rec
for recursive evaluationby Mirko Luedde
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rec
. This form is a generalization and
combination of the forms rec
and
named-lambda
of [Clinger1985]. It allows the simple and
non-imperative construction of self-referential expressions. As an
important special case, it extends the A. Church form
lambda
such that it allows the direct definition of
recursive procedures without using further special forms like
let
or letrec
, without using advanced
constructions like the H. B. Curry combinator and, unlike
define
, without introducing variable bindings into the
external environment.
Let us look at the factorial function. In mathematical notation this function is expressed as
(F : N |--> 1, if N = 0; N * F(N - 1), otherwise).
This expression is a term and not a definition or proposition.
We investigate some approaches to express the factorial function in Scheme.
The simplest way perhaps is as
(define (F N) (if (zero? N) 1 (* N (F (- N 1)))))
But this expression is not a term. It binds the factorial function to
the variable F
. The expression itself may not occur in a
syntactical context where a name of the factorial is required.
We list several ways to express the factorial as a function term.
(let () (define (F N) (if (zero? N) 1 (* N (F (- N 1))))) F)
(lambda (N) (let F ( (N N) ) (if (zero? N) 1 (* N (F (- N 1))))))
(letrec ( (F (lambda (N) (if (zero? N) 1 (* N (F (- N 1)))))) ) F)
((lambda (F) (F F)) (lambda (G) (lambda (N) (if (zero? N) 1 (* N ((G G) (- N 1)))))))
All these expressions define the factorial anonymously, not binding it to a variable. However, all these expressions are more verbose than it seems necessary and they are less intuitive than it seems desirable.
A solution to our problem was already provided in [Clinger1985] by the form
named-lambda
. An even earlier solution with a slightly
different syntax was implemented in Kent Dybvig's Chez Scheme system.
Using this special form, we can denote the factorial simply by
(named-lambda (F N) (if (zero? N) 1 (* N (F (- N 1)))))
This expression is a function term that denotes the factorial in the appropriate brevity.
However, the form named-lambda
has been dropped from
later versions of the Scheme Report. Also it is missing in
state-of-the-art implementations such as Chez Scheme (6.0a) and MIT
Scheme (7.7.0). (The latter actually knows a form
named-lambda
with different semantics).
(define S (cons 1 (delay S)))As in the case of the factorial, we are able to define the recursive object at the price of spending an externally bound name. Remedying this with
let
or letrec
leads to similar
objections as above.
This particular case of the self-referential problem was solved by the
rec
form in [Clinger1985].
This form allows writing
(rec S (cons 1 (delay S)))
This expression is non-imperative and does not introduce an external variable binding.
Also this form has been dropped from later versions of the Scheme Report. Moreover, from our point of view this form alone is not capable of solving Problem 1. The respective definition would look like
(rec F (lambda (N) (if (zero? N) 1 (* N (F (- N 1))))))
This again does not seem quite as simple and intuitive as the mathematical notation.
rec
special form in
a generalized way that combines the advantages of the
named-lambda
and rec
forms.
The factorial function could be written
(rec (F N) (if (zero? N) 1 (* N (F (- N 1)))))
<derived expression> --> <rec expression>
<rec expression> --> (rec <variable>
<expression>)
<rec expression> --> (rec (<variable>+)
<body>)
define-syntax
, syntax-rules
,
letrec
and lambda
might implement
rec
as follows.
(define-syntax rec (syntax-rules () ((rec (NAME . VARIABLES) . BODY) (letrec ( (NAME (lambda VARIABLES . BODY)) ) NAME)) ((rec NAME EXPRESSION) (letrec ( (NAME EXPRESSION) ) NAME))))
rec
allows a
tail-recursive implementation of the factorial function.
> (define F (rec (F N) ((rec (G K L) (if (zero? K) L (G (- K 1) (* K L)))) N 1))) > F #<procedure> > (F 0) 1 > (F 10) 3628800
Copyright (C) Dr. Mirko Luedde (2002). All Rights Reserved.
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