The Conception, Evolution, and Application of Functional Programming Languages
Paul Hudak Yale University
Department of Computer Science March 1989
Final Draft∗
Abstract
The foundations of functional programming languages are examined from both historical and technical perspectives. Their evolution is traced through several critical periods: early work on lambda calculus and combinatory calculus, Lisp, Iswim, FP, ML, and modern functional languages such as Miranda1 and Haskell.
The fundamental premises on which the functional programming methodology stands are critically analyzed with respect to philosophical, theoretical, and prag- matic concerns. Particular attention is paid to the main features that characterize modern functional languages: higher-order functions, lazy evaluation, equations and pattern-matching, strong static typing and type inference, and data abstrac- tion. In addition, current research areas—such as parallelism, non-determinism, input/output, and state-oriented computations—are examined with the goal of pre- dicting the future development and application of functional languages.
Categories and Subject Descriptors:D.1.1 [Programming Techniques]: Applicative (Functional) Programming; D.3.2 [Programming Languages]: Language Classifications—
applicative languages; data-flow languages; nonprocedural languages; very high- level languages; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—lambda calculus and related systems; K.2 [History of Computing]: Software.
General Terms: Languages.
Additional Key Words and Phrases: referential transparency, higher-order func- tions, types, lazy evaluation, data abstraction.
1 Introduction
The earliest programming languages were developed with one simple goal in mind: to provide a vehicle through which one could control the behavior of computers. Not
∗To appear in ACM Computing Surveys.
1Miranda is a trademark of Research Software Ltd.
surprisingly, these early languages reflected fairly well the structure of the underlying machines. Although at first blush this seems emminently reasonable, this viewpoint quickly changed, for two very good reasons. First, it became obvious that what was easy for a machine to reason about was not necessarily easy for a human to reason about. Second, as the number of different kinds of machines increased, the need arose for a common language with which to program all of them.
Thus from primitive assembly languages (which were at least a step up from raw machine code) there grew a plethora of high-level programming languages, beginning with Fortran in the 50’s. The development of these languages grew so rapidly that by the 80’s they were best characterized by grouping them into “families” that reflected a common computation model or programming “style.” Debates over which language or family of languages is best will undoubtedly persist for as long as computers need programmers.
The class of functional, orapplicativeprogramming languages, in which computa- tion is carried out entirely through theevaluation of expressions, is one such family of languages, and debates over its merits have been quite lively in recent years. Are func- tional languages toys, or are they tools? Are they artifacts of theoretical fantasy, or of visionary pragmatism? Will they ameliorate our software woes, or merely compound them? Whatever answers one might have for these questions, one cannot ignore the significant interest current researchers have in functional languages, and the impact they have had on both the theory and pragmatics of programming languages in general.
Among the claims made by functional language advocates are that programs can be written quicker, are more concise, are “higher level” (resembling more closely tra- ditional mathematical notation), are more amenable to formal reasoning and analysis, and can be executed more easily on parallel architectures. Of course, many of these features touch on rather subjective issues, which is one reason why the debates can be so lively.
It is my purpose in writing this paper to give the reader significant insight into the very essence of functional languages and the programming methodology that they support. I will do so by starting with a general discussion of the nature of functional languages, followed by a historical sketch of their development, a summary of the dis- tinguishing characteristics of modern functional languages, and a discussion of current research areas. Through this study I hope to put into perspective both the power and weaknesses of the functional programming paradigm.
A Note to the Reader: I have assumed in writing this paper that the reader has a good understanding of the fundamental issues in programming language design and use. Although reasonably self-contained, this paper should not be considered as a tu- torial on functional languages; the reader is referred to either [BW88] or [FH88] as suit- able textbooks for learning more about modern functional programming techniques, including the important ideas behind reasoning about functional programs. In addi- tion, I will say very little about how toimplementfunctional languages; see [PJ87] for
an introduction to this important area (additional references are given in Sections 2.8 and 6).
Finally, a comment on notation: unless otherwise stated, all examples will be written in Haskell, a recently proposed functional language standard [HWe88]. However, rather than explain the syntax all at once, explanations will be given “on the fly” and enclosed in square brackets [...].2
1.1 The Programming Language Spectrum
Imperative languages are characterized as having an implicit state that is modified (i.e. side-effected) by constructs (i.e. commands) in the source language. As a result, such languages generally have a notion of sequencing (of the commands) to permit precise and deterministic control over the state. Most, including the most popular, languages in existence today are imperative.
As an example, the assignment statement is a (very common) command, since its effect is to alter the underlying implicit store so as to yield a different binding for a particular variable. Thebegin...endconstruct is the prototypicalsequencerof com- mands, as are the well-knowngoto statement(unconditional transfer of control),con- ditional statement (qualified sequencer), and while loop(an example of a “structured command”). With these simple forms we can then, for example, compute the factorial of the numberx:
n := x;
a := 1;
while n>0 do begin a := a*n;
n := n-1 end;
...
After execution of this program the value of a in the implicit store will contain the desired result.
In contrast,declarative languagesare characterized as havingnoimplicit state, and thus the emphasis is placed entirely on programming with expressions (orterms). In particular,functional languagesare declarative languages whose underlying model of computation is the function(in contrast to, for example, the relation that forms the basis for logic programming languages.
2Since the Haskell Report is relatively new, some minor changes to the language may occur after this paper has appeared. An up-to-date copy of the Report may be obtained from the author.
In a declarative language state-oriented computations are accomplished by carry- ing the state around explicitly rather than implicitly, and looping is accomplished via recursionrather than by sequencing. For example, the factorial ofxmay be computed in the functional language Haskell by:
fac x 1
where fac n a = if n>0 then fac (n-1) (a*n) else a
in which the formal parameters n and a are examples of “carrying the state around explicitly,” and the recursive structure has been arranged so as to mimic as closely as possible the looping behavior of the program given earlier. Note that the conditional in this program is anexpressionrather thancommand; i.e. it denotes avalue(conditional on the value of the predicate), rather than denoting a sequencer of commands. Indeed thevalue of the programis the desired factorial, rather than it being found in an implicit store.
Functional (in general, declarative) programming is often described as expressing whatis being computed rather thanhow, although this is really a matter of degree. For example, the above program may say less about how factorial is computed than the imperative program given earlier, but is perhaps not as abstract as:
fac x
where fac n = if n==0 then 1 else n * fac (n-1)
[==is the infix operator for equality.] which appears very much like the mathematical definition of factorial, and is indeed a valid functional program.
Since most languages have expressions, it is tempting to take our definitions liter- ally and describe functional languages via derivation from conventional programming languages: simply drop the assignment statement and any other side-effecting prim- itives, and there you have it! This, of course, is very misleading. The result of such a derivation is usually far less than satisfactory, since the purely functional subset of most imperative languages is hopelessly weak (although there are important excep- tions, such as Scheme [RCe86]).
Rather than saying, then, what functional languagesdon’thave, it is better to char- acterize them by the features they do have. For modern functional languages, those features include higher-order functions, lazy evaluation, pattern-matching, and various kinds of data abstraction — all of these features will be described in detail in this paper.
Functions are treated as first-class objects, are allowed to be recursive, higher-order, and polymorphic, and in general are provided with mechanisms that ease their defini- tion and use. Syntactically, modern functional languages have an “equational” look, in which functions are defined using mutually recursive equations and pattern-matching.
This discussion suggests that what is important is the functional programming style, in which the above features are manifest, and in which side effects are strongly discouraged but not necessarily eliminated. This is the viewpoint taken by the ML com- munity, for example, and to some extent the Scheme community. On the other hand, there is a very large contingency of “purists” in the functional programming commu- nity who believe that purely functional languages are not only sufficient for general computing needs, but also better because of their “purity.” At least a dozen purely functional languages exist along with their implementations.3 The main property that is lost when side effects are introduced is referential transparency; this loss in turn impairsequational reasoning, as described below.
1.2 Referential Transparency and Equational Reasoning
The emphasis on a pure declarative style of programming is perhaps the hallmark of the functional programming paradigm. The termreferentially transparentis often used to describe this style of programming, in which “equals can be replaced by equals.” For example, consider the (Haskell) expression:
... x+x ...
where x = f a
The function application(f a)may be substituted for any free occurrence ofxin the scope created by thewhere expression, such as in the subexpressionx+x. The same cannot generally be said of an imperative language, where one must first be sure that no assignment to x is made in any of the statements intervening between the initial definition of x and one of its subsequent uses.4 In general this can be quite a tricky task, for example in the case where procedures are allowed to induce non-local changes to lexically scoped variables.
Although the notion of referential transparency may seem like a simple idea, the clean equational reasoning that it allows is very powerful, not only for reasoning for- mally about programs, but also informally in writing and debugging programs. A pro- gram in which side-effects are minimized, but not eliminated, may still benefit from equational reasoning, although naturally more care must be taken when applying such reasoning. The degree of care, however, may be much higher than one might think at first: most languages that allow “minor” forms of side-effects do not minimize their
3This situation forms an interesting contrast with the logic programming community, where Prolog is often described as declarative (whereas Lisp is usually not), and there are very few pure logic programming languages (and even fewer implementations).
4In all fairness, there are logics for reasoning about imperative programs, such as those espoused by Floyd, Hoare, Dijkstra, Wirth, and others. None of them, however, exploit any notion of referential transparency.
locality lexically—thus any call to any function in any module might conceivably intro- duce a side-effect, in turn invalidating many applications of equational reasoning.
The perils of side-effects are appreciated by the most experienced programmers in any language, although most are loathe to give them up completely. It remains the goal of the functional programming community to demonstrate that one can docom- pletelywithout side-effects, without sacrificing efficiency or modularity. Of course, as mentioned earlier, the lack of side-effects is not all there is to the functional program- ming paradigm—as we shall soon see, modern functional languages rely heavily on certain other features, most notably higher-order functions, lazy evaluation, and data abstraction.
1.3 Plan of Study
Unlike many developments in computer science, functional languages have maintained the principles on which they were founded to a surprising degree. Rather than chang- ing or compromising those ideas, modern functional languages are best classified as embellishments of a certain set of ideals. It is a distinguishing feature of modern func- tional languages that they have so effectively held on to pure mathematical principles in a way shared by very few other languages.
Because of this, one can learn a great deal about functional languages simply by studying their evolution closely. On the other hand, such a study may fail to yield a consistent treatment of any one feature that is common to most functional languages, for it will be fractured into its manifestations in each of the languages as they were historically developed. For this reason I have taken a three-fold approach to our study:
First, in the next section I will provide a historical sketch of the development of functional languages. Starting with the lambda calculus as the prototypical functional language, I will gradually embellish it with ideas as they were historically developed, leading eventually to a reasonable technical characterization of modern functional lan- guages.
In so doing, however, I will postpone detailed discussion of four important concepts—
higher-order functions, lazy evaluation, data abstraction mechanisms, and equations/pattern- matching—which are critical components of all modern functional languages and are best discussed as independent topics. Section 3 is devoted to these important ideas.
Finally, in Section 4 I will discuss more advanced ideas, and outline some critical research areas. Then to round out the paper, in Section 5 I will try to put into perspec- tive some of the limitations of functional languages by examining some of the myths that have accompanied their development.
2 The Evolution of Functional Languages
2.1 The Lambda Calculus
The development of functional languages has been influenced from time to time by many sources, but none is as paramount nor as fundamental as the work of Alonzo Church on thelambda calculus[Chu33, Chu41]. Indeed the lambda calculus is usually regarded as the first functional language, although it was certainly not thought of as programming language at the time, given that there were no computers to run the programs on! In any case, modern functional languages can be thought of as (non- trivial) embellishments of the lambda calculus.
It is often thought that the lambda calculus also formed the foundation for Lisp, but this in fact appears not to be the case [McC78]. The impact of the lambda calculus on early Lisp development was minimal, and it has only been very recently that Lisp has begun to evolve more toward lambda calculus ideals. On the other hand, Lisp had a significant impact on the subsequent development of functional languages, as will be discussed in Section 2.2.
Church’s work was motivated by the desire to create acalculus(informally, a syntax for terms and set of rewrite rules for transforming terms) that captured one’s intuition about the behavior offunctions. This approach is counter to the consideration of func- tions as, for example, sets (more precisely, sets of argument/value pairs), since the intent was to capture the computationalaspects of functions. A calculus is a formal way for doing just that.
Church’slambda calculuswas the first suitable treatment of the computational as- pects of functions. Its type-free nature yielded a particularly small and simple calculus, and it had one very interesting property, capturing functions in their fullest generality:
functions could be applied to themselves. In most reasonable theories of functions as sets, this is impossible, since it requires the notion of a set containing itself, result- ing in well-known paradoxes. This ability of self-application is what gives the lambda calculus its power, since it allows one to gain the effect of recursion without explic- itly writing a recursive definition. Despite this powerful ability, the lambda calculus is consistentas a mathematical system—no contradictions or paradoxes arise.
Because of the relative importance of the lambda calculus to the development of functional languages, I will describe it in some detail in the remainder of this section, using modern notational conventions.
2.1.1 Pure Untyped Lambda Calculus
The abstract syntax of thepure untyped lambda calculus(a name chosen to distinguish it from other versions developed later) embodies what are called lambda expressions,
defined by:5
x ∈ Id Identifiers
e ∈ Exp Lambda Expressions
wheree::=x |e1e2 |λx.e
Expressions of the form λx.e are calledabstractions, and of the form (e1 e2) are called applications. It is the former that captures the notion of a function, and the latter that captures the notion of application of a function. By convention, application is assume to be left associative, so that(e1 e2 e3)is the same as((e1e2) e3).
The rewrite rules of the lambda calculus depend on the notion ofsubstitutionof an expressione1 for all “free” occurrences of an identifier x in an expression e2, which we write as [e1/x]e2.6 Most systems, including both the lambda calculus and pred- icate calculus, that utilize substitution on identifiers must be careful to avoid name conflicts. Thus, although the intuition behind substitution is strong, its formal defini- tion can be somewhat tedious—the reader who is comfortable with his/her intuition about substitution may skip over the next paragraph, which is included primarily for completeness.
To understand substitution we must first understand the notion of thefree variables of an expressione, which we write asfv(e), and define by the following simple rules:
fv(x) = {x}
fv(e1 e2) = fv(e1)∪fv(e2) fv(λx.e) = fv(e)− {x}
We say that x is free in e iff x ∈ fv(e). The substitution [e1/x]e2 is then defined inductively by:
[e/xi]xj =
e, ifi=j xj, ifi=j
[e1/x](e2 e3) = ([e1/x]e2) ([e1/x]e3)
[e1/xi](λxj.e2) =
λxj.e2, ifi=j
λxj.[e1/xi]e2, ifi=jandxj ∈fv(e1)
λxk.[e1/xi]([xk/xj]e2), otherwise, wherek=i, k=j, andxk ∈fv(e1)∪fv(e2)
The last rule is the subtle one, since it is where a name conflict could occur, and is resolved by making a name change. The following example demonstrates application
5The notation “d ∈ D” means thatd is a “typical element” of the setD, whose elements may be distinguished by subscripting. In the case of identifiers, we assume that eachxiis unique; i.e.,xi=xjif i=j. The notation “d::=alt1|alt2| · · · |altn” is standard BNF syntax.
6In denotational semantics the notatione[v/x]is used to denote the function ethat is just likee except thatex=v. Our notation of placing the brackets in front of the expression is to emphasize that [v/x]eis asyntactictransformation on the expressioneitself.
of all three rules:
[y/x]((λy.x)(λx.x)x) ≡ (λz.y)(λx.x)y
To complete the lambda calculus we define three simple rewrite rules on lambda expressions:
1. α-conversion (“renaming”): λxi.e⇔λxj.[xj/xi]e, where xj∈fv(e).
2. β-conversion (“application”): (λx.e1)e2⇔[e2/x]e1. 3. η-conversion: λx.(e x)⇔e, ifx∈fv(e).
These rules, together with the standard equivalence relation rules for reflexivity, sym- metricity, and transitivity, induce a theory of convertibility on the lambda calculus, which can be shown to be consistent as a mathematical system.7 The well-known Church-Rosser Theorem [CR36] (actually two theorems) is what embodies the strongest form of consistency, and has to do with a notion of reduction, which is the same as convertibility but restricted so thatβ-conversion andη-conversion only happen in one direction:
1. β-reduction: (λx.e1)e2⇒[e2/x]e1. 2. η-reduction: λx.(e x)⇒e, ifx∈fv(e).
We write “e1∗⇒ e2” if e2 can be derived from zero or more β- or η-reductions or α- conversions; in other words∗⇒ is the reflexive, transitive closure of ⇒including α- conversions. Similarly,⇔∗ is the reflexive, transitive closure of⇔. In summary,∗⇒ cap- tures the notion of “reducibility,” and⇔∗ captures the notion of “intra-convertibility.”
Definition: A lambda expression is innormal formif it cannot be further reduced using β- orη-reduction.
Note that some lambda expressions havenonormal form, such as:
(λx. (x x)) (λx. (x x))
where the only possible reduction leads to an identical term, and thus the reduction process is non-terminating.
Nevertheless, the normal form appears to be an attractive canonical form for a term, has a clear sense of “finality” in a computational sense, and is what we intuitively think of as thevalueof an expression. Obviously we would like for that value to be unique, and we would like to be able to find it whenever it exists. The Church-Rosser Theorems give us positive results for both of these desires.
7The lambda calculus as we have defined it here is what Barendregt [Bar84] calls theλKη-calculus, and is slightly more general than Church’s originalλK-calculus (which did not include η-conversion).
Furthermore, Church originally showed the consistency of theλI-calculus [Chu41], an even smaller subset (it only allowed abstraction ofxfromeifxwas free ine). We will ignore the subtle differences between these calculi here—our version is the one most often discussed in the literature on functional languages.
2.1.2 The Church-Rosser Theorems
Church-Rosser Theorem I: If e0 ⇔∗ e1 then there exists an e2 such that e0∗⇒ e2 and e1∗⇒ e2.8
In other words, if e0 and e1 are intra-convertible, then there exists a third term (possible the same ase0 ore1) to which they can both be reduced.
Corollary: No lambda expression can be converted to two distinct normal forms (ignor- ing difference due toα-conversion).
One consequence of this result is that how we arive at the normal form does not matter; i.e. the order of evaluation is irrelevant (this has important consequences for parallel evaluation strategies). The question then arises as to whether or not it is always possible to find the normal form (assuming it exists). We begin with some definitions.
Definition:Anormal-order reductionis a sequential reduction in which, whenever there is more than one reducible expression (called aredex), the left-most one is chosen first.
In contrast, anapplicative-order reductionis a sequential reduction in which the left- mostinnermostredex is chosen first.
Church-Rosser Theorem II: If e0∗⇒ e1 and e1 is in normal form, then there exists a normal-order reduction frome0toe1.
This is a very satisfying result; it says that if a normal form exists, we can always find it; i.e. just use normal-order reduction. To see why applicative-order reduction is not always adequate, consider the following example:
applicative-order: normal-order:
(λx. y) ((λx. x x) (λx. x x)) (λx. y) ((λx. x x) (λx. x x))
⇒ (λx. y) ((λx. x x) (λx. x x)) ⇒ y
⇒ ...
We will return to the trade-offs between normal- and applicative-order reduction in Section 3.2. For now we simply note that the strongest completeness and consistency results have been achieved with normal-order reduction.
In actuality, one of Church’s (and others’) motivations for developing the lambda calculus in the first place was to form a foundation for all of mathematics (in the way that, for example, set theory is claimed to provide such a foundation). Unfortunately, all attempts to extend the lambda calculus sufficiently to form such a foundation failed to yield a consistent theory. Church’s original extended system was shown inconsistent by the Kleene-Rosser Paradox [KR35]; a simpler inconsistency proof is embodied in what is known as the Curry Paradox [Ros82]. The only consistent systems that have
8Church and Rosser’s original proofs of their theorems are rather long, and many have tried to improve on them since. The shortest proof I am aware of for the first theorem is fairly recent, and aptly due to Rosser [Ros82].
been derived from the lambda calculus are much too weak to claim as a foundation for mathematics, and the problem remains open today.
These inconsistencies, although disappointing in a foundational sense, did not slow down research on the lambda calculus, which turned out to be quite a nice model of functions and of computation in general. The Church-Rosser Theorem was an ex- tremely powerful consistency result for a computation model, and in fact rewrite sys- tems completely different from the lambda calculus are often described as “possessing the Church-Rosser property” or even anthropomorphically as being “Church-Rosser.”
2.1.3 Recursion,λ-Definability, and Church’s Thesis
Another nice property of the lambda calculus is embodied in the following theorem:
Fixpoint Theorem: Every lambda expressione has a fixpointesuch that(e e)⇔∗ e. Proof: Takeeto be(Y e)whereY, known as theY combinator, is defined by:
Y ≡λf .(λx.f (x x))(λx.f (x x)) Then we have:
(Y e) = (λx.e(x x))(λx.e(x x))
= e((λx.e(x x))(λx.e(x x)))
= e(Y e)
This surprising theorem (and equally surprising simple proof) is what has earned Y the name “paradoxical combinator.” The theorem is quite significant—it means that any recursive function may be written non-recursively (and non-imperatively!). To see how, consider a recursive functionf defined by:
f ≡ · · ·f· · · This could be rewritten as:
f ≡ (λf . · · ·f· · ·)f
where note that the inner occurrence off is now bound. This equation essentially says thatf is a fixpoint of the lambda expression(λf . · · ·f· · ·)—but that is exactly what Y computes for us, so we arrive at the followingnon-recursivedefinition forf:
f ≡ Y (λf . · · ·f · · ·) As a concrete example, the factorial function:
fac ≡ λn.if(n=0)then1else(n∗fac(n−1)) can be written non-recursively as:
fac ≡ Y(λfac. λn.if (n=0)then 1 else(n∗fac(n−1)))
The ability of the lambda calculus to “simulate recursion” in this way is the key to its power, and accounts for its persistence as a useful model of computation. Church recognized this, and is perhaps best expressed in his now famous thesis:
Church’s Thesis: Effectively computable functions from positive integers to positive in- tegers are just those definable in the lambda calculus.
This is quite a strong claim! Although the notion of “functions from positive inte- gers to positive integers” can be formalized very precisely, the notion of “effectively computable” cannot; thus no proof can be given for the thesis. However, it gained support from Kleene who in 1936 [Kle36] showed that “λ-definability” was precisely equivalent to Gödel and Herbrand’s notions of “recursiveness.” Meanwhile, Turing had been working on his now famous Turing Machine [Tur36], and in 1937 [Tur37]
he showed that “Turing computability” was also precisely equivalent toλ-definability.
These were quite satisfying results.9
The lambda calculus and the Turing machine were to have profound impacts on programming languages and computational complexity,10 respectively, and computer science in general. This influence was probably much greater than Church or Turing could have imagined, which is perhaps not surprising given that computers did not even exist yet!
In parallel with the development of the lambda calculus, Schönfinkel and Curry were busy foundingcombinatory logic. It was Schönfinkel [Sch24] who discovered the surprising result that any function could be expressed as the composition of only two simple functions, K and S. Curry proved the consistency of a pure combinatory calculus [Cur30], and with Feys elaborated the theory considerably [CF58]. Although this work deserves as much attention from a logician’s point of view as the lambda calculus, and in fact its origins predate that of the lambda calculus, we will not pursue it further here, since it did not contribute directly to the development of functional languages in the way that the lambda calculus did. On the other hand, the combinatory calculus was to eventually play a surprising role in theimplementationof functional languages, beginning with [Tur79] and summarized in [PJ87, Chapter 16].
Another noteworthy attribute of the lambda calculus is its restriction to functions of one argument. That it suffices to consider only such functions was first suggested by Frege in 1893 [vH67] and independently by Schönfinkel in 1924 [Sch24]. This restriction was later exploited by Curry [CF58], who used the notation(f x y)to denote((f x) y), and which previously would have been written f (x, y). This notation has become known as “currying,” and f is said to be a “curried function.” As we will see, the
9Much later Post [Pos43] and Markov [Mar51] proposed two other formal notions of effective com- putability, but these also were shown to be equivalent toλ-definability.
10Although the lambda calculus and the notion ofλ-definability pre-dated the Turing Machine, com- plexity theorists latched onto the Turing Machine as their fundamental measure of decidability. This is probably because of the appeal of the Turing Machine as amachine, giving it more credibility in the emerging arena of electronic digital computers. See [Tra88] for an interesting discussion of this issue.
notion of currying has carried over today as a distinguishing syntactic characteristic of modern functional languages.
There are several variations and embellishments of the lambda calculus that his- torically could be mentioned here, but instead of doing that I will postpone their dis- cussion until the point at which functional languages exhibited similar characteristics.
In this way we can see more clearly the relationship between the lambda calculus and functional languages.
2.2 Lisp
A discussion of the history of functional languages would certainly be remiss if it did not include a discussion of Lisp, beginning with McCarthy’s seminal work in the late 1950’s.
Although lambda calculus is often considered as the foundation of Lisp, by Mc- Carthy’s own account [McC78] the lambda calculus actually played a rather small role.
Its main impact came through McCarthy’s desire to represent functions “anonymously,”
and Church’s λ-notation was what he chose: a lambda abstraction written λx.e in lambda calculus would be written(lambda (x) e)in Lisp.
Beyond that, the similarity wanes. For example, rather than use the Y combinator to express recursion, McCarthy invented the conditional expression,11 with which re- cursive functions could be defined explicitly (and, arguably, more intuitively). As an example, the non-recursive factorial function given in the lambda calculus in the last section would be written recursively in Lisp in the following way:
(define fac (n) (if (= n 0)
1
(* n (fac (- n 1))) ))
This and other ideas were described in two landmark papers in the early 60’s [McC60, McC63], papers that inspired work on Lisp for many years to come.
McCarthy’s original motivation for developing Lisp was the desire for an algebraic list processing language for use in artificial intelligence research. Although symbolic processing was a fairly radical idea at the time, his aims were quite pragmatic. One of the earliest attempts at designing such a language was suggested by McCarthy, and resulted in FLPL (for Fortran-compiled List Processing Language), implemented in 1958 on top of the Fortran system on the IBM 704 [GHG60]. Over the next few years McCarthy designed, refined, and implemented Lisp. His chief contributions during this period were:
11The conditional in Fortran (essentially the only other programming language in existence at the time) was astatement, not an expression, and was forcontrol, not value-defining, purposes.
1. The conditional expression and its use in writing recursive functions.
2. The use of lists and higher-order operations over lists such asmapcar.
3. The central idea of a “cons cell,” and the use of garbage collection as a method of reclaiming unused cells.
4. The use of S-expressions (and abstract syntax in general) to represent both pro- gram and data.12
All four of these features are essential ingredients of any Lisp implementation today, and the first three are essential to functional language implementations as well.
A simple example of a typical Lisp definition is the following one formapcar:
(define mapcar (fun lst) (if (null lst)
nil
(cons (fun (car lst)) (mapcar fun (cdr lst))) )) This example demonstrates all of the points mentioned above. Note that the function fun is passed as an argument to mapcar. Although such higher-order programming was very well known in lambda calculus circles, it was certainly a radical departure from Fortran, and has become one of the most important programming techniques in Lisp and functional programming (higher-order functions are discussed more in Section 3.1). The primitive functionscons,car,cdr, andnull are the well-known operations on lists whose names are still used today. conscreates a new list cell without burdoning the user with explicit storage management; similarly, once that cell is no longer needed a “garbage collector” will come along and reclaim it, again without user involvement.
For example, sincemapcarconstructs a new list from an old one, in the call:
(mapcar fun (cons a (cons b nil)))
the list(cons a (cons b nil))will become garbage after the call and will automat- ically be reclaimed. Lists were to become the paradigmatic data structure in Lisp and early functional languages.
The definition of mapcar in a modern functional language such as Haskell would appear similarly, except that pattern-matching would be used to “destructure” the list:
mapcar fun [] = []
mapcar fun (x:xs) = fun x : mapcar fun xs
12Interestingly, McCarthy claims that it was the read and print routines that influenced this notation most! [McC78]
[[]is the null list, and: is the infix operator forcons; also note that function appli- cation has higher precedence than any infix operator.]
McCarthy was also interested in designing a practical language, and thus Lisp had many pragmatic features—in particular, sequencing, the assignment statement, and other primitives that induced side-effects on the store. Their presence undoubtedly had much to do with early experience with Fortran. Nevertheless, McCarthy emphasized the “mathematical elegance” of Lisp in his early papers, and in a much later paper his student Cartwright demonstrated the ease with which one could prove properties about pure Lisp programs [Car76].
Despite its impurities, Lisp had a great influence on functional language develop- ment, and it is encouraging to note that modern Lisps (especially Scheme) have returned more to the purity of the lambda calculus rather than thead hocery that plagued the Maclisp era. This return to purity includes the “first-class” treatment of functions and the lexical scoping of identifiers. Furthermore, the preferred modern style of Lisp pro- gramming, such as espoused by Abelson and Sussman [ASS85], can be characterized as being predominantly side-effect free. And finally, we point out that Henderson’s Lispkit Lisp [Hen80] is a purely functional version of Lisp that uses an infix, algebraic syntax.
2.2.1 Lisp in Retrospect
Before continuing our historical development it is helpful to consider some of the design decisions McCarthy made and how they would be formalized in terms of the lambda calculus. It may seem that conditional expressions, for example, are an obvious feature to have in a language, but I think that only reflects our familiarity with mod- ern high-level programming languages, most of which have them. In fact the lambda calculus version of the factorial example given in the previous section used a condi- tional (not to mention arithmetic operators) yet most readers probably understood it perfectly, and did not object to my departure from precise lambda calculus syntax!
The effect of conditional expressions can in fact be achieved in the lambda calculus by encoding the true and false values as functions, as well as defining a function to emulate the conditional:
true ≡ λx.λy.x false ≡ λx.λy.y
cond ≡ λp.λc.λa.(p c a)
In other words,(condp c a) ≡ (ifpthenc elsea). One can then define, for example, the factorial function by:
fac ≡ λn.cond(=n0)1(∗n (fac(−n1)))
where=is defined by:
(=n n) ⇒ true
(=n m) ⇒ false, ifm=n
wheremandnrange over the set of integer constants. However, I am still cheating a bit by not explaining the nature of the objects−,∗, 0, 1, etc. in pure lambda calculus terms. It turns out that they can be represented in a variety of ways, essentially using functions to simulate the proper behavior, just as for true, false, and the conditional (for the details, see [Chu41]). In factanyconventional data or control structure can be
“simulated” in the lambda calculus; if this were not the case, it would be difficult to believe Church’s Thesis!
However, even if McCarthy knew of these ways to express things in the lambda calculus (there is reason to believe that he did not), efficiency concerns might have rapidly led him to consider other alternatives anyway, especially since Fortran was the only high-level programming language anyone had any experience with. In particular, Fortran functions evaluated their argumentsbeforeentering the body of the function, resulting in what is often called astrict, orcall-by-value, evaluation policy, correspond- ing roughly to applicative-order reduction in the lambda calculus. With this strategy extended to the primitives, including the conditional, one cannot easily define recur- sive functions. For example, in the above definition of factorial all three arguments to condwould be evaluated, includingfac(−n1), resulting in non-termination.
Non-strict evaluation, corresponding to the normal-order reduction that is essen- tial to the lambda calculus in realizing recursion, was not very well understood at the time—it was not at all clear how to implement it efficiently on a conventional von Neumann computer—and we would have to wait another 20 years or so before such an implementation was even attempted.13 The conditional expression essentially al- lowed one toselectivelyinvoke normal-order, or non-strict, evaluation. Stated another way, McCarthy’s conditional, although an expression, was compiled into code that es- sentially controlled the “reduction” process. Most imperative programming languages today that allow recursion do just that, and thus even though such languages are of- ten referred to as strict, they all rely critically on at least one non-strict construct: the conditional.
2.2.2 The Lambda Calculus With Constants
The conditional expression is actually only one example of very many primitive func- tions that were included in Lisp. Rather than explain them in terms of the lambda cal- culus by a suitable encoding (i.e. compilation), it is perhaps better to formally extend the lambda calculus by adding a set ofconstants along with a set of what are usually
13On the other hand, thecall-by-nameevaluation strategy invented in Algol had very much of a normal- order reduction flavor. See [Weg68, Wad71] for early discussions of these issues.
calledδ-ruleswhich state relationships between constants and which effectively extend the basis set of α-, β-, and η-reduction rules. For example, the reduction rules for = that I gave earlier (and which are repeated below) areδ-rules. This new calculus, often called thelambda calculus with constants, can be given a precise abstract syntax:
x ∈ Id Identifiers
c ∈ Con Constants
e ∈ Exp Lambda Expressions
wheree::=x |c |e1e2|λx.e for which variousδ-rules apply, such as the following:
(= 0 0) ⇒ True (= 0 1) ⇒ False
· · · (+0 0) ⇒ 0 (+0 1) ⇒ 1
· · · (+27 32) ⇒ 59
· · · (If Truee1e2) ⇒ e1
(If Falsee1e2) ⇒ e2
· · · (Car (Cons e1e2)) ⇒ e1
(Cdr (Cons e1e2)) ⇒ e2
· · ·
where =, +, 0, 1,If,True,False,Cons, etc. are elements ofCon.
The above rules can be shown to be aconservative extensionof the lambda calculus, a technical term that in our context essentially means that convertible terms in the original system are still convertible in the new, and (perhaps more importantly) incon- vertible terms in the original system are still inconvertible in the new. In general, care must be taken when introducingδ-rules, since all kinds of inconsistencies could arise.
For a quick and dirty example of inconsistency, we might define a primitive function over integers calledbrokenwith the followingδ-rules:
(broken0) ⇒ 0 (broken0) ⇒ 1
which immediately implies that more than one normal form exists for some terms, violating the first Church-Rosser theorem (see Section 2.1)!
As a more subtle example, suppose we define the logical relationOrby:
(Or Truee) ⇒ True (OreTrue) ⇒ True (Or False False) ⇒ False
Even though these rules form a conservative extension of the lambda calculus, a com- mitment to evaluate either of the arguments may lead to non-termination, even though the other argument may reduce toTrue. In fact it can be shown that with the above rules there does not exist a deterministic sequential reduction strategy that will guarantee that the normal formTruewill be found for all terms having such a normal form, and thus the second Church-Rosser property is violated. This version ofOris often called the “parallel or,” since a parallel reduction strategy is needed to implement it properly (and with which the first Church-Rosser Theorem will at least hold). Alternatively, we could define a “sequential or” by:
(Or Truee) ⇒ True (Or Falsee) ⇒ e
which can be shown to satisfy both Church-Rosser theorems.
2.3 Iswim
Historically speaking, Peter Landin’s work in the mid 60’s was the next significant impe- tus to the functional programming paradigm. Landin’s work was deeply influenced by that of Curry and Church. His early papers discussed the relationship between lambda calculus and both machines and high-level languages (specifically Algol 60). In [Lan64]
Landin discussed how one could mechanize the evaluation of expressions, through an abstract machine called the SECD machine; and in [Lan65] he formally defined a non-trivial subset of Algol 60 in terms of the lambda calculus.
It is apparent from his work that Landin regarded highly the expressiveness and purity of the lambda calculus, and at the same time recognized its austerity. Undoubt- edly as a result of this work, in 1966 Landin introduced a language (actually a family of languages) called Iswim (for “If you See What I Mean”), which included a number of significant syntactic and semantics ideas [Lan66]. Iswim, according to Landin, “can be looked on as an attempt to deliver Lisp from its eponymous committment to lists, its reputation for hand-to-mouth storage allocation, the hardware dependent flavor of its pedagogy, its heavy bracketing, and its compromises with tradition.” When all is said and done, I think the primary contributions of Iswim, with respect to the development of functional languages, are the following:
1. Syntactic innovations:
(a) The abandonment of prefix syntax in favor of infix.
(b) The introduction ofletandwhereclauses, including a notion of simultaneous and mutually recursive definitions.
(c) The use of an “off-side rule” based on indentation rather than separators (such as commas or semicolons) to scope declarations and expressions. For example (using Haskell syntax), the program fragment:
e where f x = x a b = 1
cannot be confused with:
e where f x = x a b = 1
and is equivalent to what might otherwise be written as:
e where { f x = x;
a b = 1 }
2. Semantic innovations:
(a) An emphasis on generality. Landin was half-way serious in hoping that the Iswim family could serve as the “next 700 programming languages.” Central to his strategy was the idea of defining a syntactically rich language in terms of a very small but expressive core language.
(b) An emphasis on equational reasoning (i.e. the ability to replace equals with equals). This elusive idea was backed up with four sets of rules for rea- soning about expressions, declarations, primitives, and “problem-oriented”
extensions.
(c) The SECD machine as a simple abstract machine for executing functional programs.
We can think of Landin’s work as extending the “lambda calculus with constants”
defined in the last section so as to include more primitives, each with its own set ofδ-rules, but more importantly let and whereclauses, for which it is convenient to introduce the syntactic category ofdeclarations:
e ∈ Exp Expressions
wheree ::= · · · |e whered1 · · · dn
|letd1 · · · dn ine
d ∈ Decl Declarations
whered ::= x=e
| x x1 · · · xn=e
and for which we then need to add some axioms (i.e. reduction rules) to capture the desired semantics. Landin proposed special constructs for defining simultaneous and mutually recursive definitions, but we will take a simpler and more general approach
here: We assume that a block of declarationsd1· · ·dn always is potentially mutually recursive—if it isn’t, our rules still work:
(letd1 · · · dnine) ⇒ (ewhered1 · · · dn) (x x1 · · · xn=e) ⇒ (x=λx1.λx2.· · ·λxn. e)
(ewherex1=e1) ⇒ (λx1.e)(Y λx1.e1)
(e where(x1=e1)· · ·(xn=en)) ⇒ (λx1.e)(Y λx1.e1)where x2=(λx1.e2)(Y λx1.e1)
· · ·
xn=(λx1.en)(Y λx1.e1) These rules are semantically equivalent to Landin’s, but they avoid the need for a tu- pling operator to handle mutual recursion, and they use the Y combinator (defined in Section 2.1) instead of an iterative “unfolding” step.
We will call this resulting system the recursive lambda calculus with constants, or justrecursive lambda calculus.
Landin’s emphasis on expressingwhat the desired result is, as opposed to saying howto get it, and his claim that Iswim’s declarative14 style of programming was bet- ter than the incremental and sequential imperative style, were ideas to be echoed by functional programming advocates to this day. On the other hand, it took another ten years before interest in functional languages was to be substantially renewed. One of the reasons is that there were no decent implementations of Iswim-like languages around; this reason, in fact, was to persist into the 80’s.
2.4 APL
Iverson’s APL [Ive62], although not a purely functional programming language, is worth mentioning because its functional subset is an example of how one could achieve func- tional programming without relying on lambda expressions. In fact Iverson’s design of APL was motivated out of his desire to develop analgebraicprogramming language for arrays, and his original work used an essentially functional notation. Subsequent de- velopment of APL resulted in several imperative features, but the underlying principles should not be overlooked.
APL was also unique in its goal of succinctness, in that it used a specially designed alphabet to represent programs—each letter corresponding to one operator. That APL became popular is apparent in the fact that many keyboards, both for typewriters and computer terminals, carried the APL alphabet. Backus’ FP, which came after APL, was certainly influenced by the APL philosophy, and its abbreviated publication form also used a specialized alphabet (see the example in Section 2.5). In fact FP has much in common with APL, the primary difference being that FP’s fundamental data structure is thesequence, whereas in APL it is thearray.
14Landin actually disliked the term “declarative,” preferring instead “denotative.”
It is worth noting that recent work on APL has revived some of APL’s purely func- tional foundations. The most notable work is that of Tu [Tu86, TP86], who designed a language called FAC, for Functional Array Calculator (presumably a take-off on Turner’s KRC, Kent Recursive Calculator). FAC is a purely functional language that adopts most of the APL syntax and programming principles, but in addition has special features to allow programming withinfinitearrays; naturally, lazy evaluation plays a major role in the semantics of the language. Another interesting approach is that of Mullin [Mul88].
2.5 FP
Backus’ FP was one of the earliest functional languages to receive wide-spread atten- tion. Although most of the features in FP are not found in today’s modern functional languages, Backus’ Turing Award lecture [Bac78] in 1978 was one of the most influen- tial and now most-often cited papers extolling the functional programming paradigm.
It not only stated quite eloquently why functional programming was “good,” but also quite vehemently why traditional imperative programming was “bad.”15 Backus’ coined the term “word-at-a-time programming” to capture the essence of imperative languages, showed how such languages were inextricably tied to the “von Neumann machine,” and gave a convincing argument why such languages were not going to meet the demands of modern software development. That this argument was being made by the person who is given the most credit for designing Fortran, and who also had significant influ- ence on the development of Algol, led substantial weight to the “functional thesis.” The exposure given to Backus’ paper was one of the best things that could have happened to the field of functional programming, which at the time was certainly not considered main-stream.
Despite the great impetus Backus’ paper gave to functional programming, it is inter- esting to note that in the same paper Backus also said that languages based on lambda calculus would have problems, both in implementation and expressiveness, because the model was not suitably history-sensitive (more specifically, it did not handle large data structures, such as databases, very easily). With regard to implementation this argument is certainly understandable, because it was not clear how to implement the notion of substitution in an efficient manner, nor was it clear how to structure data in such ways that large data structures could be implemented efficiently (both of these issues are much better understood today). And with regard to expressiveness, well, that argument is still a matter of debate today. In any case, these problems were the motivation for Backus’ Applicative State Transition (AST) Systems, in which state is introduced as something on which purely functional programs interact with in a more traditional (i.e. imperative) way.
Perhaps more surprising, and an aspect of the paper that is usually overlooked, Backus had this to say about “lambda-calculus based systems:”
15Ironically, the Turing Award was given to Backus in a large part because of his work on Fortran!
“An FP system is founded on the use of a fixed set of combining forms called functional forms.. . .In contrast, a lambda-calculus based system is founded on the use of the lambda expression, with an associated set of substitution rules for variables, for building new functions. The lambda expression (with its substitution rules) is capable of defining all possible computable func- tions of all possible types and of any number of arguments. This freedom and power has its disadvantages as well as its obvious advantages. It is analogous to the power of unrestricted control statements in conventional languages: with unrestricted freedom comes chaos. If one constantly in- vents new combining forms to suit the occasion, as one can in the lambda calculus, one will not become familiar with the style or useful properties of the few combining forms that are adequate for all purposes.”
Backus’ argument, of course, was in the context of garnering support for FP, which had a small set of combining forms that were claimed to be sufficient for most pro- gramming applications. One of the advantages of this approach is that each of these combining forms could be named with particular brevity, and thus programs become quite compact—this was exactly the approach taken by Iverson in designing APL (see Section 2.4). For example, an FP program for inner product looks like:
DefIP ≡(/+)◦(α×)◦Trans
where/, ◦, and αare combining forms called “insert,” “compose,” and “apply-to-all,”
respectively. In a modern functional language such as Haskell this would be written with slightly more verbosity as:
ip l1 l2 = foldl (+) 0 (map2 (*) l1 l2)
[In Haskell an infix operator such as+ may be passed as an argument by surrounding it in parentheses.] Here foldl is the equivalent of insert (/), and map2 is a two-list equivalent of apply-to-all (α), thus eliminating the need for Trans. These functions are pre-defined in Haskell, as they are in most modern functional languages, for the same reason that Backus argues—they are commonly used. If they were not they could easily be defined by the user. For example, we may wish to define an infix composition operator for functions, the first of which is binary, as follows:
(f .o. g) x y = f (g x y)
[Note how infix operators may be defined in Haskell; operators are distinguished lexi- cally by their first character being non-alphabetic.] With this we can reclaim much of FP’s succinctness in definingip:
ip = foldl (+) 0 .o. map2 (*)
[Recall that in Haskell function application has higher precedence than infix operator application.] It is for this reason, together with the fact that FP’s specialization pre- cluded the generality afforded by user-defined higher-order functions (which is all that combining forms are), that modern functional languages did not follow the FP style. As we shall soon see, certain other kinds of syntactic sugar became more popular instead (such as pattern-matching, list comprehensions, and sections).
Many extensions to FP have been proposed over the years, including the inclusion of strong typing and abstract datatypes [GHW81]. In much more recent work, Backus, Williams and Wimmers have designed the language FL [BWW86], which is strongly (although dynamically) typed, and in which higher-order functions and user-defined datatypes are allowed. Its authors still emphasize the algebraic style of reasoning that is paramount in FP, although it is also given a denotational semantics which is provably consistent with respect to the algebraic semantics.
2.6 ML
In the mid 70’s, at the same time Backus was working on FP at IBM, several research projects were underway in the United Kingdom that related to functional programming, most notably work at Edinburgh. There Gordon, Milner, and Wadsworth had been work- ing on a proof generating system called LCF for reasoning about recursive functions, in particular in the context of programming languages [GMW79]. The system consisted of a deductive calculus called PPλ (polymorphic predicate calculus) together with an interactive programming language called ML, for “meta-language” (since it served as the command language for LCF).
LCF is quite interesting as a proof system, but its authors soon found that ML was also interesting in its own right, and they proceded to develop it as a stand-alone functional programming language [GMM*78]. That it was, and still is, called afunctional language is actually somewhat misleading, since it has a notion ofreferenceswhich are locations that can be stored into and read from, much as variables are assigned and read. Its I/O system also induces side-effects, and is not referentially transparent.
Nevertheless, the style of programming that it encourages is still functional, and that is the way it was promoted (the same is somewhat true for Scheme, although to a lesser extent).
More recently a standardization effort for ML has taken place, in fact taking some of the good ideas of Hope [BMS80] along with it (such as pattern-matching), yielding a language now being calledStandard ML, or just SML [Mil84, Wik88].
ML is a fairly complete language—certainly the most practical functional language at the time it appeared—and SML is even more so. It has higher-order functions, a simple I/O facility, a sophisticated module system, and even exceptions. But by far the most significant aspect of ML is itstype system, which is manifested in several ways:
1. It isstronglyandstaticallytyped.
2. It usestype inferenceto determine the type of every expression, instead of relying on explicit type declarations.
3. It allowspolymorphic functions and data structures; that is, functions may take arguments of arbitrary type if in fact the function does not depend on that type (similarly for data structures).
4. It has user-defined concrete and abstract datatypes (an idea actually borrowed from Hope, and not present in the initial design of ML).
ML was the first language to utilize type inference as a semantically integrated compo- nent of the language, and at the same time its type system was richer than any previous statically typed language in that it permitted true polymorphism. It seemed that the best of two worlds had been achieved—not only is making explicit type declarations (a sometimes burdensome task) not required, but in addition a program that successfully passes the type inferencer is guaranteed not to have any type errors. Unfortunately, this idyllic picture is not completely accurate (although it is close!), as we shall soon see.
We shall next discuss the issue of types in a foundational sense, thus continuing our plan of elaborating the lambda calculus to match the language features being discussed.
This will lead us to a reasonable picture of ML’s Hindley-Milner type system, the rich polymorphic type system that was mentioned above, and that was later adopted by every other statically-typed functional language, including Miranda and Haskell. Aside from the type system, the two most novel features in ML are itsreferencesandmodules, which are also covered in this section. Discussion of ML’s data abstraction facilities will be postponed until Section 3.3.
2.6.1 The Hindley-Milner Type System
We can introduce types into the pure lambda calculus by first introducing a domain of basic types, say BasTyp, as well as a domain of derived types, say Typ, and then requiring that every expression be “tagged” with a member of Typ, which we do by superscripting, as in “eτ.” The derived type “τ2→τ1” denotes the type of all functions from values of typeτ2to values of typeτ1, and thus a proper application will have the form “(eτ12→τ1 eτ22)τ1.” Modifying the pure lambda calculus in this way, we arrive at the
