31: A special form `rec` for recursive evaluation

by Mirko Luedde

Status

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Draft: 2002-05-24--2002-08-24
Revised: 2002-08-12
Final: 2002-12-02

Abstract

We propose the implementation of a special form called 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.

Rationale

General

Among the prominent features of the Scheme programming language as defined in [KCR1998] are the following.

It has simple syntax.
It encourages recursive definitions, e.g. by ensuring memory efficient tail recursion.
It supports non-imperative programming.

Nevertheless Scheme does not provide a syntax for recursive evaluations with the properties of

being as simple, intuitive and close to the mathematical standard notation as possible,
allowing general recursion,
being non-imperative.

Example

Problem 1

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.

Solution 1

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).

Problem 2

The constant stream of ones can be defined via

(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.

Solution 2

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.

Proposal

We therefore propose to implement the 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)))))

Specification

Syntax

The following production rules are to be added to those of [KCR1998] (we reuse names of non-terminals).

<derived expression> --> <rec expression>
<rec expression> --> (rec <variable> <expression>)
<rec expression> --> (rec (<variable>+) <body>)

Semantics

Scheme versions such as [KCR1998] providing 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))))

Test

The following session shows in which way 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

Acknowledgements

The author thanks Al Petrofsky for the final solution and Hal Abelson, Chris Hanson and others for their input. The work of the maintainers of the SRFI forum is highly appreciated.

Bibliography

[Clinger1985] EDITOR = {Clinger, W.}, TITLE = {Draft of Report of the October 1984 Workshop on Scheme}, JOURNAL = {Proceedings Brandeis Workshop Oct.~22--23, 1984}, MONTH = {Mar}, YEAR = {1985}, URL = {http://www.swiss.ai.mit.edu/ftpdir/scheme-mail/HTML/rrrs-1985}
[KCR1998] AUTHOR = {Kelsey, R. and Clinger, W. and Rees, J.}, TITLE = {Revised (5) Report on the Algorithmic Language Scheme}, JOURNAL = {Higher-Order and Symbolic Computation}, VOLUME = {11}, NUMBER = {1}, MONTH = {Sep}, YEAR = {1998}

Copyright

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Author: Mirko Luedde

Editor: Francisco Solsona

31: A special form rec for recursive evaluation