# 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.
1. It has simple syntax.
2. It encourages recursive definitions, e.g. by ensuring memory efficient tail recursion.
3. It supports non-imperative programming.
Nevertheless Scheme does not provide a syntax for recursive evaluations with the properties of
1. being as simple, intuitive and close to the mathematical standard notation as possible,
2. allowing general recursion,
3. 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.

1. ```(let ()
(define (F N)
(if (zero? N) 1
(* N (F (- N 1)))))
F)
```
2. ```(lambda (N)
(let F ( (N N) )
(if (zero? N) 1
(* N (F (- N 1))))))
```
3. ```(letrec ( (F (lambda (N)
(if (zero? N) 1
(* N (F (- N 1)))))) )	F)
```
4. ```((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).
1. `<derived expression> --> <rec expression>`
2. ```<rec expression> --> (rec <variable> <expression>)```
3. ```<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}