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 h
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: