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<!DOCTYPE html>
<html lang="en">
<!--
SPDX-FileCopyrightText: 2023 Marc Nieper-Wißkirchen
SPDX-License-Identifier: MIT
-->
<head>
<meta charset="utf-8">
<title>SRFI 247: Syntactic Monads</title>
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<h1><a href="https://srfi.schemers.org/"><img class="srfi-logo"
src="https://srfi.schemers.org/srfi-logo.svg" alt="SRFI surfboard
logo" /></a>247: Syntactic Monads</h1>
<p>by Marc Nieper-Wißkirchen</p>
<h2 id="status">Status</h2>
<p>This SRFI is currently in <em>final</em> status. Here is <a href="https://srfi.schemers.org/srfi-process.html">an explanation</a> of each status that a SRFI can hold. To provide input on this SRFI, please send email to <code><a href="mailto:srfi+minus+247+at+srfi+dotschemers+dot+org">srfi-247@<span class="antispam">nospam</span>srfi.schemers.org</a></code>. To subscribe to the list, follow <a href="https://srfi.schemers.org/srfi-list-subscribe.html">these instructions</a>. You can access previous messages via the mailing list <a href="https://srfi-email.schemers.org/srfi-247/">archive</a>.</p>
<ul>
<li>Received: 2023-10-08</li>
<li>Draft #1 published: 2023-10-12</li>
<li>Draft #2 published: 2023-12-13</li>
<li>Finalized: 2023-12-24</li>
<li>Revised to fix errata:
<ul>
<li>2024-02-11 (Fixed error (wrong meta-variable name) in
description of <a href="#let-loop">let loops</a>
in <code>define-syntactic-monad</code>.)</li></ul></li></ul>
<h2 id="abstract">Abstract</h2>
<p>This SRFI extends Scheme with a simple mechanism to implicitly add
formal arguments to procedure definitions and to implicitly add
arguments to procedure calls. Contrary to parameters (also known as fluids
or dynamically bound variables), which can be used for the same
purpose, no runtime overhead is generated.</p>
<h2 id="rationale">Rationale</h2>
<p>Thanks to its proper tail calls, Scheme is an excellent
programming language for describing state machines. In a finite
state machine, each state is modelled by a procedure and each
state transition by a tail call to one of these procedures. In
more general state machines, states are parameterized by values
of a list of state variables. Each procedure accepts the
current values of the state variables as arguments and each tail
call is done with possibly updated values of the state variables.</p>
<p>Consider the following example of an interpreter for a simple
programming language dealing with three stacks:</p>
<pre class=example>(define run
(lambda (prog a b c)
(define a->b
(lambda (prog a b c)
(run prog (cdr a) (cons (car a) b) c)))
(define a->c
(lambda (prog a b c)
(run prog (cdr a) b (cons (car a) c))))
(define b->a
(lambda (prog a b c)
(run prog (cons (car b) a) (cdr b) c)))
(define b->c
(lambda (prog a b c)
(run prog a (cdr b) (cons (car b) c))))
(define c->a
(lambda (prog a b c)
(run prog (cons (car c) a) b (cdr c))))
(define c->b
(lambda (prog a b c)
(run prog a (cons (car c) b) (cdr c))))
(if (null? prog)
(values a b c)
(let ([ins (car prog)] [prog (cdr prog)])
(case ins
[(a->b) (a->b prog a b c)]
[(a->c) (a->c prog a b c)]
[(b->a) (b->a prog a b c)]
[(b->c) (b->c prog a b c)]
[(c->a) (c->a prog a b c)]
[(c->b) (c->b prog a b c)])))))</pre>
<p>In this example, the state variables
are <code><var>prog</var></code>, <code><var>a</var></code>, <code><var>b</var></code>,
and <code><var>c</var></code>, which are threaded through the
code. While the semantics of the code are very clear,
syntactically it exhibits two problems: Firstly, it doesn't
scale when we extend the interpreter with more registers, that
is state variables because we have to change every procedure
definition and every procedure call. Secondly, at each
transition step only a subset of the state variables are
updated, yet we have to list all state variables at each procedure call.</p>
<p>Using the mechanism provided by this SRFI, we can rewrite the
interpreter with the following code, which does not exhibit the
two problems. At runtime, the code behaves exactly as the
previous code. By convention, the names for syntactic monads
begin with the character <code>$</code>. (Locally defined
syntactic monads are often just named <code>$</code>.)</p>
<pre class=example>(define-syntactic-monad $ prog a b c)
(define run
($ lambda ()
($ define (a->b)
($ run ([a (cdr a)] [b (cons (car a) b)])))
($ define (a->c)
($ run ([a (cdr a)] [c (cons (car a) c)])))
($ define (b->a)
($ run ([a (cons (car b) a)] [b (cdr b)])))
($ define (b->c)
($ run ([b (cdr b)] [c (cons (car b) c)])))
($ define (c->a)
($ run ([a (cons (car c) a)] [c (cdr c)])))
($ define (c->b)
($ run ([b (cons (car c) b)] [c (cdr c)])))
(if (null? prog)
(values a b c)
(let ([ins (car prog)] [prog (cdr prog)])
(case ins
[(a->b) ($ a->b)]
[(a->c) ($ a->c)]
[(b->a) ($ b->a)]
[(b->c) ($ b->c)]
[(c->a) ($ c->a)]
[(c->b) ($ c->b)])))))</pre>
<p>Iterative and recursive loops are a particular case of state
machines, which can also benefit from defining a syntactic monad.
As examples, we give an implementation of the <code>partition</code> procedure found in <a href="https://srfi.schemers.org/srfi-1/">SRFI 1</a> and R<sup>6</sup>RS and of the <code>factor</code> procedure found in Kent Dybvig's <i>The Scheme Programming Language</i>.</p>
<pre class=example>(define partition
(lambda (pred ls)
(define-syntactic-monad $ in out)
(let f ([ls ls])
(if (null? ls)
($ values ([in '()] [out '()]))
($ let*-values ([() (f (cdr ls))])
(let ([x (car ls)])
(if (pred x)
($ values ([in (cons x in)]))
($ values ([out (cons x out)])))))))))
(partition symbol? '(one 2 3 four five 6)) ; => (one four five) (2 3 6)</pre>
<pre class=example>(define factor
(lambda (n)
(define-syntactic-monad $ n i)
($ let f ([i 2])
(cond
[(>= i n) (list n)]
[(integer? (/ n i))
(cons i ($ f [(n (/ n i))]))]
[else ($ f [(i (+ i 1))])]))))
(factor 3628800) ; => (2 2 2 2 2 2 2 2 3 3 3 3 5 5 7)</pre>
<h2 id="specification">Specification</h2>
<p>The following definition is exported by the <code>(srfi
:247)</code> and <code>(srfi :247 syntactic-monads)</code>
libraries.</p>
<dl class=entries>
<dt class=syntax><code>(define-syntactic-monad <span class=token>syntactic
monad name</span>
<span class=token>formal</span> <span class=ellipsis>…</span>)</code></dt>
<dd>
<p>The <code>define-syntactic-monad</code> form defines a
syntactic monad.</p>
<p>A <code>define-syntactic-monad</code> form is a definition and
can appear anywhere any
other <code><span class=token>definition</span></code> can
appear.</p>
<p>
<code><span class=token>Syntactic monad name</span></code> and
the <code><span class=token>formals</span></code> must all be
identifiers.
</p>
<p><code><span class=token>Syntactic monad name</span></code> is
bound as a syntactic keyword.
The <code><span class=token>formals</span></code>, taken as
symbols, become the <dfn>state variable names</dfn> of the
syntactic monad defined. At use sites
of <code><span class=token>syntactic monad name</span></code>,
each state variable name defines a corresponding <dfn>state
variable</dfn>, which is an implicit identifier with the state
variable name as its symbolic name and which contains the same
contextual information as the
keyword <code><span class=token>syntactic monad
name</span></code> in the macro use. An identifier is
the <dfn>same</dfn> as a state variable if binding the
identifier would shadow free references to the state variable.
The list of state variables (in the order of the state variable
names) at use sites of <code><span class=token>syntactic monad
name</span></code> is denoted by <code><span class=token>state
variable</span> <span class=ellipsis>…</span></code>.
</p>
<div class="small">
<p>Example:</p>
<pre class=example>(define-syntactic-monad $ a b)</pre>
</div>
<p>The keyword <code><span class="token">syntactic monad
name</span></code> defined by the <code>define-syntactic-monad</code> definition can be used in
the following ways (every other use is a syntax violation):</p>
<ul><li>
<p><code>(<span class="token">syntactic monad
name</span> lambda <span class=token>formals</span>
<span class=token>body</span>)</code> is equivalent
to <code>(lambda
(<span class=token>state
variable</span> <span class=ellipsis>…</span>
. <span class=token>formals</span>) <span class=token>body</span>)</code>.</p>
<div class="small">
<p>Example:</p>
<pre class=example>(($ lambda (c) (list a b c)) 1 2 3) ; => (1 2 3)</pre>
</div>
</li>
<li>
<p><code>(<span class="token">syntactic monad name</span> define
(<span class=token>name</span>
. <span class=token>formals</span>) <span class=token>body</span>)</code>
is equivalent to <code>(define <span class=token>name</span>
(<span class=token>syntactic monad
name</span> lambda <span class=token>formals</span> <span class=token>body</span>)</code>.</p>
<div class="small">
<p>Example:</p>
<pre class=example>($ define (f c d) (list a b c d))
(f 1 2 3 4) ; => (1 2 3 4)</pre>
</div>
</li>
<li>
<p><code>(<span class="token">syntactic monad name</span>
case-lambda
[<span class="token">formals</span> <span class="token">body</span>] <span class=ellipsis>…</span>)</code>
is equivalent to <code>(case-lambda [(<span class=token>state
variable</span> <span class=ellipsis>…</span>
. <span class="token">formals</span>) <span class="token">body</span>] <span class=ellipsis>…</span>)</code></p>
<div class="small">
<p>Example:</p>
<pre class=example>(($ case-lambda [() (list a b)]) 1 2) ; => (1 2)</pre>
</div>
</li>
<li>
<p><code>(<span class="token">syntactic monad name</span> let*-values
([<span class="token">formals</span> <span class=token>init</span>] <span class=ellipsis>…</span>) <span class="token">body</span>)</code>
is equivalent to <code>(let*-values ([(<span class=token>state
variable</span> <span class=ellipsis>…</span>
. <span class="token">formals</span>) <span class=token>init</span>] <span class=ellipsis>…</span>) <span class="token">body</span>)</code></p>
<div class="small">
<p>Example:</p>
<pre class=example>($ let*-values ([(c) (values 1 2 3)]
[() (values a (+ b c))])
(list a b)) ; => (1 5)</pre>
</div>
</li>
<li>
<p><code>(<span class="token">syntactic monad
name</span> <span class=token>procedure expression</span>
(<span class=token>binding</span>
<span class=ellipsis>…</span>) <span class=token>argument
expression</span> <span class=ellipsis>…</span>)</code>
is equivalent to a procedure call of the
form <code>(<span class=token>procedure expression</span>
<span class=token>state variable
expression</span> <span class=ellipsis>…</span> <span class=token>argument
expression</span>
<span class=ellipsis>…</span>)</code>. Here,
each <code><span class=token>binding</span></code> is of the
form <code>[<span class=token>variable</span> <span class=token>expression</span>]</code>
where each <code><span class=token>variable</span></code> must
be the same as one of the state variables. It is also a syntax
violation if one of the <code><span class=token>variables</span></code>
appears more than once amongst
the <code><span class=token>variables</span></code>.
The <code><span class="token">state variable
expressions</span></code> are a list of expressions
corresponding to the state variables. For
each <code><span class="token">variable</span></code>, which is
a state variable, the
corresponding <code><span class="token">state variable
expression</span></code> is
the <code><span class="token">expression</span></code>
corresponding to
the <code><span class="token">variable</span></code>. For the
state variables that do not appear amongst
the <code><span class="token">variables</span></code>, the
corresponding <code><span class="token">state variable
expression</span></code> is a variable reference to the same
state variable.</p>
<div class="small"><p>Example:</p>
<pre class=example>(let ([a 1])
($ list ([b 2]) 4)) ; => (1 2 4)</pre>
</div>
</li>
<li>
<p><code>(<span class="token">syntactic monad
name</span> <span class=token>procedure expression</span>)</code>
is equivalent to
<code>(<span class=token>syntactic monad
name</span> <span class=token>procedure expression</span>
())</code>.</p>
<div class="small"><p>Example:</p>
<pre class=example>(let ([a 1] [b 6])
($ list)) ; => (1 6)</pre>
</div>
</li>
<li id="let-loop">
<p><code>(<span class="token">syntactic monad name</span>
let <span class="token">loop variable</span>
([<span class=token>variable</span> <span class=token>init</span>] <span class=ellipsis>…</span>) <span class=token>body</span>)</code>
is equivalent to
<code>(let <span class="token">loop variable</span> ([<span class="token">state variable</span> <span class="token">state variable init</span>]
<span class="ellipsis">…</span> [<span class="token">other variable</span> <span class="token">other variable init</span>] <span class="ellipsis">…</span>)
<span class="token">body</span>)</code>. Here, for
each <code><span class="token">state variable</span></code>
appearing among
the <code><span class="token">variables</span></code>, the
corresponding <code><span class="token">state variable
init</span></code> is the
corresponding <code><span class="token">init</span></code>. For each <code><span class="token">state
variable</span></code> not appearing among
the <code><span class="token">variables</span></code>, the
corresponding <code><span class="token">state variable
init</span></code> is a variable reference to the same state
variable. The <code><span class="token">other
variables</span></code> are
the <code><span class="token">variables</span></code> that are
not state variables in that order.
The <code><span class="token">other inits</span></code> are the
corresponding <code><span class="token">inits</span></code>.</p>
<div class="small"><p>Example:</p>
<pre class=example>(let ([a 1])
($ let loop ([c 3] [b 2])
(if (= a 2)
(list a b c)
(loop 2 (+ b 6) (+ c 7))))) ; => (2 8 10)</pre>
</div>
</li>
</ul>
<p><i>Note:</i> In <code>(define-syntactic-monad <span class=token>syntactic-monad-name</span> <span class=token>formal</span> <span class=ellipsis>…</span>)</code>,
only the names of
the <code><span class=token>formals</span></code> are
significant.
</p>
</dd>
</dl>
<h2 id="implementation">Implementation</h2>
<p>The sample implementation is written in portable
R<sup>6</sup>RS scheme. It can easily be ported to other Scheme
systems supporting the <code>syntax-case</code> system.</p>
<a href="lib/srfi/:247/syntactic-monads.sls">Source for the sample
implementation.</a>
<h2 id="acknowledgements">Acknowledgements</h2>
<p>This SRFI was inspired from and would not exist without the
prior appearance of a form
of <code>define-syntactic-monad</code> in the source code of
Chez Scheme.</p>
<h2 id="copyright">Copyright</h2>
<p>© 2023 Marc Nieper-Wißkirchen.</p>
<p>
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:</p>
<p>
The above copyright notice and this permission notice (including
the next paragraph) shall be included in all copies or
substantial portions of the Software.</p>
<p>
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.</p>
<hr>
<address>Editor: <a href="mailto:srfi-editors+at+srfi+dot+schemers+dot+org">Arthur
A. Gleckler</a></address></body></html>