 44..44..22 EExxttrraaccttiinngg ppaaiirr ccoommppoonneennttss P_r_o_j_e_c_t_i_o_n_ 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 user-defined 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 g_e_n_e_r_i_c_ 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. 44..55 PPaatttteerrnnss aanndd ppaatttteerrnn--mmaattcchhiinngg Patterns and pattern-matching play an important role in ML languages. They appear in all real programs and are strongly related to types (predefined or user-defined). A p_a_t_t_e_r_n_ indicates the s_h_a_p_e_ 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 m_a_t_c_h_e_d_ a_g_a_i_n_s_t_ a pattern (this is called p_a_t_t_e_r_n_-_m_a_t_c_h_i_n_g_), the pattern acts as a filter. We already used patterns and pattern-matching in all the functions we wrote: the function body (function x -> ...) does (trivial) pattern-matching. When applied to a value, the formal parameter x gets bound to that value. The function body (function (x,y) -> x+y) does also pattern-matching: when applied to a value (a pair, because of well-typing hypotheses), the x and y get bound respectively to the first and the second component of that pair. 25 All these pattern-matching 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 p_r_o_d_u_c_t_ t_y_p_e_). 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 pattern-matching 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 pattern-matching may fail (we did not consider the false case). Thus, if we apply f to a value that evaluates to true, pattern-matching will succeed (an equality test is performed during execution). #f (1<2);; - : bool = false But, if f's argument evaluates to false, a run-time error is reported: #f (1>2);; Uncaught exception: Match_failure ("", 1346, 1368) Here is a correct definition using pattern-matching: #let negate = function true -> false # | false -> true;; negate : bool -> bool = The pattern-matching 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 right-hand part of the match case. Since in our example we do not use the value of the argument in the right-hand 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 pattern-matching, 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 = Pattern-matching is allowed on any type of value (non-trivial pattern-matching is only possible on types with d_a_t_a_ c_o_n_s_t_r_u_c_t_o_r_s_). 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 44..66 FFuunnccttiioonnss 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. 44..66..11 FFuunnccttiioonnaall ccoommppoossiittiioonn 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 m_o_s_t_ g_e_n_e_r_a_l_ 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 44..66..22 CCuurrrryyiinngg 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 t_h_e_ c_u_r_r_i_e_d_ v_e_r_s_i_o_n_ of plus (in honor of the logician Haskell Curry). 28 The currying transformation may be written in Caml under the form of a higher-order 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 = EExxeerrcciisseess EExxeerrcciissee 44..11 Define functions that compute the surface area and the volume of well-known geometric objects (rectangles, circles, spheres, ...). EExxeerrcciissee 44..22 What would happen in a language submitted to call-by-value with recursion if there was no conditional construct (if ... then ... else ...)? EExxeerrcciissee 44..33 Define the factorial function such that: factorial n= n*(n -1)* ... *2* 1 Give two versions of factorial: recursive and tail recursive. EExxeerrcciissee 44..44 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 EExxeerrcciissee 44..55 What are the types of the following expressions? - uncurry compose 29 - compose curry uncurry - compose uncurry curry 30 CChhaapptteerr 55 LLiissttss Lists represent an important data structure, mainly because of their success in the Lisp language. Lists in ML are h_o_m_o_g_e_n_e_o_u_s_: 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). 55..11 BBuuiillddiinngg lliissttss Lists are empty or non empty sequences of elements. They are built with two v_a_l_u_e_ c_o_n_s_t_r_u_c_t_o_r_s_: - [], the empty list (read: n_i_l_); - ::, the non-empty list constructor (read: c_o_n_s_). It takes an element e and a list l, and builds a new list whose first element (h_e_a_d_) is e and rest (t_a_i_l_) 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 ``l_i_s_t_ o_f_ a_n_y_t_h_i_n_g_''. 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 r_e_c_u_r_s_i_v_e_ type. The :: constructor receives two arguments; the second argument is itself a list. 55..22 EExxttrraaccttiinngg eelleemmeennttss ffrroomm lliissttss:: ppaatttteerrnn--mmaattcchhiinngg 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 pattern-matching 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 run-time 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 s_u_m_ t_y_p_e_s_, 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 run-time tests in order to be matched by a non-variable pattern (i.e. a pattern that is not a single variable). The cartesian product is a p_r_o_d_u_c_t_ type (only one alternative). Product types do not involve run-time tests during pattern-matching, because the type of their values suffices to indicate statically what their structure is. 55..33 FFuunnccttiioonnss oovveerr lliissttss 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]. EExxeerrcciisseess EExxeerrcciissee 55..11 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 run-time error if the argument lists do not have the same length. EExxeerrcciissee 55..22 Define a function which, when given a list, returns the list of all its sublists. 35 CChhaapptteerr 66 UUsseerr--ddeeffiinneedd ttyyppeess 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 user-defined data types reflect precisely the needs of the algorithms. Types are either: - p_r_o_d_u_c_t_ types, - or s_u_m_ 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. 66..11 PPrroodduucctt ttyyppeess Product types are f_i_n_i_t_e_ l_a_b_e_l_e_d_ products of types. They are a generalization of cartesian product. Elements of product types are called r_e_c_o_r_d_s_. 66..11..11 DDeeffiinniinngg pprroodduucctt ttyyppeess 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 pattern-matching 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 4-tuples (and indeed would be the same function). This type is not precise enough since it allows for the application of the function to any 4-uple, 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 p_r_o_d_u_c_t_ 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 l_a_b_e_l_s_ (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. 66..11..22 EExxttrraaccttiinngg pprroodduuccttss ccoommppoonneennttss The canonical way of extracting product components is p_a_t_t_e_r_n_-_m_a_t_c_h_i_n_g_. Pattern-matching 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. 66..11..33 PPaarraammeetteerriizzeedd pprroodduucctt ttyyppeess It is important to be able to define parameterized types in order to define g_e_n_e_r_i_c_ 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 a_n_y_ 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 l_a_t_t_e_r_ type pair that we defined, while p's type is the f_o_r_m_e_r_ pair. 66..22 SSuumm ttyyppeess A s_u_m_ type is the f_i_n_i_t_e_ l_a_b_e_l_e_d_ disjoint union of several types. A sum type definition defines a type as being the union of some other types. 66..22..11 DDeeffiinniinngg ssuumm ttyyppeess 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 b_o_t_h_ int * int and character strings. We also want to preserve static type-checking, 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 i_n_j_e_c_t_i_o_n_s_. 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 e_n_u_m_e_r_a_t_e_d_ t_y_p_e_s_, 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)] 66..22..22 EExxttrraaccttiinngg ssuumm ccoommppoonneennttss Of course, pattern-matching is used to extract sum components. In case of sum types, pattern-matching 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 = 66..22..33 RReeccuurrssiivvee ttyyppeess Some types possess a naturally recursive structure, lists, for example. It is also the case for tree-like 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 a_b_s_t_r_a_c_t_ because they are pieces of a_b_s_t_r_a_c_t_ s_y_n_t_a_x_ contrasting with c_o_n_c_r_e_t_e_ s_y_n_t_a_x_ 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 i_n_f_i_n_i_t_e_ 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 w_e_l_l_-_f_o_u_n_d_e_d_ (i.e. possess a non-recursive case, or b_a_s_e_ c_a_s_e_) in order to work well with call-by-value. 66..22..44 PPaarraammeetteerriizzeedd ssuumm ttyyppeess 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. 42 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. 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 pattern-matching is not restricted to function bodies, i.e. constructs such as: function P1 -> E1 | .. . | Pn -> En but there is also a construct dedicated to pattern-matching 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 = "+" 66..22..55 DDaattaa ccoonnssttrruuccttoorrss aanndd ffuunnccttiioonnss 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 i_n_j_e_c_t_i_o_n_ while a Caml function is a general function without further information. A mathematical injection f:A->B admits an inverse function f-1 from its image f(A)# B to A. From the examples above, if we consider the King constructor, then: 43 #let king c = King c;; king : suit -> card = 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 pattern-matching may fail. 66..22..66 DDeeggeenneerraattee ccaasseess:: wwhheenn ssuummss mmeeeett pprroodduuccttss 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 t_o_t_a_l_ function from int to counter1. It is thus a b_i_j_e_c_t_i_o_n_. 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 pattern-matching does not perform any run-time 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 o_b_j_e_c_t_-_l_a_n_g_u_a_g_e_s_ and Caml is said to be their m_e_t_a_l_a_n_g_u_a_g_e_. 66..33 SSuummmmaarryy - New types can be introduced in Caml. - Types may be p_a_r_a_m_e_t_e_r_i_z_e_d_ by type variables. The syntax of type parameters is: :: | 44 | ( [, ]* ) - Types can be r_e_c_u_r_s_i_v_e_. - 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 . EExxeerrcciisseess EExxeerrcciissee 66..11 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. EExxeerrcciissee 66..22 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. 45 EExxeerrcciissee 66..33 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. 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 PPaarrtt IIII CCaammll LLiigghhtt ssppeecciiffiiccss 47 CChhaapptteerr 77 MMuuttaabbllee ddaattaa ssttrruuccttuurreess 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 i_m_p_e_r_a_t_i_v_e_ p_r_o_g_r_a_m_m_i_n_g_ 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. 77..11 UUsseerr--ddeeffiinneedd mmuuttaabbllee ddaattaa ssttrruuccttuurreess 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 n_o_t_ 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 o_n_l_y_ 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 a_l_i_a_s_ 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 s_e_q_u_e_n_c_i_n_g_ 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 s_i_d_e_-_e_f_f_e_c_t_s_ such as physical modifications and input/output, since we want to explicitly specify the order in which they are performed. 49 77..22 TThhee ref ttyyppee 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. 77..33 AArrrraayyss 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 77..33..11 AAcccceessssiinngg aarrrraayy eelleemmeennttss 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. 77..33..22 MMooddiiffyyiinngg aarrrraayy eelleemmeennttss 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|] 77..44 LLooooppss:: while aanndd for Imperative programming (i.e. using side-effects 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 side-effects, this construct ensures that they will be performed in the specified order (from left to right). In order to emphasize sequential side-effects, 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 p_e_r_ s_e_: 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: - w_h_i_l_e_: 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. - f_o_r_: 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 (n2-n1 )+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 (n1-n2) +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|] 77..55 PPoollyymmoorrpphhiissmm aanndd mmuuttaabbllee ddaattaa ssttrruuccttuurreess 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). EExxeerrcciisseess EExxeerrcciissee 77..11 Give a mutable data type defining the Lisp type of lists and define the four functions car, cdr, rplaca and rplacd. 53 EExxeerrcciissee 77..22 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. EExxeerrcciissee 77..33 Define a quick_sort function on arrays of floating point numbers following the q_u_i_c_k_s_o_r_t_ algorithm [13]. Information about the q_u_i_c_k_s_o_r_t_ algorithm can be found in [33], for example. 54 CChhaapptteerr 88 EEssccaappiinngg ffrroomm ccoommppuuttaattiioonnss:: eexxcceeppttiioonnss 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 r_a_i_s_e_ a_n_ e_x_c_e_p_t_i_o_n_. 88..11 EExxcceeppttiioonnss An exception is a Caml value of the built-in type exn, very similar to a sum type. An exception: - has a n_a_m_e_ (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 88..22 RRaaiissiinngg aann eexxcceeppttiioonn 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 t_r_a_p_ exceptions: we want to trap possible errors, but we also want to get our result in the case of the find_index function. 56 88..33 TTrraappppiinngg eexxcceeppttiioonnss 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 control-C! #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 88..44 PPoollyymmoorrpphhiissmm aanndd eexxcceeppttiioonnss 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 ill-typed 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. EExxeerrcciisseess EExxeerrcciissee 88..11 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. EExxeerrcciissee 88..22 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 CChhaapptteerr 99 BBaassiicc iinnppuutt//oouuttppuutt 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 [21]. Caml Light has an imperative input/output model: an IO operation should be considered as a side-effect, and is thus dependent on the order of evaluation. IOs are performed onto c_h_a_n_n_e_l_s_ with types in_channel and out_channel. These types are a_b_s_t_r_a_c_t_, 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. 99..11 PPrriinnttaabbllee ttyyppeess 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 p_r_i_n_t_a_b_l_e_ t_y_p_e_ 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 non-printable 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 o_r_-_p_a_t_t_e_r_n_s_, 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, or-patterns 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 [21]. 99..22 OOuuttppuutt 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 b_u_f_f_e_r_e_d_, i.e. the effect of a call to a printing function may not be seen immediately: f_l_u_s_h_i_n_g_ 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 = 99..33 IInnppuutt 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. 99..44 CChhaannnneellss oonn ffiilleess When programs have to read from or print to files, it is necessary to open and close channels on these files. 99..44..11 OOppeenniinngg aanndd cclloossiinngg cchhaannnneellss 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. WWaarrnniinngg:: - 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! 99..44..22 RReeaaddiinngg oorr wwrriittiinngg ffrroomm//ttoo ssppeecciiffiieedd cchhaannnneellss 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 = 99..44..33 FFaaiilluurreess 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 [21]. EExxeerrcciisseess EExxeerrcciissee 99..11 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. EExxeerrcciissee 99..22 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. NNoottee:: 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 CChhaapptteerr 1100 SSttrreeaammss aanndd ppaarrsseerrss 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: s_t_r_e_a_m_s_. Their main usage is to be interfaced to input channels or strings and to be matched against s_t_r_e_a_m_ p_a_t_t_e_r_n_s_. 1100..11 SSttrreeaammss 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. 1100..11..11 TThhee stream ttyyppee 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 s_u_b_s_t_r_e_a_m_s_ 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 d_o_e_s_ n_o_t_ c_o_p_y_ 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 = 1100..11..22 SSttrreeaammss aarree llaazziillyy eevvaalluuaatteedd Stream expressions are submitted to l_a_z_y_ e_v_a_l_u_a_t_i_o_n_, 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 end-of-file 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 side-effects will occur on demand, i.e. when tests will be needed by a matching function on streams. 1100..22 SSttrreeaamm mmaattcchhiinngg aanndd ppaarrsseerrss The syntax for building streams can be used for pattern-matching over them. However, stream matching is more complex than the usual pattern matching. 1100..22..11 SSttrreeaamm mmaattcchhiinngg iiss ddeessttrruuccttiivvee 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 a_n_ i_n_i_t_i_a_l_ s_e_g_m_e_n_t_ 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 p_h_y_s_i_c_a_l_l_y_ m_o_d_i_f_i_e_d_ 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;; Toplevel input: 68 >next ints; next ints; next ints;; >^^^^^^^^^ Warning: this expression has type int, but is used with type unit. Toplevel input: >next ints; next ints; next ints;; > ^^^^^^^^^ Warning: this expression has type int, but is used with type unit. 0 1 2 - : int = 2 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 non-terminals, 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 non-empty sequence of `a` starts with an `a`, and is followed by a possibly empty sequence of `a`. 69 Another example is the recognition of a non-empty 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. 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 recursive-descent parsing with one look-ahead 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. 1100..22..22 SSeeqquueennttiiaall bbiinnddiinngg iinn ssttrreeaamm ppaatttteerrnnss 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. WWaarrnniinngg:: 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. 1100..33 PPaarraammeetteerriizzeedd ppaarrsseerrss 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 right-hand side of stream-matching rules simulate i_n_h_e_r_i_t_e_d_ a_t_t_r_i_b_u_t_e_s_ of attribute grammars. Parameters used as parsers in stream patterns allow for the implementation of h_i_g_h_e_r_-_o_r_d_e_r_ parsers. We will use the next example to motivate the introduction of parameterized parsers. 70 1100..33..11 EExxaammppllee:: aa ppaarrsseerr ffoorr aarriitthhmmeettiicc eexxpprreessssiioonnss 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. 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 a_l_i_a_s_ p_a_t_t_e_r_n_s_. 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 >] 71 #| [< '`*`; 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: Expr ::= Expr + Expr | Expr - Expr | Expr * Expr | Expr / Expr | ( Expr ) | INT The values computed by the parser will be a_b_s_t_r_a_c_t_ s_y_n_t_a_x_ t_r_e_e_s_ (by contrast with c_o_n_c_r_e_t_e_ s_y_n_t_a_x_, 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 well-known 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 non-terminal for each precedence level, and removing left-recursion. We obtain: Expr ::= Mult | Mult + Expr | Mult - Expr Mult ::= Atom | Atom * Mult | Atom / Mult Atom ::= INT | ( Expr ) 72 While removing left-recursion, 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 (left-factor). We obtain: Expr ::= Mult RestExpr 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 73 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 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) 1100..33..22 PPaarraammeetteerrss ssiimmuullaattiinngg iinnhheerriitteedd aattttrriibbuutteess 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 74 calls such as (restexpr (Plus(e1,e2))). If we see such parsers as non-terminals (RestExpr from the grammar above) this parameter acts as an inherited attribute of the non-terminal. Synthesized attributes are simulated by the right hand sides of stream matching rules. 1100..33..33 HHiigghheerr--oorrddeerr ppaarrsseerrss 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 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 left-associative 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;; 75 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 #| [< '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! 1100..33..44 EExxaammppllee:: ppaarrssiinngg aa nnoonn ccoonntteexxtt--ffrreeee llaanngguuaaggee As an example of parsing with parameterized parsers, we shall build a parser for the language {wCw | w#(A|B)*}, which is known to be non context-free. 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 w_o_r_d_ d_e_f_i_n_i_t_i_o_n_) returning information in order to build a parser recognizing only w; - then to recognize C; 76 - 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 #| [< >] -> [];; wd : token stream -> (token stream -> string) list = The wu function (for w_o_r_d_ u_s_a_g_e_) 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 = "" 77 Here, wd is made local to w, and takes as parameter wr (for w_o_r_d_ r_e_c_o_g_n_i_z_e_r_) 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. 1100..44 FFuurrtthheerr rreeaaddiinngg A summary of the constructs over streams and of primitives over streams is given in [21]. 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 [24], where alternative semantics are given with some possible implementations. EExxeerrcciisseess EExxeerrcciissee 1100..11 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 EExxeerrcciissee 1100..22 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 CChhaapptteerr 1111 SSttaannddaalloonnee pprrooggrraammss aanndd sseeppaarraattee ccoommppiillaattiioonn So far, we have used Caml Light in an interactive way. It is also possible to program in Caml Light in a batch-oriented 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). 1111..11 SSttaannddaalloonnee pprrooggrraammss 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 stand-alone program produced by the compiler, all phrases are executed in order. The values of expressions and declared global variables are not printed, however. A stand-alone 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 MS-DOS, 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. 1111..22 PPrrooggrraammss iinn sseevveerraall ffiilleess 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 stand-alone 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__+. 1111..33 AAbbssttrraaccttiioonn 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, well-structured 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. EExxeerrcciisseess EExxeerrcciissee 1111..11 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 command-line arguments to the process. 83 PPaarrtt IIIIII AA ccoommpplleettee eexxaammppllee 84 CChhaapptteerr 1122 AASSLL:: AA SSmmaallll LLaanngguuaaggee We present in this chapter a simple language: ASL (A Small Language). This language is basically the l-calculus (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 call-by-value: thus, the conditional cannot be a function. ASL programs are built up from numbers, variables, functional expressions (l-abstractions), 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. 1122..11 AASSLL aabbssttrraacctt ssyynnttaaxx ttrreeeess We encode variable names by numbers. These numbers represent the b_i_n_d_i_n_g_ d_e_p_t_h_ 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. 1122..22 PPaarrssiinngg AASSLL pprrooggrraammss 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 l-calculus 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 l-calculus (except that we will use the b_a_c_k_s_l_a_s_h_ character because there is no ``l'' on our keyboards). We will use the c_u_r_r_i_e_d_ versions of equality and arithmetic functions. We will also use a p_r_e_f_i_x_ 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). 1122..22..11 LLeexxiiccaall aannaallyyssiiss The concrete aspect of ASL programs will be either declarations of the form: let i_d_e_n_t_i_f_i_e_r_ be e_x_p_r_e_s_s_i_o_n_; or: e_x_p_r_e_s_s_i_o_n_; which will be understood as: let it be e_x_p_r_e_s_s_i_o_n_; 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. 1122..22..22 PPaarrssiinngg 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 parse-time. 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 CChhaapptteerr 1133 UUnnttyyppeedd sseemmaannttiiccss ooff AASSLL pprrooggrraammss In this section, we give a semantic treatment of ASL programs. We will use d_y_n_a_m_i_c_ t_y_p_e_c_h_e_c_k_i_n_g_, i.e. we will test the type correctness of programs during their interpretation. 1133..11 SSeemmaannttiicc vvaalluueess 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 ill-typed program and will represent run-time 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 run-time 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 ] 1133..22 SSeemmaannttiicc ffuunnccttiioonnss 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 (n-1) 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 = 1133..33 EExxaammpplleess #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 CChhaapptteerr 1144 EEnnccooddiinngg rreeccuurrssiioonn 1144..11 FFiixxppooiinntt ccoommbbiinnaattoorrss We have seen that we do not have recursion in ASL. However, it is possible to encode recursion by defining a f_i_x_p_o_i_n_t_ c_o_m_b_i_n_a_t_o_r_. 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. 95 Unfortunately, when using call-by-value, any application of a fixpoint combinator F such that: F M evaluates to M (F M) leads to non-termination 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 call-by-value evaluation regime (in order to check that fact, it is necessary to know the evaluation rules of l-calculus). 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 = () 96 1144..22 RReeccuurrssiioonn aass aa pprriimmiittiivvee ccoonnssttrruucctt 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! 97 CChhaapptteerr 1155 SSttaattiicc ttyyppiinngg,, ppoollyymmoorrpphhiissmm aanndd ttyyppee ssyynntthheessiiss We now want to perform static typechecking of ASL programs, that is, to complete typechecking b_e_f_o_r_e_ evaluation, making run-time type tests unnecessary during evaluation of ASL programs. Furthermore, we want to have p_o_l_y_m_o_r_p_h_i_s_m_ (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 t_y_p_e_ s_y_s_t_e_m_, 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]. 1155..11 TThhee ttyyppee ssyysstteemm 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 first-order 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 l-calculus). A t_y_p_e_ 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 u_n_k_n_o_w_n_, i.e. a type that we are 98 computing. We will use t, t', t1, ..., as m_e_t_a_v_a_r_i_a_b_l_e_s_(1) representing types. This notation is important: we shall use other greek letters to denote other notions in the following sections. EExxaammppllee (a->Number)->b->b is a type. O A t_y_p_e_ s_c_h_e_m_e_, is a type where some variables are distinguished as being g_e_n_e_r_i_c_. We can represent type schemes by: for all a1,.. .,an.t where t is a type. EExxaammppllee 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 u_n_k_n_o_w_n_s_ occurring in the type scheme s. Unknowns are also called f_r_e_e_ v_a_r_i_a_b_l_e_s_ (they are not bound by a for all quantifier). We also write BV(s) (b_o_u_n_d_ t_y_p_e_ v_a_r_i_a_b_l_e_s_ o_f_ 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 g_e_n_e_r_i_c_. EExxaammppllee 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 s_u_b_s_t_i_t_u_t_i_o_n_ i_n_s_t_a_n_c_e_ s' of a type scheme s is the type scheme S(s) where S is a substitution of types for f_r_e_e_ 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. EExxaammppllee - 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 v_a_r_i_a_b_l_e_ or a t_y_p_e_ v_a_r_i_a_b_l_e_. 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 g_e_n_e_r_i_c_ i_n_s_t_a_n_c_e_ 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. EExxaammppllee 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 i_n_f_e_r_e_n_c_e_ r_u_l_e_s_. An inference rule is written as a fraction: - the numerator is called the p_r_e_m_i_s_s_e_s_; - the denominator is called the c_o_n_c_l_u_s_i_o_n_. An inference rule: P1 ... Pn ----------- C may be read in two ways: - ``IIff P1, ...aanndd Pn , tthheenn C''. - ``IInn oorrddeerr ttoo pprroovvee C, iitt iiss ssuuffffiicciieenntt ttoo pprroovvee P1, ...aanndd Pn ''. An inference rule may have no premise: such a rule will be called an a_x_i_o_m_. 100 A complete proof will be represented by a p_r_o_o_f_ t_r_e_e_ 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 j_u_d_g_e_m_e_n_t_s_ 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 t_y_p_i_n_g_ h_y_p_o_t_h_e_s_e_s_ 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 l-calculus terms, the typing environment is managed as a s_t_a_c_k_ (because identifiers possess local scopes). We represent that fact in the presentation of the type system by r_e_m_o_v_i_n_g_ the typing hypothesis concerning an identifier name x (if such a typing hypothesis exists) before adding a new typing hypothesis concerning x. We write G-G(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 s_e_m_a_n_t_i_c_a_l_l_y_ s_o_u_n_d_, i.e. the semantic value of a well-typed expression (an expression admitting a type) cannot be an e_r_r_o_r_ v_a_l_u_e_ due to a type error. This is usually expressed as: Well-typed programs cannot go wrong. This fact implies that a clever compiler may produce code without any dynamic type test for a well-typed expression. EExxaammppllee 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 102 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 m_o_s_t_ g_e_n_e_r_a_l_. 1155..22 TThhee aallggoorriitthhmm 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 t_y_p_e_s_ 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 u_n_i_f_i_c_a_t_i_o_n_ [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 103 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: - s_y_n_t_a_c_t_i_c_a_l_l_y_ s_o_u_n_d_: when the algorithm succeeds on an expression e and returns a type t, then e:t. - c_o_m_p_l_e_t_e_: 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. EExxaammppllee 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 104 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 s_y_n_t_a_x_-_d_i_r_e_c_t_e_d_: since, for a given expression, a type deduction for that expression uses exactly one rule per sub-expression, 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 ``types-as-propositions and programs-as-proofs'' paradigm, also known as the ``Curry-Howard 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. 1155..33 TThhee AASSLL ttyyppee--ssyynntthheessiizzeerr 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. 1155..33..11 RReepprreesseennttaattiioonn ooff AASSLL ttyyppeess aanndd ttyyppee sscchheemmeess 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 = 1155..33..22 DDeessttrruuccttiivvee uunniiffiiccaattiioonn ooff AASSLL ttyyppeess 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 o_c_c_u_r_-_c_h_e_c_k_. The following function implements occur-check: #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 = 1155..33..33 RReepprreesseennttaattiioonn ooff ttyyppiinngg eennvviirroonnmmeennttss 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 type-checked. 107 #let global_typing_env = ref init_typing_env;; 1155..33..44 FFrroomm ttyyppeess ttoo ttyyppee sscchheemmeess:: ggeenneerraalliizzaattiioonn 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 = 1155..33..55 FFrroomm ttyyppee sscchheemmeess ttoo ttyyppeess:: ggeenneerriicc iinnssttaannttiiaattiioonn 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 = 1155..33..66 TThhee AASSLL ttyyppee ssyynntthheessiizzeerr 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 = 1155..33..77 TTyyppiinngg,, ttrraappppiinngg ttyyppee ccllaasshheess aanndd pprriinnttiinngg AASSLL ttyyppeess Now, we define some auxiliary functions in order to build a ``good-looking'' 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 = 1155..33..88 TTyyppiinngg AASSLL pprrooggrraammss 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" 1155..33..99 TTyyppiinngg aanndd rreeccuurrssiioonn 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 CChhaapptteerr 1166 CCoommppiilliinngg AASSLL ttoo aann aabbssttrraacctt mmaacchhiinnee ccooddee 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 pattern-matching 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 C_a_t_e_g_o_r_i_c_a_l_ A_b_s_t_r_a_c_t_ M_a_c_h_i_n_e_ (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, 25]. 1166..11 TThhee AAbbssttrraacctt MMaacchhiinnee The execution model is a s_t_a_c_k_ m_a_c_h_i_n_e_ (i.e. a machine using a stack). In this section, we define in Caml the s_t_a_t_e_s_ of the CAM and its instructions. A state is composed of: - a r_e_g_i_s_t_e_r_ (holding values and environments), - a p_r_o_g_r_a_m_ c_o_u_n_t_e_r_, represented here as a list of instructions whose first element is the current instruction being executed, - and a s_t_a_c_k_ 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 higher-level 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 o_p_e_r_a_t_i_o_n_a_l_ s_e_m_a_n_t_i_c_s_ 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(n-m); 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=(Plus|Minus|Times|Divide|Equal)::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) r_e_t_u_r_n_ instruction. We could argue that pattern-matching in the Camlstep function implements a kind of dynamic typechecking. In fact, in a concrete (low-level) 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 b_u_g_ in one of the above processes and would lead to memory errors or illegal instructions (usually detected by the computer's operating system). 1166..22 CCoommppiilliinngg AASSLL pprrooggrraammss iinnttoo CCAAMM ccooddee 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]], Environ- ment []); Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Times]], Environ- ment []); Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Divide]], Environ- ment []); Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Equal]], Environ- ment [])] #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]], Environ- ment []); Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Divide]], Environment []); Closure (Address [Clos [Push; Nth 2; Swap; Nth 1; Equal]], Environ- ment [])] As an example, here is the code for some ASL expressions. 118 #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] #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] 1166..33 EExxeeccuuttiioonn ooff CCAAMM ccooddee 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; 119 # 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();; 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: 120 > with [Clos(C)] -> Closure(Address(C), Environment []) > ^ Warning: the variable C starts with an upper case letter in this pattern. - : unit = () #run (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 = () #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 [25] for efficient compilation of recursion, data structures etc. EExxeerrcciissee 1166..11 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 CChhaapptteerr 1177 AAnnsswweerrss ttoo eexxeerrcciisseess 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. 33..11,, pp.. 1177 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 = 33..22,, pp.. 1177 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 33..33,, pp.. 1177 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 44..11,, pp.. 2299 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 = 44..22,, pp.. 2299 In a call-by-value language without conditional construct (and without sum types), all programs involving a recursive definition never terminate. 44..33,, pp.. 2299 #let rec factorial n = if n=1 then 1 else n*(factorial(n-1));; factorial : int -> int = #factorial 5;; - : int = 120 #let tail_recursive_factorial n = # let rec fact n m = if n=1 then m else fact (n-1) (n*m) # in fact n 1;; tail_recursive_factorial : int -> int = #tail_recursive_factorial 5;; - : int = 120 44..44,, pp.. 2299 #let rec fibonacci n = # if n=1 then 1 # else if n=2 then 1 # else fibonacci(n-1) + fibonacci(n-2);; fibonacci : int -> int = #fibonacci 20;; - : int = 6765 124 44..55,, pp.. 2299 #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 = 55..11,, pp.. 3344 #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" 55..22,, pp.. 3355 #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"]] 66..11,, pp.. 4455 #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] 66..22,, pp.. 4455 #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 = 66..33,, pp.. 4455 We need to give a functional interpretation to btree data constructors. We use f (resp. g) to denote the function associated to the Leaf (resp. Btree) data constructor, obtaining the following Caml definition: #let rec btree_it f g = function # Leaf x -> f x # | Btree{Op=op; Son1=s1; Son2=s2} # -> g op (btree_it f g s1) (btree_it f g s2) #;; btree_it : ('a -> 'b) -> ('c -> 'b -> 'b -> 'b) -> ('c, 'a) btree -> 'b = 126 #btree_it (function x -> x) # (function "+" -> prefix + # | _ -> raise (Failure "Unknown op")) # (Btree {Op="+"; Son1=Leaf 1; Son2=Leaf 2});; - : int = 3 77..11,, pp.. 5533 #type ('a,'b) lisp_cons = {mutable Car:'a; mutable Cdr:'b};; Type lisp_cons defined. #let car p = p.Car #and cdr p = p.Cdr #and rplaca p v = p.Car <- v #and rplacd p v = p.Cdr <- v;; car : ('a, 'b) lisp_cons -> 'a = cdr : ('a, 'b) lisp_cons -> 'b = rplaca : ('a, 'b) lisp_cons -> 'a -> unit = rplacd : ('a, 'b) lisp_cons -> 'b -> unit = #let p = {Car=1; Cdr=true};; p : (int, bool) lisp_cons = {Car = 1; Cdr = true} #rplaca p 2;; - : unit = () #p;; - : (int, bool) lisp_cons = {Car = 2; Cdr = true} 77..22,, pp.. 5544 #let stamp_counter = ref 0;; stamp_counter : int ref = ref 0 #let stamp () = # stamp_counter := 1 + !stamp_counter; !stamp_counter;; stamp : unit -> int = #stamp();; - : int = 1 #stamp();; - : int = 2 77..33,, pp.. 5544 #let exchange t i j = 127 # let v = t.(i) in vect_assign t i t.(j); vect_assign t j v #;; exchange : 'a vect -> int -> int -> unit = #let quick_sort t = # let rec quick lo hi = # if lo < hi # then begin # let i = ref lo # and j = ref hi # and p = t.(hi) in # while !i < !j # do # while !i < hi & t.(!i) <=. p do incr i done; # while !j > lo & p <=. t.(!j) do decr j done; # if !i < !j then exchange t !i !j # done; # exchange t hi !i; # quick lo (!i - 1); # quick (!i + 1) hi # end # else () # in quick 0 (vect_length t - 1) #;; quick_sort : float vect -> unit = #let a = [| 2.0; 1.5; 4.0; 0.0; 10.0; 1.0 |];; a : float vect = [|2.0; 1.5; 4.0; 0.0; 10.0; 1.0|] #quick_sort a;; - : unit = () #a;; - : float vect = [|0.0; 1.0; 1.5; 2.0; 4.0; 10.0|] 88..11,, pp.. 5588 #let rec find_succeed f = function # [] -> raise (Failure "find_succeed") # | x::l -> try f x; x with _ -> find_succeed f l #;; find_succeed : ('a -> 'b) -> 'a list -> 'a = #let hd = function [] -> raise (Failure "empty") | x::l -> x;; hd : 'a list -> 'a = #find_succeed hd [[];[];[1;2];[3;4]];; - : int list = [1; 2] 128 88..22,, pp.. 5588 #let rec map_succeed f = function # [] -> [] # | h::t -> try (f h)::(map_succeed f t) # with _ -> map_succeed f t;; map_succeed : ('a -> 'b) -> 'a list -> 'b list = #map_succeed hd [[];[1];[2;3];[4;5;6]];; - : int list = [1; 2; 4] 99..11,, pp.. 6655 The first function (copy) that we define assumes that its arguments are respectively the input and the output channel. They are assumed to be already opened. #let copy inch outch = # (* inch and outch are supposed to be opened channels *) # try (* actual copying *) # while true # do output_char outch (input_char inch) # done # with End_of_file -> (* Normal termination *) # raise End_of_file # | sys__Sys_error msg -> (* Abnormal termination *) # prerr_endline msg; # raise (Failure "cp") # | _ -> (* Unknow exception, maybe interruption? *) # prerr_endline "Unknown error while copying"; # raise (Failure "cp") #;; copy : in_channel -> out_channel -> unit = The next function opens channels connected to its filename arguments, and calls copy on these channels. The advantage of dividing the code into two functions is that copy performs the actual work, and can be reused in different applications, while the role of cp is more ``administrative'' in the sense that it does nothing but opening and closing channels and printing possible error messages. #let cp f1 f2 = # (* Opening channels, f1 first, then f2 *) # let inch = # try open_in f1 # with sys__Sys_error msg -> # prerr_endline (f1^": "^msg); # raise (Failure "cp") # | _ -> prerr_endline ("Unknown exception while opening "^f1); # raise (Failure "cp") 129 # in # let outch = # try open_out f2 # with sys__Sys_error msg -> # close_in inch; # prerr_endline (f2^": "^msg); # raise (Failure "cp") # | _ -> close_in inch; # prerr_endline ("Unknown exception while opening "^f2); # raise (Failure "cp") # in (* Copying and then closing *) # try copy inch outch # with End_of_file -> close_in inch; close_out outch # (* close_out flushes *) # | exc -> close_in inch; close_out outch; raise exc #;; cp : string -> string -> unit = Let us try cp: #cp "/etc/passwd" "/tmp/foo";; - : unit = () #cp "/tmp/foo" "/foo";; /foo: /foo: Permission denied Uncaught exception: Failure "cp" The last example failed because a regular user is not allowed to write at the root of the file system. 99..22,, pp.. 6655 As in the previous exercise, the function count performs the actual counting. It works on an input channel and returns a pair of integers. #let count inch = # let chars = ref 0 # and lines = ref 0 in # try # while true do # let c = input_char inch in # chars := !chars + 1; # if c = `\n` then lines := !lines + 1 else () # done; # (!chars, !lines) # with End_of_file -> (!chars, !lines) #;; count : in_channel -> int * int = The function wc opens a channel on its filename argument, calls count and prints the result. 130 #let wc f = # let inch = # try open_in f # with sys__Sys_error msg -> # prerr_endline (f^": "^msg); # raise (Failure "wc") # | _ -> prerr_endline ("Unknown exception while opening "^f); # raise (Failure "wc") # in let (chars,lines) = count inch # in print_int chars; # print_string " characters, "; # print_int lines; # print_string " lines.\n" #;; wc : string -> unit = Counting /etc/passwd gives: #wc "/etc/passwd";; 16624 characters, 203 lines. - : unit = () 1100..11,, pp.. 7788 Let us recall the definitions of the type token and of the lexical analyzer: #type token = # PLUS | MINUS | TIMES | DIV | LPAR | RPAR #| INT of int;; Type token defined. #(* Spaces *) #let rec spaces = function # [< '` `|`\t`|`\n`; spaces _ >] -> () #| [< >] -> ();; spaces : char stream -> unit = #(* 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 # [< ' `0`..`9` as c; (integer (10*n + int_of_digit c)) r >] -> r #| [< >] -> n;; integer : int -> char stream -> int = #(* The lexical analyzer *) #let rec lexer s = match s with 131 # [< '`(`; 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 = The parser has the same shape as the grammar: #let rec expr = function # [< 'INT n >] -> n #| [< 'PLUS; expr e1; expr e2 >] -> e1+e2 #| [< 'MINUS; expr e1; expr e2 >] -> e1-e2 #| [< 'TIMES; expr e1; expr e2 >] -> e1*e2 #| [< 'DIV; expr e1; expr e2 >] -> e1/e2;; expr : token stream -> int = #expr (lexer (stream_of_string "1"));; - : int = 1 #expr (lexer (stream_of_string "+ 1 * 2 4"));; - : int = 9 1100..22,, pp.. 7788 The only new function that we need is a function taking as argument a character stream, and returning the first identifier of that stream. It could be written as: #let ident_buf = make_string 8 ` `;; ident_buf : string = " " #let rec ident len = function # [< ' `a`..`z`|`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 #| [< >] -> sub_string ident_buf 0 len;; ident : int -> char stream -> string = The lexical analyzer will first try to recognize an alphabetic character c, then put c at position 0 of ident_buf, and call ident 1 on the rest of the character stream. Alphabetic characters encountered will be stored in the string buffer ident_buf, up to the 8th. Further alphabetic characters will be skipped. Finally, a substring of the buffer will be returned as result. 132 #let s = stream_of_string "toto 1";; s : char stream = #ident 0 s;; - : string = "toto" #(* Let us see what remains in the stream *) #match s with [< 'c >] -> c;; - : char = ` ` #let s = stream_of_string "LongIdentifier ";; s : char stream = #ident 0 s;; - : string = "LongIden" #match s with [< 'c >] -> c;; - : char = ` ` The definitions of the new token type and of the lexical analyzer is trivial, and we shall omit them. A slightly more complex lexical analyzer recognizing identifiers (lowercase only) is given in section 12.2.1 in this part. 1111..11,, pp.. 8833 (* main.ml *) let chars = counter__new 0;; let lines = counter__new 0;; let count_file filename = let in_chan = open_in filename in try while true do let c = input_char in_chan in counter__incr chars; if c = `\n` then counter__incr lines done with End_of_file -> close_in in_chan ;; for i = 1 to vect_length sys__command_line - 1 do count_file sys__command_line.(i) done; print_int (counter__read chars); print_string " characters, "; print_int (counter__read lines); print_string " lines.\n"; exit 0;; 133 CChhaapptteerr 1188 CCoonncclluussiioonnss aanndd ffuurrtthheerr rreeaaddiinngg We have not been exhaustive in the description of the Caml Light features. We only introduced general concepts in functional programming, and we have insisted on the features used in the prototyping of ASL: a tiny model of Caml Light typing and semantics. 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