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Date: Sat, 23 Nov 2013 21:46:23 -0800
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To: PHP internals list <internals@lists.php.net>
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Subject: Re: [PHP-DEV] rfc: pow-operator
From: ww.galen@gmail.com (Galen Wright-Watson)

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On the left-vs-right-vs-non-associative debate: it seems to me that the
reason for supporting associativity for any binary operator is to cut down
on the number of parentheses that must be typed, which can help cut down on
visual clutter. Otherwise, you might as well use a Lisp. (What are other
reasons?) If that's true, then whether to pick left or right association
should depend on the most common use-case.

Personally, I'd be much more likely to mean 2**(3**2), as (2**3)**2 is
equivalent to 2 ** (3*2) but is less performant. Also, by making **
right-associative, 2**3**2 is visually more distinct from 2 ** (3*2); if
left-associative, 2 ** (3 ** 2) and 2 ** (3*2) would be more easily
confused when reading or writing.

Closely related to the power-tower argument is the relation between
associativity and hyper
operations<https://en.wikipedia.org/wiki/Hyperoperation>,
which offers another reason why right association is more natural in
mathematics. Defining hyper operations in terms of hypo operations
(successor -> addition -> multiplication -> exponentiation -> tetration
...) requires right associativity. However, as addition and multiplication
are associative (and the computational versions of these operations aren't
usually implemented in terms of each other), it doesn't much matter whether
they're made left- or right-associative. For reasons others have stated in
this conversation, computational + and * are generally implemented as
left-associative. Mathematical exponentiation isn't associative (hence this
whole debate), so the reasons for allowing a switch from right- to left-
association simply isn't present. Of course, the computational operators
aren't the same as the mathematical, so this isn't actually a rigorous
argument in favor of right association.

Finally, here's a graphic depicting the "exponentiation is
right-associative in math" argument (sorry text-readers, you'll have to go
to http://i.imgur.com/G9aIn6X.png if you want to see the image):
[image: Inline image 2]

Of course, this doesn't help at all to determine whether ** should be
associative or not.

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<div dir=3D"ltr"><div class=3D"gmail_extra">On the left-vs-right-vs-non-ass=
ociative debate: it seems to me that the reason for supporting associativit=
y for any binary operator is to cut down on the number of parentheses that =
must be typed, which can help cut down on visual clutter. Otherwise, you mi=
ght as well use a Lisp. (What are other reasons?) If that&#39;s true, then =
whether to pick left or right association should depend on the most common =
use-case.</div>

<div class=3D"gmail_extra"><br></div><div class=3D"gmail_extra">Personally,=
 I&#39;d be much more likely to mean 2**(3**2), as (2**3)**2 is equivalent =
to 2 ** (3*2) but is less performant. Also, by making ** right-associative,=
 2**3**2 is visually more distinct from 2 ** (3*2); if left-associative, 2 =
** (3 ** 2) and 2 ** (3*2) would be more easily confused when reading or wr=
iting.</div>

<div class=3D"gmail_extra"><br></div><div class=3D"gmail_extra">Closely rel=
ated to the power-tower argument is the relation between associativity and =
<a href=3D"https://en.wikipedia.org/wiki/Hyperoperation">hyper operations</=
a>, which offers another reason why right association is more natural in ma=
thematics. Defining hyper operations in terms of hypo operations (successor=
 -&gt; addition -&gt; multiplication -&gt; exponentiation -&gt; tetration .=
..) requires right associativity. However, as addition and multiplication a=
re associative (and the computational versions of these operations aren&#39=
;t usually implemented in terms of each other), it doesn&#39;t much matter =
whether they&#39;re made left- or right-associative. For reasons others hav=
e stated in this conversation, computational + and * are generally implemen=
ted as left-associative. Mathematical exponentiation isn&#39;t associative =
(hence this whole debate), so the reasons for allowing a switch from right-=
 to left- association simply isn&#39;t present. Of course, the computationa=
l operators aren&#39;t the same as the mathematical, so this isn&#39;t actu=
ally a rigorous argument in favor of right association.</div>

<div class=3D"gmail_extra"><br></div><div class=3D"gmail_extra">Finally, he=
re&#39;s a graphic depicting the &quot;exponentiation is right-associative =
in math&quot; argument (sorry text-readers, you&#39;ll have to go to <a hre=
f=3D"http://i.imgur.com/G9aIn6X.png">http://i.imgur.com/G9aIn6X.png</a> if =
you want to see the image):</div>

<div class=3D"gmail_extra"><img src=3D"cid:ii_14288640a1b17da9" alt=3D"Inli=
ne image 2"><br></div><div class=3D"gmail_extra"><br></div><div class=3D"gm=
ail_extra">Of course, this doesn&#39;t help at all to determine whether ** =
should be associative or not.</div>

</div>

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