Scheme Programming Language Essay, Research Paper

The Scheme programming language

Ken Dickey

An Alternate World View

Each programming language presents a particular world view in the

features it allows, supports, and forbids. This series of

articles describes the world view of the Scheme Programming

Language. This view contains many elements desired in a modern

programming language: multi-paradigm support, composable,

reusable abstractions, ability to create languages specialized

for particular applications, clean separation of the generic and

the implementation specific, and scalability from stand alone

utilities to major software systems.

Scheme started as an experiment in programming language design

by challanging some fundamental design assumptions. It is

currently gaining favor as a first programming language in

universities and is used in industry by such companies as DEC,

TI, Tektronix, HP, and Sun.

WHAT IS SCHEME?

Scheme is a small, exceptionally clean language which is, very

importantly, fun to use. The language was designed to have very

few, regular constructs which compose well to support a variety

of programming styles including functional, object-oriented, and

imperative. The language standard is only about 50 pages,

including a formal, denotational definition of its semantics.

Scheme is based on a formal model (the lambda calculus), so there

are plenty of nice properties for the theoreticians and one can

build knowledgeable code transformation tools reliably.

Scheme has lexical scoping, uniform evaluation rules, and uniform

treatment of data types. Scheme does not have the concept of a

pointer, uninitialized variables, specialized looping constructs,

or explicit storage management.

So what does Scheme look like? Well, it looks a lot like Lisp.

Don’t let this put you off. We can change how it looks (and will

in a future article). But what is important are the concepts

behind it and what you can say with it. So let me make a few

comparisons between Scheme and, say C. You already know that

Scheme is prefix and C is infix:

Scheme C

(+ 2 3 4) (2 + 3 + 4)

(* low x high) ((low * x) && (x * high))

(+ (* 2 3) (* 4 5)) ((2 * 3) + (4 * 5))

(f x y) f(x, y)

(define (sq x) (* x x)) int sq(int x) { return (x * x) }

In Scheme, all data types are equal. What one can do to one data

type, one can do to all data types. This is different from most

languages which have some data types that can do special things

and other data types which are specially restricted. In most

languages numbers have special rights because they can be used

without having names (imagine having to name each number before

using it!). Functions are specially restricted because they

cannot be created without a name.

In Scheme, unnamed functions are created with the key word

lambda.

(lambda (x) (* x x)) -* a function

(define sq (lambda (x) (* x x))

(sq 9) -* 27

((lambda (x) (* x x)) 9) -* 27

((if (foo? x) * +) 2 3 4) -* if (foo? x) is true,

then (* 2 3 4)

else (+ 2 3 4)

(define (curried-add x) (lambda (y) (+ x y))

(define add3 (curried-add 3)) ;; add3 is a funciton

(add3 7) -* 10

((curried-add 3) 7) -* 10

Rubrics:

- Variables can hold values of any type.

- Names refer to values; names are not required.

- An expression is one or more forms between parentheses.

- To evaluate an expression, evaluate all the terms first and

apply the value of the first form to the values of the other

forms. A nested expression counts as a form.

- A comment starts at a semicolon (;) and goes to the end of the

line it is on.

- When a function is evaluated, it remembers the environment

where it was evaluated. {So add3 remembers that X has the value

3 when it evaluates ((lambda (y) (+ x y)) 7).}

(define (sq x) (* x x)) is just syntactic sugar for

(define sq (lambda (x) (* x x))

There are seven kinds of expressions:

Constant: ‘foo #\Z 3 “a string”

Variable reference: foo joe a-long-name-for-an-identifier +

Procedure creation: (lambda (z) (* z z z))

Procedure application: (cube 37)

Conditional: (if (* x 3) sqrt modulo)

Assignment: (set! x 5)

Sequence: (begin (write x) (write y) (newline))

Scheme has the usual assortment of data types:

Characters: #\a #\A #\b #\B #\space #\newline

Strings: “A little string”

Arrays (called vectors): #(1 2 “string” #\x 5)

Lists: (a little (list of) (lists))

Numbers: 47 1/3 2.3 4.3e14 1+3i

Functions (also called procedures)

Booleans: #t #f

Ports (e.g. open files)

Symbols: this-is-a-symbol foo a32 c$23*4&7+3-is-a-symbol-too!

Rubrics:

- A vector’s contents can be any data objects.

- Symbols may include the characters + – . * / * = * ! ? : $ % _

& ~ and ^.

- Symbols are case insensitive.

- Symbols are used for identifiers, so identifiers (variable

names) are case insensitive.

- By convention predicates end in a question mark {e.g. string?}

and side effecting procedures end in an exclimation mark {e.g.

set!}.

Numbers are especially interesting in that an integer is a

rational is a real is a complex. There is no classification of

numbers based on their storage class {e.g. fixed, float, bignum,

…} but on whether or not they are exact.

(exact? 3) -* #t

(exact? 1/3) -* #t

(exact? 2.3##) -* #f

(+ 2/3 1/2 5/6) -* 2

(integer? 2) -* #t

(integer? 3/7) -* #f

(real? 2) -* #t

The ZEN of SCHEME

One of the joys of Scheme which initially confuses some people is

the lack of inhibition. Scheme is very expressive, which means

that one can say “the same thing” in many ways. In Scheme one

has the freedom–and the responsibility–of saying exactly what

is desired. We are all used to working around certain language

features to get something done. Scheme gets in the way very

little.

As a warming up exercise, let’s build a pair. A pair consists of

2 elements obtained by the access routines FIRST and SECOND. The

constructor is called PAIR. The relations to be maintained are

(first (pair 1 2)) -* 1 ; (second (pair 1 2)) -* 2. For those of you

who know LISP, this is not much to get worked up over. But how

many ways can we implement the trio: pair, first, second? There

is a virtually endless variety.

;; vector

(define (PAIR a b) (vector a b)) ;; or just: (define PAIR vector)

(define (FIRST aPair) (vector-ref aPair 0))

(define (SECOND aPair) (vector-ref aPair 1))

——

;; selector function

(define (PAIR a b) (lambda (bool) (if bool a b)))

(define (FIRST aPair) (aPair #t))

(define (SECOND aPair) (aPair #f))

——

;; message passing

(define (PAIR (a b)

(lambda (msg)

(case msg ;; we’ll implement CASE in the next article

((first) a) ;; when the message is the symbol first, return a

((second) b)

) ) )

(define (FIRST aPair) (aPair ‘first))

(define (SECOND aPair) (aPair ’second))

——

;; alias

(define PAIR cons)

(define FIRST car)

(define SECOND cdr)

——

;; pure functions

(define (PAIR a b) (lambda (proc) (proc a b)))

(define (FIRST aPair) (aPair (lambda (x y) x)))

(define (SECOND aPair) (aPair (lambda (x y) y)))

The moral of the above is not to assume anything about the

implementation of interfaces–even simple ones.

Now that we are warmed up, let’s take a look at ye ol’ factorial

function on integers.

First, the recursive definition:

(define (fact n)

(if (* n 2)

1

(* n (fact (sub1 n)) ;; (define (sub1 n) (- n 1))

)

When I first learned about recursive definitions like the one

above, the hard thing to get used to was the how the hidden state

kept on the stack. A transformation of the above code makes the

stack visible.

;; the identity function just returns its argument

(define (identity value) value)

;; keep invocation of fact the same, but do something different

(define (fact n) (cfact n identity))

;; the transformed recursive factorial

(define (cfact n k)

(if (* n 2)

(k 1)

(cfact (sub1 n)

(lambda (result) (k (* n result))))

) )

Cfact is the continuation-passing version of fact. The basic

idea here is that instead of returning a result each function

takes an extra argument, the continuation, which is invoked with

the result of the function.

Let’s look at what happens for (fact 3).

(fact 3) is (cfact 3 identity)

(cfact 3 identity) is

(cfact 2

(lambda (result) (identity (* 3 result)))) ;; k’

which is

(cfact 1

(lambda (result^) ;; k”

((lambda (result) (identity (* 3 result))) ; fun k’

(* 2 result^)) ; argument to k’

-*

((lambda (result^) ;; k”

((lambda (result) (identity (* 3 result)))(* 2 result^)))

1)

-*

((lambda (result) (identity (* 3 result))) (* 2 1))

-*

(identity (* 3 (* 2 1)))

-*

(* 3 (* 2 1))

or {as we knew all along}

6

The point of this is not that we can take something simple and

make it look bizarre. So why did I do it? The point is that we

can take control which is typically hidden on the stack at run

time and study and manipulate it as source code. This lets us do

some interesting things. For example, we can ignore the

continuation passed in and use another one. If we are writing a

game or an AI search routine or a mathematical routine which

converges on a result, we can monitor our progress. If our

progress is slow, we can save our continuation–”where we are

now”–and try a different approach. If the newer approach is

even worse, we can go back to our saved continuation and try

again from there. We can of course save our attempt at doing

better to try again just in case it really was better…

So continuation passing can let us build some pretty interesting

control structures. Let’s take another look at the simple

function, fact. When we look at (fact 4), (fact 5), and so on,

we see a common pattern. Each call just stacks up a

multiplication. Since we can see this, we can eliminate the

stack and do the multiplication directly by introducing an

accumulator. We take care of initializing the accumulator by

noting that anything times one is the same as itself {i.e. the

multiplicitive identity: x * 1 = x}.

(define (cfact n k)

;; lexical scope means we can nest functions

(define (ifact x acc k^)

(if (* x 2)

(k^ acc)

(ifact (sub1 x) (* x acc) k^)

) )

(ifact n 1 k)

)

Now this looks a bit strange too. But there is an interesting

detail here. We did not have to build any new continuations!

The continuation k^ which is given the final result is exactly

the original continuation k. This means that the call of ifact,

which looks recursive, is really iterative. When it “calls

itself” it does not add to the stack. The “recursive” call to

ifact is just a goto.

Scheme requires no special looping constructs. Any function which

calls itself in the “tail” position {i.e. as the last thing to be

done in the function} is just a loop. Most procedural languages

do too much work. They “push a return address” even when they

don’t have to. Scheme doen’t. In Scheme, function calls can be

thought of as gotos which can pass parameters–one of which may

be a “return address”.

This means that we can simplify cfact to:

(define (cfact n k)

(define (ifact n acc)

(if (* n 2)

(k acc)

(ifact (sub1 n) (* n acc))

)

(ifact n 1)

)

Taking this a step further, we can redefine fact to call ifact

directly:

(define (fact n) (ifact n acc))

(define (ifact n acc)

(if (* n 2) acc (ifact (sub1 n) (* n acc)) )

Or taking advantage of Scheme’s lexical scoping:

(define (fact n)

(define (ifact n^ acc)

(if (* n^ 2)

acc

(ifact (sub1 n^) (* n^ acc))

) )

(ifact n 1)

)

Now we have transformed a recursive function into an iterative

one. This can be done in a formal way, proving every code

transformation. But a nice feature of Scheme is that such

transformations can be seen and used directly. Correct programs

can be written which are simple but perhaps run slowly or are not

very space efficient and then reliably transformed into programs

which are smaller and run much faster–and are also correct.

Code transformations become second nature to experienced Scheme

programmers. When a function similar to the recursive function

fact is seen, it tends to get written down in the iterative form.

—–

Scheme has several important advantages. Its elegantly simple,

regular structure and trivial syntax avoids “special case” confusion.

Its expressiveness means that one spends little time trying to work

around the language–it lets users concentrate on *what* they want to

say rather than on *how* to say it. Its support of a variety of

styles (including OO, which has not been emphasized here) allows users

to better match their solution style to the style of the problems to

be solved. Its formal underpinnings make reasoning about programs

much easier. Its abstractive power makes it easy to separate system

specific optimizations from reusable code. Its composability makes

it easy to construct systems from well-tested components.

If you want to write complex, correct programs quickly, scheme for

success!

;;========================MACROS=================================

Syntax Extension: MACROS

Just as functions are semantic abstractions over operations,

macros are textual abstractions over syntax. Managing complex

software systems frequently requires designing specialized

“languages” to focus on areas of interest and hide superfluous

details. Macros have the advantage of expanding the syntax of

the base language without making the native compiler more complex

or introducing runtime penalties.

Macros have always been a source of great power and confusion.

Scheme has perhaps the most sophisticated macro system of any

programming language in wide use today. How has macro technology

advanced? What are the problems which have been solved?

PROBLEMS WITH MACROS

Macros are frequently considered an advanced topic. Non-trivial

macros are notoriously hard to get right. Because macros can be

used to extend the base language, doing macro design can also be

viewed as doing language design.

In the early days, macros were based on simple text substitution.

A problem with text substitution is that tokenization rules of

the programming language are not respected. Indeed the

tokenization rules of the preprocessor may not match the rules of

the target language. For example, using the m4 preprocessor:

define( glerph, `”ugly’ )

the token glerph followed by the non-token glop”

glerph glop”

becomes the string token

“ugly glop”

In addition to tokenization confusion, there are problems where

macro expansion does not respect the structure of source language

expressions. For example, a well known problem with the C

preprocessor:

#define mean( a, b ) a + b / 2

mean( x+y, x*y )

becomes

x+y+x*y/2

and is interpreted by the compiler as

x + y + ((x * y) / 2)

There are frequently problems relating to the accidental

collision of introduced names. Even obscure names may collide

where there are multiple macro writers or recursive macros.

Again, using the C preprocessor:

#define swap( a, b ) { int temp; temp = a; a = b; b = temp }

…

swap( temp, x )

becomes

{int temp; temp = temp; temp = b; b = temp}

Free names may collide with those used locally:

#define clear(addr,len) fill(addr,len,0)

…

{int fill = 0×5a5aa5a5L;

…

clear(mem_ptr, mem_size); /* local fill shadows hidden global fill */

In general, macro systems have done poorly on name conflicts

where lexical scoping and recursion are involved.

So what do we want out of a macro system? We want respect for

the syntax, expressions, and name bindings. We want error

checking. We want a convenient notation for specifying syntactic

transcriptions. And of course we want this in linear processing

time.

SOME SOLUTIONS

One thing to notice is that as macro systems have become more

sophisticated, they have moved closer to the semantic analysis

phase of the compiler. LISP systems achieved respect for target

language syntax by doing macro transformations on parse trees

rather than source text. Scheme’s system takes things a step

further by respecting lexical scoping and name bindings. In

addition, the standard high-level part of Scheme’s macro system

specifies syntactic rewrite rules in a pattern directed fashon

which includes error checks.

To contrast Scheme’s macro system with those of older LISPs, here

is a brief historical view of the LET macro. LET is a construct

for introducing new name bindings. It has the form:

(let ( ( ) … )

… )

The semantics of LET is to evaluate the expressions in an

environment extended by the names which have initial

values obtained by evaluating the expressions . An

example is: (let ( (a 1) (b 2) ) (+ a b) ), which binds the value 1 to

a, 2 to b and then evaluates the expression (+ a b). LET is

syntactic sugar for lambda binding:

( (lambda ( …) …) … )

So (let ( (a 1) (b 2) ) (+ a b) ) rewrites to

( (lambda (a b) (+ a b)) 1 2 )

Early LISP systems operated directly on the list structures of

the parsed source expression using low-level operations:

(macro LET

(lambda (form)

(cons (cons ‘lambda

(cons (map car (cadr form))

(cddr form)))

(map cadr (cadr form)))))

Later, arguments and “backquotes” were added, making things much

more readable, but without error checking. The backquote (`)

indicates an “almost constant” list expression, where comma (,)

or comma-splice (,@) are used to force evaluation of a

subexpression. The comma replaces the evaluated expression

directly, where comma-splice splices it into the list.

So `(lambda ,(cdr (cons ‘a ‘b)) ,@(cdr (cons ‘a ‘b)))

becomes (lambda (b) b) .

Here is LET with argument binding and backquote:

(define-syntax (LET def-list . expressions)

`((lambda ,(map car def-list) ,@expressions)

,@(map cadr def-list)))

And here is the Scheme version using pattern maching with error

checks:

(define-syntax LET

(syntax-rules ()

( (let ( ( ) …) …) ; before

; rewrites to

((lambda ( …) …) …) ; after

) )

)

This next example demonstrates both macro names shadowing local

variables and locals shadowing macros. The outer binding of CAR

is to a function which returns the first element of a list.

;; from “Macros That Work”

(let-syntax ( (first (syntax-rules () ((first ?x) (car ?x))))

(second (syntax-rules () ((second ?x) (car (cdr ?x)))))

)

(let ( (car “Packard”) )

(let-syntax ( (classic (syntax-rules () ((classic) car))) )

(let ( (car “Matchbox”) )

(let-syntax ( (affordable (syntax-rules () ((affordable) car))) )

(let ( (cars (list (classic) (affordable))) )

(list (second cars) (first cars)) ; result

) ) ) ) ) )

The above evaluates to: (”Matchbox” “Packard”)

PRACTICUM: extending core Scheme to standard scheme

Scheme can remain such a small language because one can extend

the syntax of the language without making the compiler more

complex. This allows the compiler to know a great deal about the

few (7) basic forms. Most compiler optimizations then fall out

as general rather than special cases.

The following syntax definitions from the current Scheme standard

are directly (and simply) implemented.

Form: (or …)

Example: (or (= divisor 0) (/ number divisor))

Semantics: OR evaluates the expressions from left to right. The

value of the first true (not #f) expression is returned. Any

remaining expressions are not evalusted.

(define-syntax OR

(syntax-rules ()

((or) ;=*

#f

)

((or ) ;=*

)

((or …) ;=*

(let ( (temp ) )

(if temp temp (or …))

) )

) )

Form: (and …)

Example: (and (input-port? p) (read p))

Semantics: AND evaluates the expressions from left to right.

The value of the first false expression is returned. Any

remaining expressions are not evalusted.

(define-syntax AND

(syntax-rules ()

((and) ;=*

#t

)

((and ) ;=*

)

((and …) ;=*

(if (and …) #f)

))

) )

Forms: (let ( ( ) …) …)

(let ( ( ) …) …)

Examples:

(define A 37)

(let ( (a 2) (b (+ a 5)) ) (* a b)) ;=* 84

(let loop ( (count N) (accum 0) )

(if (* n 2)

accum

(loop (- count 1) (* count accum))

Semantics: LET evaluates the s in the enclosing environment

in some unspecified order and then extends the environment by

binding the values to the s and evaluates the expressions

from left to right, returning the value of the last expression as

the value of the LET. LET can be thought of as a “parallel

assignment”. Note that the value of B in the first example

depends on the value of A in the outer environment.

The second form is known as NAMED LET and allows recursion within

the let form. For the example above, the call to LOOP acts as a

goto which rebinds the s to fresh values and “starts over

at the top of the loop”. Note that this is a functional

construct: there are no unbound variables introduced and no

assignment is used.

(define-syntax LET

(syntax-rules ()

((let ( ( ) …) …)

;=*

((lambda ( …) …) …)

)

((let ( ( ) …) …)

;=*

((letrec ( (

(lambda ( …) …))

)

)

…)

)

) )

Form: (LET* ( ( ) …) …)

Example:

(define A 37)

(let ( (a 2) (b (+ a 5)) ) (* a b)) ;=* 14

Semantics: Like un-named LET except that the expressions

are evaluated sequentially and each “sees” the value of the

previous name bindings.

(define-syntax LET*

(syntax-rules ()

((let* () …) ;=*

(let () …)

)

((let* ( (var1* ) ( ) … )

…)

;=*

(let ( ( ) )

(let* ( ( ) … )

…))

)

) )

Form: (LETREC ( ( ) …) …)

Example:

(letrec ( (EVEN?

(lambda (n)

(if (zero? n) #t (odd? (sub1 n)))))

(ODD?

(lambda (n)

(if (zero? n) #f (even? (sub1 n)))))

)

(even? 14)) ;;=* #t

Semantics: Mutually recursive form of let. All name bindings are

visible to all s. Because the order of evaluation of the

s is unspecified, it must be possible to evaluate each init

without refering to the *value* of any . Note that if the

values are all lambda expressions, this condition is always

satisfied.

(define-syntax LETREC

(syntax-rules ()

((letrec ( ( ) … )

…)

;=*

(let ( ( #f) … )

(set! ) …

…)

))

) )

Forms: (cond ( …) … )

(cond ( …) … (else …))

Examples: (cond ((* x 0) ‘negative)

((* x 0) ‘positive)

(else ‘neither))

(cond ((assq key alist) =* cdr)

(else search-failure-value))

Semantics: Each expression is evaluated in turn. The

first which evaluates to true causes the s to be

evaluated. The value of the last is returned in this

case, or the value of if there are no s. If no

is true, the s of the else clause are evaluated and

the value of the last is the value of the COND

expression. If not is true and there is no else clause,

the result is unspecified. If a is true and followed by

‘=* then the following must evaluate to a procedure of one

argument and the procedure is called with the value of the

expression as an argument.

(define-syntax COND

(syntax-rules ( else =* ) ;; ‘else and ‘=* must match exactly

((cond) ;=*

#f

)

((cond (else …))

(begin …)

)

((cond ( =* ) …)

;=*

(let ( (result ) )

(if result

( result)

(cond …))

))

((cond () …)

(or (cond …))

)

((cond ( …) …)

(if (begin …) (cond …))

)

) )

Form: (case (( …) …) …

(else …))

Example: (case (peak-char (current-input-port))

((# #\?) (print-help-text))

((#\space #\newline) (keep-looking))

(else (read-command)))

Semantics: The expression is evaluated and compared in turn

to each . If it is equivalent (eqv?), then the s of

the first clause containing the datum are evaluated and the value

of the last one is returned as the value of the CASE expression.

If no match is found, then the else expression(s) are evaluated

and the last value returned, otherwise the value of the CASE is

unspecified.

(define-syntax CASE

(syntax-rules ( else )

((case )

)

((case (else …))

(begin …)

)

((case (( …) …) …)

;=*

(let ( (key ) )

(if (memv key ‘( …))

(begin …)

(case key …))

))

) )

PROBLEMS WHICH REMAIN:

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