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<!--
SPDX-FileCopyrightText: 2012 Alan Manuel K. Gloria
SPDX-FileCopyrightText: 2012 David A. Wheeler
SPDX-License-Identifier: MIT
-->
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN">
<html>
<head>
<title>SRFI 105: Curly-infix-expressions</title>
<meta content="text/html; charset=us-ascii" http-equiv="content-type">
</head>
<body>
<!-- This commented out text is for the brittle SRFI tools -->
<!--
<H1>Title</H1>
Curly-infix-expressions
<H1>Author</H1>
David A. Wheeler, Alan Manuel K. Gloria
<H1>Status</H1>
This SRFI is currently in ``draft'' status.
-->
<h1 id="title">Title</h1>
<p>Curly-infix-expressions</p>
<h1 id="author" id="authors">Authors</h1>
<p><a href="http://www.dwheeler.com">David A. Wheeler</a></p>
<p>Alan Manuel K. Gloria</p>
<h1 id="status">Status</h1>
<p>
This SRFI is currently in ``draft'' status. To see an explanation of
each status that a SRFI can hold, see <a
href="http://srfi.schemers.org/srfi-process.html">here</a>.
To provide input on this SRFI, please
<a href="mailto:srfi minus 105 at srfi dot schemers dot org">mail to
<code><srfi minus 105 at srfi dot schemers dot org></code></a>. See
<a href="../../srfi-list-subscribe.html">instructions here</a> to
subscribe to the list. You can access previous messages via
<a href="mail-archive/maillist.html">the archive of the mailing list</a>.
</p>
<ul>
<li>
Received: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.2.html">2012/08/22</a></li>
<li>Draft: 2012/08/22-2012/10/22</li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.3.html">2012/09/03</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.4.html">2012/09/08</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.5.html">2012/09/17</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.6.html">2012/09/19</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.7.html">2012/09/27</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.8.html">2012/10/02</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.9.html">2012/10/15</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.10.html">2012/10/16</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.11.html">2012/10/20</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.12.html">2012/10/21</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.13.html">2012/10/25</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.14.html">2012/10/27</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.15.html">2012/10/30</a></li>
<li>Revision: <a href="http://srfi.schemers.org/srfi-105/srfi-105-1.16.html">2012/10/31</a></li>
</ul>
<h1><a name="related-srfis">Related SRFIs</a></h1>
<p>None</p>
<h1><a name="abstract">Abstract</a></h1>
<p>Lisp-based languages, like Scheme, are almost the only
programming languages in modern use that do not support infix notation.
In addition, most languages allow infix expressions to be combined
with function call notation of the form f(x).
This SRFI provides these capabilities, both for
developers who already use Scheme and want these conveniences,
and also for other developers who may choose to use other languages
in part because they miss these conveniences.
Scheme currently reserves <code>{</code>...<code>}</code>
“for possible future extensions to the language”.
We propose that <code>{</code>...<code>}</code> be used
to support “curly-infix” notation as a homoiconic
infix abbreviation,
as a modification of the Scheme reader.
It is an abbreviation in much the same way that
<samp>'x</samp> is an abbreviation for <samp>(quote x)</samp>.
</p>
<p>
A
<a href="#specification"><dfn>curly-infix list</dfn></a>
introduces a list whose visual presentation
is in infix order instead of prefix order.
For example,
<samp>{n > 5}</samp> ⇒ <samp>(> n 5)</samp>,
and
<samp>{a + b + c}</samp> ⇒
<samp>(+ a b c)</samp>.
By intent, there is no precedence, but e.g.,
<samp>{x + {y * z}}</samp>
maps cleanly to
<samp>(+ x (* y z))</samp>.
Forms with mixed infix operators and other complications have
“<var>$nfx$</var>” prepended to
enable later macro processing, e.g.,
<samp>{4 + 5 * 6}</samp> ⇒
<samp>($nfx$ 4 + 5 * 6)</samp>.
Also, inside a curly-infix list (recursively),
expressions of the form <samp>f(...)</samp> are simply
an abbreviation for <samp>(f ...)</samp>.</p>
<p>Note that this is derived from the
“<a href="http://readable.sourceforge.net/">readable</a>”
project.
We intend to later submit at least one additional SRFI that will build on
top of this SRFI, but curly-infix-expressions are useful on their own.
</p>
<h1><a name="rationale">Rationale</a></h1>
<p>Lisp-based languages, like Scheme, are almost the only
programming languages in modern use that do not support infix
notation. Even some Lisp advocates, like Paul Graham, admit that
they “don’t find prefix math expressions
natural” (<a href="http://www.paulgraham.com/popular.html"
rel="nofollow">http://www.paulgraham.com/popular.html</a>) even after
decades of experience with Lisp-based languages.
Paul Prescod has said, “I have more faith that you could convince
the world to use Esperanto than prefix notation”
(<a href="http://people.csail.mit.edu/gregs/ll1-discuss-archive-html/msg01571.html" rel="nofollow">http://people.csail.mit.edu/gregs/ll1-discuss-archive-html/msg01571.html</a>).
Infix is not going away; standard mathematical notation uses infix,
infix notation is taught to most people (programmers or not) in school,
and nearly all new programming languages include infix.
</p>
<p>
Adding infix support to Scheme would be a useful convenience for
some existing developers who use Scheme, and it would also
eliminate a common complaint
by developers who currently choose to use other languages instead.
</p>
<p>Scheme currently reserves
<code>{</code>...<code>}</code> “for possible future
extensions to the language”. We propose that
<code>{</code>...<code>}</code> be used to
support “curly-infix” notation as a reader
abbreviation, just as
<samp>'x</samp> is an abbreviation for <samp>(quote x)</samp> and
<samp>(x y z)</samp>
is an abbreviation for
<samp>(x . (y . (z . ())))</samp>.
</p>
<p>This proposal is an extremely <em>simple</em> and
<em>straightforward</em> technique for supporting infix notation.
There is no complex precedence system, all other Scheme
capabilities (including macros and quasiquoting) work unchanged,
any symbol can be used as an infix operation where desired,
and Scheme remains general and homoiconic.
Curly-infix-expressions (also known as c-expressions)
are just a convenient reader abbreviation for infix notation.</p>
<p>At its core, this SRFI provides
the <em>simple</em> curly-infix list,
a list whose visual presentation is in infix order instead of prefix order.
The simple curly-infix list
<samp>{operand-1 operator operand-2 operator operand-3 operator ...}</samp>
is mapped to
<samp>(operator operand-1 operand-2 operand-3 ...)</samp> so that two or
more operands are handled cleanly.
E.g., <samp>{a + b + c}</samp> ⇒
<samp>(+ a b c)</samp>.</p>
<p>
More examples of c-expressions and their mappings are given below in the
<a href="#specification">specification</a>.
See the <a href="#design-rationale">design rationale</a>
for details on why the notation is designed the way it is.
</p>
<h1><a name="specification">Specification</a></h1>
<p>
The key words
“<em>MUST</em>”,
“<em>MUST NOT</em>”,
“<em>REQUIRED</em>”,
“<em>SHALL</em>”,
“<em>SHALL NOT</em>”,
“<em>SHOULD</em>”,
“<em>SHOULD NOT</em>”,
“<em>RECOMMENDED</em>”,
“<em>MAY</em>”,
and “<em>OPTIONAL</em>” in this
document are to be interpreted as described in
<a href="http://www.ietf.org/rfc/rfc2119.txt">RFC 2119</a>.
</p>
<p>“<dfn>Curly-infix-expressions</dfn>” (aka “<dfn>c-expressions</dfn>”) are
s-expressions with an additional supported notation:
The curly-infix list.
A curly-infix list is syntactically almost identical to a regular list,
but it is surrounded by a matched pair of braces instead of by a pair of parentheses, and instead of
a sequence of s-expressions it contains a sequence of
neoteric-expressions (which add support for formats like <samp>f(x)</samp>).
Once a curly-infix list is read, it is mapped differently than a
regular list by a curly-infix reader:</p>
<ol>
<li>A <dfn>simple</dfn> curly-infix list has an odd number of
parameters, at least three parameters, and all even parameters are
“<var>equal?</var>”.
If there is more than one even parameter, and
an even parameter contains a cycle, then
the <var>equal?</var> comparison <em>MUST</em> terminate
if <var>equal?</var> terminates (otherwise
the comparison <em>MAY</em> terminate).
A simple curly-infix list is mapped by
the reader into a list with the first even parameter
followed by the odd parameters.
E.g.,
<samp>{n <= 5}</samp> ⇒
<samp>(<= n 5)</samp>,
and
<samp>{4 * 5 * 6}</samp> ⇒
<samp>(* 4 5 6)</samp>.</li>
<li>The <dfn>empty</dfn> curly-infix list
<samp>{}</samp> is mapped to the empty list <samp>()</samp>.
An implementation <em>MUST</em> permit, and not require,
whitespace between the braces in an empty curly-infix list.</li>
<li>An <dfn>escaping</dfn> curly-infix list
<samp>{e}</samp> is mapped to <samp>e</samp>.
E.g., <samp>{5}</samp> is mapped to <samp>5</samp>.</li>
<li>A <dfn>unary-operation</dfn> curly-infix list
<samp>{e1 e2}</samp>
is mapped to
<samp>(e1 e2)</samp>.
E.g.,
<samp>{- x}</samp> ⇒
<samp>(- x)</samp>.</li>
<li>The mapping of a curly-infix list beginning with the symbol
“<var>.</var>” is unspecified.
(Note: the reference implementation maps
<samp>{. e}</samp> to <samp>e</samp>.)
</li>
<li>Any other curly-infix list (including all other improper lists) is
<dfn>mixed</dfn>. A mixed curly-infix list <em>MUST</em> be mapped to that list with “<var>$nfx$</var>” added to its front.
E.g.,
<samp>{q + r * s}</samp>
is mapped to
<samp>($nfx$ q + r * s)</samp>, and
<samp>{q + r . s}</samp>
is mapped to
<samp>($nfx$ q + r . s)</samp>.
</li>
</ol>
<p>
Here is the syntax of a curly-infix list
(which is nearly identical to a traditional list):
</p>
<!--
Note: CSS "style" is not included in HTML 3.2, so we'll use
a table to force a reasonable format.
-->
<table>
<tr valign="top">
<td>curly‑infix‑list →</td>
<td>“<tt>{</tt>”
<whitespace>*
[ <n-expression>
[ <whitespace>+ <n-expression> ]*
[ <whitespace>+ <b>.</b>
<whitespace>+ <n-expression> ]
<whitespace>* ]
“<tt>}</tt>”
</td>
</tr>
</table>
<p>
A “<dfn>neoteric-expression</dfn>” (aka “<dfn>n-expression</dfn>”)
is a curly-infix-expression, with the following additional syntaxes
and mappings
for a datum (where <var>e</var> is any datum expression):
</p>
<ol>
<li>
<samp>e(...)</samp> ⇒
<samp>(e ...)</samp>.
E.g.,
<samp>cos(x)</samp> ⇒
<samp>(cos x)</samp>,
<samp>f(a b)</samp> ⇒
<samp>(f a b)</samp>,
<samp>exit()</samp> ⇒
<samp>(exit)</samp>, and
<samp>read(. options)</samp> ⇒
<samp>(read . options)</samp>.</li>
<li>
<samp>e{}</samp> ⇒ <samp>(e)</samp>
when there are zero or more whitespace characters
within the braces;
otherwise,
<samp>e{...}</samp> ⇒ <samp>(e {...})</samp>.
E.g.,
<samp>f{n - 1}</samp>
⇒
<samp>(f {n - 1})</samp>
⇒
<samp>(f (- n 1))</samp>,
and
<samp>g{- x}</samp>
⇒
<samp>(g (- x))</samp>.</li>
<li><samp>e[...]</samp> ⇒
<samp>($bracket-apply$ e ...)</samp></li>
<li>The above mappings <em>MUST NOT</em> be applied
if one or more whitespace characters are present between e and the open
paired character.</li>
<li>An unprefixed <samp>( . e)</samp> <em>MUST</em> evaluate as <samp>e</samp>.</li>
<li>
These <em>MUST</em> recurse within lists and vectors,
so any list or vector
in a position that accepts a neoteric expression
<em>MUST</em> accept a sequence
of zero or more <em>neoteric</em> expressions,
not just s-expressions.
(Note that this occurs if they are directly or indirectly
within a curly-infix list or a neoteric-expression.)
Any other implementation-specific constructs
in a position that accepts a neoteric expression
<em>SHOULD</em> accept neoteric-expressions within it.
</li>
<li>
These <em>MUST</em> recurse left-to-right.
E.g.,
<samp>f{n - 1}(x)</samp>
⇒
<samp>(f {n - 1})(x)</samp>
⇒
<samp>(f (- n 1))(x)</samp>
⇒
<samp>((f (- n 1)) x)</samp>
</li>
</ol>
<p>
Where datum comments are supported using <samp>#;</samp>,
datum comments <em>SHOULD</em> comment the datum as defined above.
(Datum comments are defined in SRFI-62; they are also included in
R6RS and R7RS draft 6.)
Note that any s-expression is also an n-expression,
because n-expressions include c-expressions
and c-expressions include s-expressions.
</p>
<p>Here are some examples of c-expressions (note that all operators in curly-infix are delimited):</p>
<ol>
<li><samp>{n <= 5}</samp> ⇒
<samp>(<= n 5)</samp></li>
<li><samp>{x + 1}</samp>
⇒
<samp>(+ x 1)</samp></li>
<li><samp>{a + b + c}</samp>
⇒
<samp>(+ a b c)</samp></li>
<li><samp>{x ,op y ,op z}</samp> ⇒
<samp>(,op x y z)</samp></li>
<li><samp>{x eqv? `a}</samp> ⇒ <samp>(eqv? x `a)</samp></li>
<li><samp>{'a eq? b}</samp> ⇒ <samp>(eq? 'a b)</samp></li>
<li>
<samp>{n-1 + n-2}</samp>
⇒
<samp>(+ n-1 n-2)</samp>
</li>
<li><samp>{a * {b + c}}</samp> ⇒ <samp>(* a (+ b c))</samp></li>
<li>
<samp>{a + {b - c}}</samp>
⇒
<samp>(+ a (- b c))</samp>
</li>
<li>
<samp>{{a + b} - c}</samp>
⇒
<samp>(- (+ a b) c)</samp>
</li>
<li><samp>{{a > 0} and {b >= 1}}</samp> ⇒
<samp>(and (> a 0) (>= b 1))</samp></li>
<li>
<samp>{}</samp>
⇒
<samp>()</samp>
</li>
<li>
<samp>{5}</samp>
⇒
<samp>5</samp>
</li>
<li>
<samp>{- x}</samp>
⇒
<samp>(- x)</samp>
</li>
<li>
<samp>{length(x) >= 6}</samp>
⇒
<samp>(>= (length x) 6)</samp></li>
<li>
<samp>{f(x) + g(y) + h(z)}</samp>
⇒
<samp>(+ (f x) (g y) (h z))</samp>
</li>
<li><samp>{(f a b) + (g h)}</samp> ⇒ <samp>(+ (f a b) (g h))</samp></li>
<li><samp>{f(a b) + g(h)}</samp> ⇒ <samp>(+ (f a b) (g h))</samp> as well</li>
<li><samp>'{a + f(b) + x}</samp> ⇒ <samp>'(+ a (f b) x)</samp></li>
<li><samp>{(- a) / b}</samp> ⇒ <samp>(/ (- a) b)</samp></li>
<li><samp>{-(a) / b}</samp> ⇒ <samp>(/ (- a) b)</samp> as well</li>
<li><samp>{cos(q)}</samp> ⇒ <samp>(cos q)</samp></li>
<li>
<samp>{e{}}</samp> ⇒
<samp>(e)</samp>
</li>
<li>
<samp>{pi()}</samp> ⇒
<samp>(pi)</samp>
</li>
<li><samp>{'f(x)}</samp> ⇒ <samp>'(f x)</samp></li>
<li><samp>{#1=f(#1#)}</samp> ⇒ <samp>#1=(f #1#)</samp>
if there is support for the SRFI-38 external representation
for data with shared structure</li>
<li>
<samp>{ (f (g h(x))) }</samp>
⇒
<samp>(f (g (h x)))</samp>
... note that this is not
<samp>(f (g h (x)))</samp>
</li>
<li>
<samp>{#(1 2 f(a) 4)}</samp>
⇒
<samp>#(1 2 (f a) 4)</samp></li>
<li>
<samp>{(f #;g(x) h(x))}</samp>
⇒
<samp>(f (h x))</samp>
if datum comments are supported... note that this is not
<samp>(f (x) (h x))</samp>
</li>
<li>
<samp>{(map - ns)}</samp>
⇒
<samp>(map - ns)</samp>
</li>
<li>
<samp>{map(- ns)}</samp>
⇒
<samp>(map - ns)</samp> as well
</li>
<li>
<samp>{n * factorial{n - 1}}</samp>
⇒
<samp>(* n (factorial (- n 1)))</samp>
</li>
<li>
<samp>{2 * sin{- x}}</samp>
⇒
<samp>(* 2 (sin (- x)))</samp>
</li>
<li>
<samp>{3 + 4 +}</samp>
⇒
<samp>($nfx$ 3 + 4 +)</samp>
</li>
<li>
<samp>{3 + 4 + 5 +}</samp>
⇒
<samp>($nfx$ 3 + 4 + 5 +)</samp>
</li>
<li>
<samp>{a . z}</samp>
⇒
<samp>($nfx$ a . z)</samp>
</li>
<li>
<samp>{a + b - c}</samp>
⇒
<samp>($nfx$ a + b - c)</samp>
</li>
<li>
<samp>{read(. options)}</samp> ⇒
<samp>(read . options)</samp>
</li>
<li>
<samp>{a(x)(y)}</samp>
⇒
<samp>((a x) y)</samp>
<li>
<samp>{x[a]}</samp>
⇒
<samp>($bracket-apply$ x a)</samp>
</li>
<li>
<samp>{y[a b]}</samp>
⇒
<samp>($bracket-apply$ y a b)</samp>
</li>
<li>
<samp>{f{n - 1}(x)}</samp>
⇒
<samp>((f (- n 1)) x)</samp>
</li>
<li>
<samp>{f{n - 1}{y - 1}}</samp>
⇒
<samp>((f (- n 1)) (- y 1))</samp>
</li>
<li>
<samp>{f{- x}[y]}</samp>
⇒
<samp>($bracket-apply$ (f (- x)) y)</samp>
</li>
</ol>
<p>A <dfn>curly-infix reader</dfn> is a datum reader
that can correctly read and map curly-infix-expressions.
A curly-infix reader <em>MUST</em>
include the braces “<code>{</code>” and
“<code>}</code>” as delimiters.</p>
<p>
An implementation of this SRFI <em>MUST</em> accept
the marker <var>#!curly-infix</var> followed by a whitespace character
in its standard datum readers (e.g., <var>read</var> and, if applicable,
the default implementation REPL).
This marker (including the trailing whitespace character)
<em>MUST</em> be consumed and considered whitespace.
After reading this marker, the reader <em>MUST</em> accept
curly-infix expressions in subsequent datums
read from the same port until some other conflicting marker is given
(no conflicting marker is specified here).
</p>
<p>
Implementations of this SRFI <em>SHOULD</em>
implement curly-infix expressions in their datum readers by default,
even when the marker is not received.
Portable applications <em>SHOULD</em> include this marker before
using curly-infix expressions, typically near the top of a file.
Portable applications <em>SHOULD NOT</em>
use this marker as the very first characters
of a file (e.g., it could be preceded by a newline),
because they might be misinterpreted on some platforms
as an executable script header.
</p>
<p>
An implementation <em>MUST NOT</em>
bind the symbols
“<var>$nfx$</var>” or
“<var>$bracket-apply$</var>”
by default to a procedure, macro, or syntax
that <em>cannot</em> be overridden.
An implementation <em>SHOULD NOT</em> bind the symbols
“<var>$nfx$</var>” or
“<var>$bracket-apply$</var>”
to a procedure, macro, or syntax in the
default environment, with the exception that it
<em>MAY</em> bind them by default to something that produces an error.
These two symbols are reserved for use by library writers
(in the case of a library-based implementation of this SRFI,
these symbols are
reserved for use by <em>other</em> libraries)
and application writers.
</p>
<p>
<em>However</em>, an implementation <em>MAY</em>
provide one or more
<em>libraries</em> that when imported
bind the “<var>$nfx$</var>” and/or
“<var>$bracket-apply$</var>”
symbols
(as it is then a library, this case actually falls under
the “reserved for use by library writers” clause above).
Application writers and other library writers
using that implementation are then free to use or not use
the implementation’s provided
“<var>$nfx$</var>” and/or
“<var>$bracket-apply$</var>”
as provided by those libraries.
</p>
<p>
Implementations <em>MAY</em> provide the procedure
<var>curly-infix-read</var> as a
curly-infix reader.
If provided, this procedure
<em>SHOULD</em> support an optional port parameter.
</p>
<p>
Security implication: If the implementation does not check for
circularity when doing equality comparisons, and a supplier of
malicious data can specify a circularity,
the reader could fail to terminate when comparing infix operators.
In the worst case this could cause a denial of service.
A solution is to check for circularity when comparing operators.
</p>
<p>Note that, by definition, this SRFI modifies lexical syntax.</p>
<h1><a name="design-rationale">Design Rationale</a></h1>
<p>
This SRFI design
rationale is unusually long, especially when you compare it to
the simplicity of its specification.
However, the notation described in this SRFI builds on the
lessons learned from the many previous infix mechanisms that have been
developed for Scheme and related Lisp-based languages.
The authors believe that it is important to document why various
decisions were made, in particular, to show why this approach is
an improvement over past approaches and more likely to gain wide acceptance.
We have separated the design rationale from the overall rationale,
as was previously done by SRFI-26, because it is easier to
understand the design rationale after reading the specification.
</p>
<h2><a name="rationale_changereader">Why not macros? Why modify the reader?</a></h2>
<p>Many previous systems have implemented “infix”
systems as a named macro or procedure (e.g., <var>INFIX</var>).
This looks ugly, and it does the wrong thing — the resulting list
always has <var>INFIX</var> at the beginning,
not the actual infix operator, so this
approach can interfere with quoting, macros, and other capabilities.
In particular,
consider the following <var>syntax-rules</var> macro
for function composition:
</p>
<pre>
(define-syntax o
(syntax-rules ()
({f o g}
(lambda args
(f (apply g args))))
({f o g o h o ...}
{(lambda (x)
(f (g x))) o h o ...})))
</pre>
<p>
This example takes advantage of the fact that
<samp>{f o g o h o ...}</samp>
⇒ <samp>(o f g h ...)</samp>.
Infix cannot be implemented as a macro alone,
as the <var>syntax-rules</var> form has
a particular treatment for the pattern.
A macro for infix would very likely confuse
the <var>syntax-rules</var> form.
</p>
<p>
A reader notation that
maps to a simple and obvious s-expression structure
also allows
notations such as
<samp>(map . {as + bs})</samp>
⇒
<samp>(map + as bs)</samp>.
For example,
in combination with
<a href="http://srfi.schemers.org/srfi-26/srfi-26.html">SRFI-26</a>,
you can express
templated procedures:
<samp>(cut . {<> < 42})</samp>
</p>
<h2><a name="rationale_braces">Why use brace characters for infix?</a></h2>
<p>
There is no perfect character, but braces (aka curly braces) are pretty close.
A key issue is that you want a balanced pair of characters to identify infix,
since you can have infix-in-infix.
The curly braces are visually pleasant pairs, so
it makes sense to use these precious characters on something
extremely common: infix notation.
</p>
<p>
All other character pairs other than braces have serious problems.
Parentheses are already spoken for, of course.
R6RS Scheme already uses up square brackets as a synonym for parentheses.
Angle brackets are already used for comparison.
Paired characters outside the ASCII set have other problems:
some Schemes do not support characters outside the ASCII character set,
such characters are not as well supported by other tools
(and are sometimes corrupted by such tools),
they are more complicated to deal with due to
character encoding problems, and they are harder to enter on many keyboards.
</p>
<p>
In contrast, curly braces are in the ASCII character set and are
already available for this purpose.
They do not have a standardized meaning in any Scheme specification.
They are also widely available in many other Lisp-derived languages,
such as Common Lisp
(as we’d like this notation to be widely useful across Lisps,
even beyond Scheme).
</p>
<p>
Although curly braces <em>can</em> be used as local Scheme extensions,
there are few Scheme implementations which do so.
On
<a href="http://srfi.schemers.org/srfi-105/mail-archive/msg00063.html">
September 5, 2012, John Cowan posted</a> the
<a href="http://trac.sacrideo.us/wg/wiki/BracketsBraces">results</a>
analyzing the
meanings of square brackets and curly braces in his Scheme test suite
(of 45 Scheme implementations).
Only 2 (Chibi and RScheme) of 45 currently do something
special with braces; “the other implementations treat them
as either synonyms for parentheses, lexical syntax errors,
or identifier characters”.
That is a remarkably small number of Scheme implementations where this
use of curly braces would conflict with some special semantic.
Donovan Kolbly reported that
“RScheme uses braces to delimit C code embedded in Scheme code...
that said, a scanner hack could
easily mode-switch to SRFI-105 interpretation where needed.”
The Chibi extension using braces can be added or removed through a
compile-time option, so not even all Chibi executables have a
conflicting use of brace characters.
</p>
<p>
It’s true that {...} are often used in math for set notation. But
infix notation is far more basic, and common, than sets.
Also, traditional function call notation and infix are helpful when working
with sets, so infix notation is the more important need.
Once you allow neoteric-expressions,
the notation set(...) is a reasonable alternative.
</p>
<h2><a name="rationale_notdifferentsyntax">Why not use a completely different notation inside the expression?</a></h2>
<p>
Some past systems have built infix notations into the reader
in which the infix notation was radically different from normal Lisp notation.
For example, the symbol for procedure calls might change, the names of
variables or procedures might be spelled differently (at least in some cases),
and so on.
The result, in some cases, would be that these notations would
simultaneously lose Lisp’s abilities for
quoting, quasiquoting, and so on, and these notations were
not homoiconic.
It may become impossible to refer to certain symbols, since their
names might include a character that is interpreted as an infix operator.
They can also be confusing; the same symbols (e.g., parentheses) would have
a completely different meaning inside and outside the parentheses.
</p>
<p>
In contrast, this curly-infix-expression proposal avoids these problems.
The syntax for list creation, quasiquotation, and so on is almost
identical in a curly-infix-expression when compared to traditional notation.
For example, in curly-infix,
<samp>`{,a + ,b}</samp>
maps cleanly to
<samp>`(+ ,a ,b)</samp>,
which works as expected with all macros.
The main difference is that, in a curly-infix expression, the position of
the operator in its surface syntax may be in a different location (infix)
than its actual final location.</p>
<h2><a name="rationale_noautodetect">Why not autodetect infix?</a></h2>
<p>Some past efforts tried to automatically detect infix operators,
but this turns out to not work well. It’s hard to express
good rules for detecting infix operators, and the rules become too
complex for users (e.g., “punctuation-only symbols”
doesn’t detect “<var>and</var>” or “<var>or</var>”).
And in
any case, if they were automatically detected, an escape mechanism
would be needed anyway -
consider <samp>(map - ns)</samp>
for getting a new list with
the numbers in <var>ns</var> negated.
Allowing the user to expressly notate when
infix was intended, using <code>{</code>...<code>}</code>,
turns out to be clearer
and more intuitive. In particular, curly-infix allows the use of
infix with any symbol, whenever you want... and where it’s
not convenient, you don’t need to use it. It is also very
backwards-compatible: Normal lists work normally, and if you want
infix, use <code>{</code>...<code>}</code>.</p>
<h2><a name="rationale_equal">Why use <var>equal?</var> to compare operators in a “simple” curly-infix-list for equality?</a></h2>
<p>
Operators are compared using <var>equal?</var> so that
constructs like <samp>,op</samp> are legal operators,
e.g., <samp>{x ,op y ,op z}</samp>.
Note that unfortunately if the operator construct contains a cycle, it might not terminate
if <var>equal?</var> does not terminate in the presence of cycles.
This was specified this way so that implementers
could use the normal Scheme <var>equal?</var> comparison instead of
having to implement a special comparison operator just for this
particular case.
</p>
<h2><a name="rationale_delimiters">Why must infix operators be delimited?</a></h2>
<p>Curly-infix requires that the infix operators be delimited (e.g., by
spaces). This is consistent with Lisp history and current practice.
Currently, in Lisp, operators are always
delimited in traditional s-expressions (typically by left
parentheses on the left, and by whitespace on the right).
It’s impractical to do otherwise today; most Lisps,
including Scheme, allow and predefine symbols that include characters (like
“<code>-</code>”) that are typically used for infix operators.
If infix operators were not delimited, it would be impractical or
complicated to refer to standard Scheme identifiers.
By requiring delimiters (as is already true for the rest of Scheme),
any procedure may be used as an infix operator, not just a fixed list.
Many developers put space around infix operators even in languages
that don’t require them, so syntactically requiring them is no burden.
There are even other existing languages
that also require infix operators be delimited,
such as SNOBOL4,
<a href="http://users.rcn.com/david-moon/PLOT/page-20.html">PLOT</a>,
and
<a href="http://www.rebol.com/docs/expert-intro.html">REBOL</a>.
In short, it is difficult to allow infix operators
without delimiters, and the visual results are the same as many
real-world uses in other languages, so the result appears quite
customary to typical software developers.</p>
<h2><a name="rationale_noprecedence">Why isn’t precedence part of this SRFI?</a></h2>
<p>Many past “infix” systems for Lisp build in
precedence. However, Lisp systems often process other languages,
and they may freely mix these different languages.
Thus, the same symbol may have different meanings
and precedence levels in different contexts.
The symbol might not even be defined where it is being
used, and allowing precedence definitions would create subtle errors
if files are read in a different order.
If users hook in their own precedence system into a reader, it could
even become difficult to combine code written for different precedence systems.
In short, building precedence into a
Lisp reader creates many complexities.</p>
<p>Yet the complexity of precedence systems is often unnecessary.
In practice, we’ve found that simple infix is all
that’s needed most of the time in Lisp-based languages.
Even in other languages, many developers unnecessarily use
grouping symbols with infix operators to make their order clear.
An examination of two Scheme programs written using curly-infix
(posted 2012-09-14) found that 55/78 (71%) of the top-level
c-expressions do not embed an opening brace; this means that precedence
is irrelevant for more than two-thirds of these top-level c-expressions.
Thus, requiring grouping symbols is less of a
hardship than it might appear.</p>
<p>By intentionally <em>not</em>
building a precedence system into the reader,
a very simple yet useful infix system results.
We don’t need to register procedures, ensure that declarations
of precedence precede their use, gain widespread agreement on some
precedence order, or anything like it.
We also ensure that the notation is clearly homoiconic.</p>
<p>Instead, where precedence is desired, application and library writers
can implement precedence by defining and controlling the scope of an
“<var>$nfx$</var>” macro or procedure, or by later postprocessing
of that symbol.
Scheme macros are already quite powerful and capable of handling this;
in these cases, <code>{</code>...<code>}</code> provides a more
convenient notation.
The curly-infix approach, instead of trying to manage both infix
and precedence, handles simple cases and then
takes advantage of the existing Scheme scoping rules and macro system for
more complex cases (in the rare cases where they are needed).</p>
<p>Note that curly-infix includes support for unary operators, but again,
they are without precedence.
As a result, they must be grouped separately.
This does not lead to hard-to-read expressions, however.
Examples of simple curly-infix lists combining infix and unary operations
include
<code>{-(x) * -(y)}</code>
and
<code>{-{x} * -{y}}</code>
(the notation is designed so that both work).
</p>
<p>At first <a href="http://www.dwheeler.com">David A. Wheeler</a>,
who started this project,
considered reporting an error if a simple infix
expression isn’t provided.
However, prepending “<var>$nfx$</var>”
is much more flexible.</p>
<h2><a name="rationale_precedence_addable">Could precedence be added?</a></h2>
<p>It would be <em>possible</em>
to extend curly-infix to provide a fixed
precedence system (e.g., if an expression is mixed,
attempt to use various precedence rules).
Here is a discussion how this could be accomplished in the future
(should that be necessary), which may also show why such systems were
not proposed in this SRFI.
It is important to understand that
such capabilities would be extensions <em>beyond</em> this SRFI.
It would be best if the precedence rules (if any) were absolutely fixed;
otherwise, subtle bugs would happen if files were read after the
precedence was declared, and code would be hard to combine
if it used different precedence rules.
However, the precedence rules can be fixed while still allowing
arbitrary new symbols, as shown below.
Here is an example of such an extension, called a “math” extension.
In this extension, if a mixed curly-infix list is seen,
it first attempts to apply the “math” ruleset, and only
prepends “$nfx$” if it does not meet the requirements. In this extension:
</p>
<ol>
<li>All even-numbered parameters must be symbols, there must be an odd number of parameters, and there must be at least five parameters (the minimum to have more than one operator).
</li>
<li>A fixed set of operators is supported, in a fixed order of precedence;
the operators are compared to the table below to determine precedence.
</li>
<li>
If an operator is not in the fixed table, but it is a symbol,
a new symbol is created by
removing from the symbol all “-” surrounded by alphanumerics,
removing all alphanumerics, and then removing all characters not in
an operator of the precedence table; that new symbol is then compared to
the table.
This allows the use of many more operators, e.g.,
“char-ci<=?” would be considered an operator with the
same precedence as “<=”.
If there’s still no match for any operator,
the expression does not meet the “math” ruleset, and
the normal mixed rules are used (<var>$nfx$</var> is inserted at the front).
</li>
<li>
When the same operator is repeated, the operator simply gains another operand, so {a + b + c * d} is the “+” operator applied to three operands: a, b, and (* c d).
</li>
<li>
Other different operators of the same precedence are interpreted in left-to-right order, so {a + b - c} maps to {{a + b} - c}.
</li>
</ol>
<p>
But there would be substantial arguments about the semantics of
any precedence system.
For example, should there be support for combining different ranged
comparisons, to support notations such as {a < b <= c}?
Should some operators be right-associative, and if so, which ones?
There would also be substantial disagreement on exactly what operators
should be in the precedence table (including which combinations and if
Unicode characters should be included), their order, and whether
the table should be “big” or “small”
(since it is fixed, there are arguments for a larger one).
Below is one possible order, in high-to-low precedence order
(where ... matches 0 or more characters), presuming all is left-to-right;
the point in part is to show that it
would be challenging to get agreement on such a list:
</p>
<ol>
<li>Subscript/down-arrow: sub {Unicode: ↓, ⇓}</li>
<li>Exponentiation/superscript/up-arrow: exp..., **, ^, sup {Unicode: ↑, ⇑}</li>
<li>Multiplication/division: *, /, div..., mod..., quo... {Unicode: ÷, ×}</li>
<li>Addition/subtraction: +, -</li>
<li>Bitwise and: bit...and, log...and, &</li>
<li>Bitwise or/xor: bit...or, log...or, |</li>
<li>Comparison:
<ul>
<li>
Ranged: < , <= , >=, > {Unicode: ≥, ≤, ∋, ∉, ⊄, ⊂, ⊆, ⊃, ⊇}
</li>
<li>Unranged: =, ==, !=, <>, =/=, eq..., in, is {Unicode: ≠, ≈, ≅}</li>
</ul>
<li>Logical conjunction: and {Unicode: ∩, ∧}</li>
<li>Logical/exclusive disjunction: or, xor, eor {Unicode: ∪, ∨, ⊕}</li>
<li>Implication/right arrows/doubled arrows: ->, =>, <->, <=>, -->, ==>, <-->, <==> {Unicode: ↔, ⇔, →, ⇒}</li>
<li>Definition/assignment/left arrow: <-, <--, <==, :=, ::= {Unicode: ≡, ←, ⇐}</li>
</ol>
<p>
Think that is too complicated?
Many other variations are possible.
A “simple math” ruleset could be devised instead,
e.g., perhaps it has a shorter list of built-in operators.
The extant code using curly-infix tends to combine these:
*, /; +, -; < , <= , >=, >, =, <>, eq...;
<var>and</var>; and <var>or</var>;
adding exponentiation (e.g., ** and exp...) at a higher precedence level,
and implication (e.g., -> and =>) would make sense for a short list.
(The <> isn’t in many Scheme specifications, but it is odd to omit
it from a precedence list.)
</p>
<p>
The key point is that although this SRFI does not include a precedence system,
one (such as these) could be added later.
If a precedence system were added, all existing code using simple
curly-infix expressions, or 0..2 parameter expressions, would work unchanged.
Even when it’s not, many “$nfx$” processors would likely generate the
same order in most actual cases.
Since it would be difficult to gain such agreement, and the value of
such a system is doubtful, it is better to provide a much simpler
system that does not include precedence.
Again, any support for precedence is an extension beyond this SRFI.
</p>