While working on a CSS layout for the J Dictionary, I thought it might be even more flexible/useful to describe the definition pages of J primitives using a more formal XML structure. Below is my attempt.
I'm not very experienced at creating XML structures, and have only had minimal experience using XSLT to format such structures (I haven't even attempted to create XSLT code to format the primitives below!) so I'm not sure how difficult it would be to work with this structure.
I'd be interested in thoughts on whether an XML structure for J primitives would be a desirable/sensible approach? Would it complicate things too much for too little reward? Would other descriptive formats (e.g. wiki markup, JSON) be better?
«primitive.xml»= Example XML structure to describe J Primitives
<?xml version="1.0" encoding="utf-8"?>
<dictionary>
<primitive type="verb">
<symbol>^</symbol>
<form>
<syntax></syntax>
<definition valence="monad">
<name>Exponential</name>
<rank>0</rank>
<description>
<p><tt>^y </tt>is equivalent to<tt> e^y</tt>,<tt> </tt>where
<tt> e </tt>is <i>Euler's number</i><tt> ^1 </tt>
(approximately 2.71828). The <i>natural logarithm</i><tt> </tt>
(<tt>^.</tt>)<tt> </tt>is inverse to<tt> ^ </tt>(that is,<tt> y=^.^y </tt>
and <tt>y=^^.y</tt>).</p>
<p>The monad<tt> x&^ </tt>is inverse to the monad <tt>x&^.</tt> .<tt> </tt>
For example:</p>
<pre>
10&^ 10&^. 1 2 3 4 5
1 2 3 4 5
10&^. 10&^ 1 2 3 4 5
1 2 3 4 5
</pre>
</description>
</definition>
<definition valence="dyad">
<name>Power</name>
<rank>0 0</rank>
<description>
<p><tt>x^2 </tt>and<tt> x^3</tt> and<tt> x^0.5</tt> are the <i>square</i>,
<i>cube</i>, and <i>square root</i> of<tt> x</tt> .</p>
<p>In general,<tt> x^y </tt>is<tt> ^y*^.x</tt>,<tt> </tt>applying for
complex numbers as well as real. </p>
<p>For a non-negative integer<tt> y</tt>,<tt> </tt>the phrase<tt>x ^ y </tt>
is equivalent to<tt> */y # x</tt>;<tt> </tt>in particular,<tt> */ </tt>on
an empty list is<tt> 1</tt>,<tt> </tt>and<tt> x^0 </tt>is<tt> 1 </tt>for
any<tt> x</tt>,<tt> </tt>including<tt> 0</tt> .</p>
<p>The fit conjunction applies to<tt> ^ </tt>to yield a stope defined
as follows:<tt> x^!.k n </tt>is<tt> */x + k*i. n</tt> .<tt> </tt>
In particular,<tt> ^!._1 </tt>is the <i>falling factorial</i> function.</p>
</description>
</definition>
<examples>
<p>The last result in the first example below illustrates the
falling factorial function, formed by the fit conjunction.
See <a href="ref.htm#14">Chapter 5 of [14]</a>
for the use of stope functions, stope polynomials,
and Stirling numbers in the difference calculus:</p>
<pre>
e=: ^ 1 [ x=: 4 [ y=: 0 1 2 3
,.&.> x (e"_ ; e&^@] ; ^ ; ^@(] * ^.@]) ; (]^]) ; ^!._1) y
+-------+-------+--+--+--+--+
|2.71828| 1| 1| _| 1| 1|
| |2.71828| 4| 1| 1| 4|
| |7.38906|16| 4| 4|12|
| |20.0855|64|27|27|24|
+-------+-------+--+--+--+--+
S2=: %.@S1=: (^!._1/~ %. ^/~) @ i. @ x:
(S1;S2) 8
+---------------------------+-------------------+
|1 0 0 0 0 0 0 0|1 0 0 0 0 0 0 0|
|0 1 _1 2 _6 24 _120 720|0 1 1 1 1 1 1 1|
|0 0 1 _3 11 _50 274 _1764|0 0 1 3 7 15 31 63|
|0 0 0 1 _6 35 _225 1624|0 0 0 1 6 25 90 301|
|0 0 0 0 1 _10 85 _735|0 0 0 0 1 10 65 350|
|0 0 0 0 0 1 _15 175|0 0 0 0 0 1 15 140|
|0 0 0 0 0 0 1 _21|0 0 0 0 0 0 1 21|
|0 0 0 0 0 0 0 1|0 0 0 0 0 0 0 1|
+---------------------------+-------------------+
</pre>
<p><tt>S1 </tt>gives (signed) Stirling numbers of the first kind
and<tt> S2 </tt>gives Stirling numbers of the second kind.
They can be used to transform between ordinary and stope polynomials.
Note that<tt> <a href="dxco.htm">x:</a> </tt>gives
<a href="dictg.htm">extended precision</a>.</p>
</examples>
</form>
</primitive>
<primitive type="conjunction">
<symbol>^:</symbol>
<form>
<syntax>u^:n</syntax>
<definition valence="monad">
<name>Power</name>
<rank>_</rank>
<description>
<p><tt>n </tt>may be integer, boxed, or a gerund.</p>
<p><b>Integer.</b> The verb<tt> u </tt>is applied<tt> n </tt>times.
An infinite power<tt> n </tt>produces the limit of the application
of<tt> u</tt> .<tt> </tt>For example,<tt> (2&o.^:_)1 </tt>is<tt> 0.73908</tt> ,<tt> </tt>
the solution of the equation<tt> y=Cos y</tt> .<tt> </tt>If<tt> n </tt>is negative,
the <i>obverse</i><tt> u^:_1 </tt>(see below) is applied<tt> |n </tt>times.
Finally,<tt> u^:n y </tt>for an array<tt> n </tt>is produced by assembling<tt> u^:a y </tt>
(for all the atoms<tt> a </tt>in<tt> n</tt>)<tt> </tt>into an overall result.</p>
<p>The obverse is used in<tt> <a href="d631.htm">u&.v</a> </tt>and is
produced by<tt> <a href="dbdotu.htm">v b. _1</a></tt> .
Repeated application of a verb is also provided by
<a href="d630n.htm">Bond (<tt>&</tt>)</a>.</p>
<p><b>Boxed.</b> If<tt> n </tt>is boxed it must be an atom, and<tt> u^:(<m)</tt></p>
<table>
<tr>
<td><big>↔</big><tt> u^:(i.m) y</tt></td>
<td>if<tt> m </tt>is a non-negative integer</td>
</tr>
<tr>
<td><big>↔</big><tt> u^:(i.k) y</tt></td>
<td>if<tt> m </tt>is<tt> _ </tt>or<tt> '' </tt>,<tt> </tt>where<tt> k </tt>is the smallest
positive integer such that<tt> (u^:(k-1) y) -: u^:k y</tt></td>
</tr>
<tr>
<td><big>↔</big><tt> u^:_1^:(<|m) y</tt></td>
<td>if<tt> m </tt>is negative</td>
</tr>
</table>
<p><b>Gerund.</b> See on the right.</p>
</description>
</definition>
<definition valence="dyad">
<name>Power</name>
<rank>_ _</rank>
<description>
<p><tt>n </tt>may be integer, boxed, or a gerund.</p>
<p><b>Integer or Boxed.</b><tt> x u^:n y </tt><big>↔</big><tt> x&u^:n y </tt></p>
<p><b>Gerund.</b> (Compare with the gerund cases of the <a href="d530n.htm">merge adverb<tt> }</tt></a>)<br/>
<tt> x u^:(v0`v1`v2)y </tt><big>↔</big><tt> (x v0 y)u^:(x v1 y) (x v2 y) </tt><br/>
<tt> x u^:( v1`v2)y </tt><big>↔</big><tt> x u^:([`v1`v2) y </tt><br/>
<tt> u^:( v1`v2)y </tt><big>↔</big><tt> u^:(v1 y) (v2 y)</tt></p>
</description>
</definition>
<examples>
<p>The obverse (which is normally the inverse) is specified for six cases:</p>
<ol>
<li>The self-inverse functions<tt> + - -. % %. |. |: /: [ ] C. p.</tt></li>
<li>The pairs in the following table:
<table>
<tr>
<td><tt><</tt></td>
<td><tt>></tt></td>
</tr>
<tr>
<td><tt><:</tt></td>
<td><tt>>:</tt></td>
</tr>
<tr>
<td><tt>+.</tt></td>
<td><tt>j./"1"_</tt></td>
</tr>
<tr>
<td><tt>+:</tt></td>
<td><tt>-:</tt></td>
</tr>
<tr>
<td><tt>*.</tt></td>
<td><tt>r./"1"_</tt></td>
</tr>
<tr>
<td><tt>*:</tt></td>
<td><tt>%:</tt></td>
</tr>
<tr>
<td><tt>^</tt></td>
<td><tt>^.</tt></td>
</tr>
<tr>
<td><tt>$.</tt></td>
<td><tt>$.^:_1</tt></td>
</tr>
<tr>
<td><tt>,:</tt></td>
<td><tt>{.</tt></td>
</tr>
<tr>
<td><tt>;:</tt></td>
<td><tt>;@(,&' '&.>"1)</tt></td>
</tr>
<tr>
<td><tt>#.</tt></td>
<td><tt>#:</tt></td>
</tr>
<tr>
<td><tt>!</tt></td>
<td><tt>3 : '(-(!-y"_)%1e_3&* !"0 D:1 ])^:_^.y'</tt></td>
</tr>
<tr>
<td><tt>3!:1</tt></td>
<td><tt>3!:2</tt></td>
</tr>
<tr>
<td><tt>3!:3</tt></td>
<td><tt>3!:2</tt></td>
</tr>
<tr>
<td><tt>\:</tt></td>
<td><tt>/:@|.</tt></td>
</tr>
<tr>
<td><tt>".</tt></td>
<td><tt>":</tt></td>
</tr>
<tr>
<td><tt>j.</tt></td>
<td><tt>%&0j1</tt></td>
</tr>
<tr>
<td><tt>o.</tt></td>
<td><tt>%&1p1</tt></td>
</tr>
<tr>
<td><tt>p:</tt></td>
<td>π<tt>(n)</tt></td>
</tr>
<tr>
<td><tt>q:</tt></td>
<td><tt>*/</tt></td>
</tr>
<tr>
<td><tt>r.</tt></td>
<td><tt>%&0j1@^.</tt></td>
</tr>
<tr>
<td><tt>s:</tt></td><td><tt>5&s:</tt></td></tr>
<tr>
<td><tt>u:</tt></td><td><tt>3&u:</tt></td></tr>
<tr>
<td><tt>x:</tt></td><td><tt>_1&x:</tt></td></tr>
<tr>
<td><tt>+~</tt></td>
<td><tt>-:</tt></td>
</tr>
<tr>
<td><tt>*~</tt></td>
<td><tt>%:</tt></td>
</tr>
<tr>
<td><tt>^~</tt></td>
<td><tt>3 : '(- -&b@(*^.) % >:@^.)^:_ b=.^.y'"0</tt></td>
</tr>
<tr>
<td><tt>,~</tt></td>
<td><tt><.@-:@# {. ]</tt></td>
</tr>
<tr>
<td><tt>,:~</tt></td>
<td><tt>{.</tt></td></tr>
<tr>
<td><tt>;~</tt></td>
<td><tt>>@{.</tt></td>
</tr>
<tr>
<td><tt>j.~</tt></td>
<td><tt>%&1j1</tt></td>
</tr>
</table>
</li>
<li>Obviously invertible bonded dyads such as<tt> -&3 </tt>and<tt> 10&^. </tt>
and<tt> 1 0 2&|: </tt>and<tt> 3&|. </tt>and<tt> 1&o. </tt>
and<tt> a.&i. </tt>as well as<tt> u@v </tt>and<tt> u&v </tt>
if<tt> u </tt>and<tt> v </tt>are invertible.
</li>
<li>Monads of the form<tt> v/\ </tt>and<tt> v/\. </tt>where<tt> v </tt>is one of<tt> + * - % = ~:</tt>
</li>
<li>Obverses specified by the conjunction<tt> :.</tt></li>
<li>The following cases merit special mention:
<ul>
<li><tt>p:^:_1 n </tt>gives the number of primes less than<tt> n</tt>,<tt> </tt>
denoted by<tt> </tt>π<tt>(n) </tt>in math</li>
<li><tt>q:^:_1 </tt>is<tt> */</tt></li>
<li><tt>b&#^:_1 </tt>where<tt> b </tt>is a boolean list is <i>Expand</i>
(whose fill atom<tt> f </tt>can be specified by
<i>fit</i>,<tt> b&#^:_1!.f </tt>or<tt> #^:_1!.f</tt> )</li>
<li><tt>a&#.^:_1 </tt>produces the base-<tt>a </tt>representation</li>
<li><tt>!^:_1 </tt>and<tt> !&n^:_1 </tt>and<tt> n&!^:_1 </tt>
produce the appropriate results</li>
<li><tt>{= </tt>and<tt> i."1&1 </tt>are inverses of each other;
these convert between integer permutation vectors and boolean permutation matrices</li>
</ul>
</li>
</ol>
<p><b>Example 1:</b></p>
<pre>
(] ; +/\ ; +/\^:2 ; +/\^:0 1 2 3 _1 _2 _3 _4) 1 2 3 4 5
+---------+-----------+------------+-------------+
|1 2 3 4 5|1 3 6 10 15|1 4 10 20 35|1 2 3 4 5|
| | | |1 3 6 10 15|
| | | |1 4 10 20 35|
| | | |1 5 15 35 70|
| | | |1 1 1 1 1|
| | | |1 0 0 0 0|
| | | |1 _1 0 0 0|
| | | |1 _2 1 0 0|
+---------+-----------+------------+-------------+
</pre>
<p><b>Example 2:</b> Fibonacci Sequence</p>
<pre>
+/\@|.^:(i.10) 0 1
0 1
1 1
1 2
2 3
3 5
5 8
8 13
13 21
21 34
34 55
{. +/\@|.^:n 0 1x [ n=:128 NB. n-th term of the Fibonacci sequence
251728825683549488150424261
{.{: +/ .*~^:k 0 1,:1 1x [ k=:7 NB. (2^k)-th term of the Fibonacci sequence
251728825683549488150424261
</pre>
<p><b>Example 3:</b> Newton Iteration</p>
<pre>
-:@(+2&%)^:(0 1 2 3) 1
1 1.5 1.41667 1.41422
-:@(+2&%)^:(_) 1
1.41421
-:@(+2&%)^:a: 1
1 1.5 1.41667 1.41422 1.41421 1.41421
%: 2
1.41421
</pre>
<p><b>Example 4:</b> Subgroup Generated by a Set of Permutations</p>
<pre>
sg=: ~. @ (,/) @ ({"1/~) ^: _ @ (i.@{:@$ , ])
sg ,: 1 2 3 0 4
0 1 2 3 4
1 2 3 0 4
2 3 0 1 4
3 0 1 2 4
# sg 1 2 3 4 5 0 ,: 1 0 2 3 4 5
720
</pre>
<p><b>Example 5:</b> Transitive Closure</p>
<pre>
x=: (#x)<. (#x),~x=: (i.20)+1+20 ?.@# 3
(i.#x) ,: x
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 4 5 5 7 6 9 9 10 12 11 14 14 15 16 18 18 18 20 20 20
{&x^:(<15) 0
0 1 4 7 9 12 14 16 18 20 20 20 20 20 20
{&x^:a: 0
0 1 4 7 9 12 14 16 18 20
x {~^:a: 0
0 1 4 7 9 12 14 16 18 20
</pre>
<p>Interpretation:<tt> x </tt>specifies a directed graph
with nodes numbered<tt> i.#x </tt>and links from<tt> i </tt>
to <tt> i{x</tt> .<tt> </tt>For example, the
links are:<tt> 0 1</tt> ,<tt> 1 4</tt> ,<tt> 2 5</tt> ,<tt> 3 5 </tt>
and so on. Then<tt> {&x^:a:0 </tt>or<tt> x{~^:a:0 </tt>computes
all the nodes reachable from node<tt> 0</tt>.</p>
<p><b>Example 6:</b> Transitive Closure</p>
<p>Each record of a file begins with a byte indicating the record length
(excluding the record length byte itself), followed by
the record contents. Given a file, the verb<tt> rec </tt>below produces
the list of boxed records.</p>
<pre>
rec=: 3 : 0
n=. #y
d=. _1 ,~ n<.1+(i.n)+a.i.y
m=. d {~^:a: 0
((i.n) e. m) <;._1 y
)
randomfile=: 3 : 0
c =. 1+y ?@$ 255 NB. record lengths
rec=. {&a.&.> c ?@$&.> 256 NB. record contents
(c{a.),&.> rec NB. records with lengths
)
boxed_rec=: randomfile 1000
$ boxed_rec
1000
file=: ; boxed_rec
$ file
132045
r=: rec file
$r
1000
r -: }.&.> boxed_rec
1
</pre>
<p>The last phrase verifies that the result of<tt> rec </tt>
are the records without the leading length bytes.</p>
</examples>
</form>
<form>
<syntax>u^:v</syntax>
<definition valence="monad">
<name>Power</name>
<rank>_</rank>
<description>
<p>The case of<tt> ^: </tt>with a verb right argument is
defined in terms of the noun right argument
case<tt> </tt>(<a href="d202n.htm"><tt>u ^: n</tt></a>)<tt> </tt>as follows:</p>
<p><tt> u ^: v y </tt><big>↔</big><tt> u^:( v y) y</tt></p>
</description>
</definition>
<definition valence="dyad">
<name>Power</name>
<rank>_ _</rank>
<description>
<p>The case of<tt> ^: </tt>with a verb right argument is
defined in terms of the noun right argument
case<tt> </tt>(<a href="d202n.htm"><tt>u ^: n</tt></a>)<tt> </tt>as follows:</p>
<p><tt>x u ^: v y </tt><big>↔</big><tt> x u^:(x v y) y</tt></p>
</description>
</definition>
<examples>
<p>For example:</p>
<pre>
x=: 1 3 3 1
y=: 0 1 2 3 4 5 6
x p. y
1 8 27 64 125 216 343
x p. ^: (]>3:)"1 0 y
0 1 2 3 125 216 343
a=: _3 _2 _1 0 1 2 3
%: a
0j1.73205 0j1.41421 0j1 0 1 1.41421 1.73205
* a
_1 _1 _1 0 1 1 1
%: ^: * " 0 a
9 4 1 0 1 1.41421 1.73205
*: a
9 4 1 0 1 4 9
</pre>
<p>The following monads are equivalent. (See the example of<tt> ^ T. _ </tt>
in the definition of
<a href="dtcapdot.htm">Taylor Approximation<tt> </tt>(<tt>T.</tt>)</a> .)</p>
<pre>
g=: u ^: p ^: _
h=: 3 : 't=. y while. p t do. t=. u t end.'
u=: -&3
p=: 0&<
(g"0 ; h"0) i. 10
+-------------------------+-------------------------+
|0 _2 _1 0 _2 _1 0 _2 _1 0|0 _2 _1 0 _2 _1 0 _2 _1 0|
+-------------------------+-------------------------+
</pre>
<p>All partitions of an integer, based on an algorithm by R.E. Boss on 2005-07-05:</p>
<pre>
partition=: [: final ] (,new)@]^:[ (<i.1 0)"_
final =: 0 <@-."1~ >@{:
new =: [: <@; (-i.)@# cat&.> ]
cat =: [ ,. (>: {."1) # ]
partition 5
+-+---+---+-----+-----+-------+---------+
|5|4 1|3 2|3 1 1|2 2 1|2 1 1 1|1 1 1 1 1|
+-+---+---+-----+-----+-------+---------+
</pre>
</examples>
</form>
</primitive>
<primitive type="adverb">
<symbol>~</symbol>
<form>
<syntax>u~</syntax>
<definition valence="monad">
<name>Reflexive</name>
<rank>_</rank>
<description>
<p><tt>u~ y </tt><big>↔</big><tt> y u y</tt> .<tt> </tt>For example,<tt> ^~ 3 </tt>
is<tt> 27</tt>,<tt> </tt>and<tt> +/~ i. n </tt>is an addition table.</p>
</description>
</definition>
<definition valence="dyad">
<name>Passive</name>
<rank>ru lu</rank>
<description>
<p><tt>~ </tt><i>commutes</i> or <i>crosses</i> connections to
arguments:<tt> x u~ y </tt><big>↔</big><tt> y u x</tt> .</p>
</description>
</definition>
<examples>
<p>Certain uses of the reflexive and passive are illustrated below:</p>
<pre>
x=: 1 2 3 4 [ y=: 4 5 6
x (,.@[ ; ^/ ; ^/~ ; ^/~@[ ; ]) y
+-+-------------+-------------+-----------+-----+
|1| 1 1 1|4 16 64 256|1 1 1 1|4 5 6|
|2| 16 32 64|5 25 125 625|2 4 8 16| |
|3| 81 243 729|6 36 216 1296|3 9 27 81| |
|4|256 1024 4096| |4 16 64 256| |
+-+-------------+-------------+-----------+-----+
into=: %~
(i. 6) % 5
0 0.2 0.4 0.6 0.8 1
5 into i. 6
0 0.2 0.4 0.6 0.8 1
from=: -~
(i.6) - 5
_5 _4 _3 _2 _1 0
5 from i.6
_5 _4 _3 _2 _1 0
(x %/ y);(x %~/ y);(x %/~ y)
+-----------------+-------------------+------------------+
|0.25 0.2 0.166667| 4 5 6|4 2 1.33333 1|
| 0.5 0.4 0.333333| 2 2.5 3|5 2.5 1.66667 1.25|
|0.75 0.6 0.5|1.33333 1.66667 2|6 3 2 1.5|
| 1 0.8 0.666667| 1 1.25 1.5| |
+-----------------+-------------------+------------------+
</pre>
</examples>
</form>
<form>
<syntax>m~</syntax>
<definition valence="monad">
<name>Evoke</name>
<rank>_</rank>
<description>
<p>If<tt> m </tt>is a name, then<tt> 'm'~ </tt>is equivalent
to<tt> m</tt> .<tt> </tt>For example:</p>
<pre>
m=: 2 3 4
'm'~
2 3 4
m=: +/
'm'~ 2 3 5 7
17
m=: /
+ 'm'~ 2 3 5 7
17
</pre>
</description>
</definition>
</form>
</primitive>
</dictionary>
I get the following error attempting to display the above XML in MSIE7:
The XML page cannot be displayed
Cannot view XML input using XSL style sheet. Please correct the error and then click the Refresh button, or try again later.
--------------------------------------------------------------------------------
The operation completed successfully. Error processing resource 'file:///C:/amisc/J/NYCJUG/200911/primitiveEG.xml'. Line 1...
<td><big>↔</big><tt> u^:(i.m) y</tt></td&...-- DevonMcCormick 2009-10-20 18:03:00
- I suspect this is because the HTML fragments are not correctly embedded in the XML file.
The document was intended as an idea of how a dictionary page could be stored as an XML structure, I didn't take this or the CSS formatting ideas any further because Roger didn't see any value in it. -- RicSherlock 2009-10-21 00:46:24
- It doesn't have to be XML: can be HTML or XHTML with well-defined elements, classes and attributes throughout so that it renderes in the browser, but also is susceptible to automated parsing.
The leading spaces should be minimized or eliminated, especially in <pre> elements.
-- OlegKobchenko 2009-10-21 07:48:05