Power-Programmierung/CD1.mdf/j/dos/what_is.j

Text File | 1992-03-24 | 18KB | 505 lines

                          What is J?

            An introduction

                 by

            Leroy J. Dickey
             University of Waterloo


This article is intended to be a first introduction to the language J
and some of its features.  J is a high powered general purpose
programming language.  This dialect of APL uses the ASCII character
set, has boxed arrays, complex numbers, the rank operator, and some
novel compositions of functions.

Like APL, J is an array language.  Any time one wants to do
calculations with more than one number at a time, or with more than one
name in a list, a collective (an array) is the right structure to use,
and J is designed to do such calculations easily.

Those who write programs for themselves and who want their answers
quickly will be happy with the swift and concise way programs can be
written in J.

This article consists of several examples that illustrate some of the
power of the language J.  Each example presents input and output from
an interactive J session, and a few lines of explanation have been
added.  Lines that are indented with four spaces are those that were
typed in to the J interpreter, and the lines that are indented with
only one space are the responses from J.  Other text lines are comments
that have been added later.  The topics that have been chosen for
inclusion do not come close to telling everything about J, but some of
them represent subjects that at one time or another the author found
new and exciting.

J is a general purpose computing language, and is well suited to a
broad spectrum of programming needs.  Because the interests of the
author are in mathematics, examples are primarily mathematical in
nature.

In this discussion about J, the words noun and verb are used to stand
for data and function, respectively.  Thus 'neon' is a noun,  1 2 3 4
is a noun, and + is a verb.  One may assign data to a name, as in the
statement    alpha5=.'neon' , and a name (such as alpha5) used to refer
to the data is said to be a pronoun.  Similarly, one may associate a
name with a verb, and such a name is said to be a proverb,
(pronounced "pro-verb", not "prah-verb").

In the assignment
    plus =. +
the name plus is a proverb used to refer to the verb +.  There are
also language elements called conjunctions, which join two verbs or
proverbs, and there are elements called adverbs, which modify the
meaning of one verb or proverb.


Example 1:  Calculating e

This example shows how calculations are done with a list of numbers and
how J uses a functional notation.  That is, each verb (function) acts on
the noun or pronoun (data) to its right.   It is easy to create a
collective (array).  Here, for instance, are 9 integers:

    i. 9
 0 1 2 3 4 5 6 7 8

    a =. i. 9
    a
 0 1 2 3 4 5 6 7 8

The factorials of these numbers are:
    ! a
 1 1 2 6 24 120 720 5040 40320

The reciprocals of the above:
    % ! a
 1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413 2.48016e_5

And the sum of reciprocals of the factorials of a is:
    +/ % ! a
 2.71828


Those who know some mathematics may recognize an approximation to the
number  e,  the base for the natural logarithms.  Of course, J has
other ways to calculate this number; the point here is the use of the
natural left-to-right execution of the meaning of the sequence symbols
+/ % ! a  as "the sum of the reciprocals of the factorials of a".

Those who have done some programming in almost any other language, will
see that the expression  +/%!i.9  is a remarkably short one for a function
that produces the sum of the first few terms of the series that
approximates  e.  Moreover, it is possible to pronounce this
program in English in a way that precisely conveys its meaning, and
those who know the meaning of the words in the sentence will be able to
understand it.  And anybody who knows the mathematics and who has
learned this much J will recognize that the program does exactly
what the English words say and that it will produce an approximation
to  e.

The author finds this expression in J much easier to think about and
understand than the corresponding program intended to compute the same
number in certain older languages.  Here, for example, is what one
might write in Fortran, to accomplish the same result:


    REAL SUM, FACT
    SUM = 1.0
    FACT = 1.0
    DO 10 N = 1, 8
        FACT = FACT * N
        SUM = SUM + 1.0 / FACT
     10 CONTINUE
    PRINT, SUM
    STOP
    END


Compare this Fortran program with the J program that uses only a few
key strokes: +/ % ! i. 9 .  Not only is the J program shorter, but if
you agree with the author that the program more closely approximates
the English description, then you will probably agree that the J
expression is easier to understand, even for someone who may have been
using an older language, and who has only just learned the meaning of
the symbols used in this example.  The use of this compact notation
brings about much shorter programs.  It is the author's experience
that for some applications, the ratio of the number of lines of
Fortran or C code to the number of lines of J code has been about
20 to 1.


Example 2:  Average

In this example, two verbs (functions) are defined.  As before, lines
indented with four spaces were given as input during a J session, and
the ones immediately after, with one leading space, were produced by
the J interpreter.


    Sum =. +/
    Average =. Sum % #
    c =. 1 2 3 4
    Average c
 2.5


The first two lines create proverbs, the third creates a pronoun,
and the fourth invokes the proverb (function) Average with the
pronoun (data) c.  The meaning of   Average c  is this:

        (+/ % #) 1 2 3 4

and this may be thought of as:  find Sum c, ( +/ c means add up the
entries in c), find # c (tally c), and then find the quotient of those
two results.

The sequence of three functions used in the definition of Average
provides an example of a fork.  A fork is a sequence of three functions
(f h g), where
    x (f g h) y  means  (x f y) g (x h y) .
This diagram may help you to understand the meaning:

         g
            / \
           f   h
          / \ / \
          | | | |
          x y x y

and the figure might suggest to you why the name fork is used.
In a similar way, used with one argument,

  (f g h) y  means  (  f y) g (  h y).

Thus, looking again at Average c, we can see that this has the
same meaning as (Sum c) % (tally c).

If the above definition of fork strikes the reader as awkward, the
reader is invited to consider a thought experiment about the meaning of
the English phrase "a less than or equal to b".  Imagine that there is
a function LT which returns the value 1 when a<b and 0 otherwise.
Imagine a function EQ which returns the value 1 when a=b, and 0
otherwise.  Imagine and a function called OR so that the expression x
OR y returns 1 when x is 1 or y is 1.  Then the meaning of the fork
(LT OR EQ) is understood by noting the equivalence of the statements:
    x (LT OR EQ) y
    (x LT y) OR (x EQ y).



Example 3:  Continued fractions

In this example, the verb  pr  (plus reciprocal), is considered.

    pr =. +%

The next two input lines show a use of this hook.  From them you
should be able to see that the meaning of "a pr b" is "a plus the
reciprocal of b", that is  5 +% 4 :

    5 pr 4
 5.25

And this next example means 4 + % 5 :

    4 pr 5
 4.2

The function pr is defined above as  +% , and is equivalent
to 4 + (% 5).  This diagram for  4 +% 5   may help to understand
its meaning.

                     +
                        /  \
                       4    %
                            |
                            5

Because of the shape of the diagram, the combination of two verbs
in this way is called a hook.  In general, x (F G) y  means
x F (G y) .

To continue with our example,

    1 pr 1 pr 1
 1.5

The above input line may be read as "1 plus the reciprocal
of (1 plus the reciprocal of 1)".  And here below is an analogous
line with four ones:

    1 pr 1 pr 1 pr 1
 1.66667

    1 pr (1 pr (1 pr 1))
 1.66667

    pr / 1 1 1 1
 1.66667

    pr / 4 $ 1
 1.66667

The above input line has exactly the same meaning as the three input
lines that come before it.  Notice that the value of the result is the
same in each case.  The / is an adverb called insert and the result
of  pr/  is a new verb, whose meaning is derived from the verb  pr.

In traditional mathematical notation the above expressions are written


        1
   1 + ------------------------
        1
    1 + ------------------
            1
        1 + ------------
            1


This particular continued fraction is a famous one because it is
known to converge to the golden ratio.  That means that if you
just keep on going, with more and more ones, the value gets closer and
closer to the golden ratio.  One can see that the values of the continued
fraction expansion seem to converge by using \, the prefix adverb.  It
creates a sequence of partial results.  Here is a nice, compact
expression that shows the values for the first 15 partial continued
fractions.

    pr /\ 15$ 1
 1 2 1.5 1.66667 1.6 1.625 1.61538 1.61905 1.61765 1.61818 1.61798
      1.61806 1.61803 1.61804 1.61803

In this, you can see that the numbers seem to be getting closer and
closer to some limit, probably between 1.61803 and 1.61804.  If you
concentrate only on the odd numbered terms, (the first, the third, the
fifth, and so on), you will see a sequence of numbers that is
monotonically increasing.  On the other hand, the even numbered terms,
( 2, 1.66667, 1.625, and so on), form a sequence of numbers that is
monotonically decreasing.  Of course these observations do not constitute
a proof of convergence, but they might convince you that a proof can
be found, and in the light of these observations, the proof probably
follows along these lines.

Can you think of a simpler way to represent and evaluate continued
fractions?  This author can not!

The golden ratio has been known for thousands of years, and is the
number, bigger than one, such that itself minus 1 is its own reciprocal.
The golden ratio is usually denoted by the lower case Greek letter tau.
It is said that pictures whose sides are in the ratio 1 to tau
or tau to 1 are the most pleasing to the eye.


Example 4.  Summing data along different dimensions

In this example, we use some of the earlier functions to illustrate
ideas about rank.  First some data are built.

    a =. 100 200
    b =. 10 20 30
    c =. 1 2 3 4

This is the beginning of the data construction.  Again, we use the verb
Sum, a modification of the verb plus, defined as before:
    Sum =. +/
But in this example, the usage is different, because we use it with a
right and a left argument.  To get an idea about the action of the verb
Sum used dyadically (that is, used with two arguments), notice what Sum
does with just b and c:

    b Sum c
 11 12 13 14
 21 22 23 24
 31 32 33 34

Once you have seen this example, you are ready to see and understand
the next step.  Here the collective  data  is built, using a, b,
and c.

    data =. a Sum b Sum c
    data
 111 112 113 114
 121 122 123 124
 131 132 133 134

 211 212 213 214
 221 222 223 224
 231 232 233 234

This rank three array may be thought of as having has two planes,
each having three rows and four columns.  The function $ (shape of)
tells us the size of the array.

    $ data
 2 3 4

Now examine what Sum does with the data:

    Sum data
 322 324 326 328
 342 344 346 348
 362 364 366 368

Can you see what is happening?  It is adding up corresponding
elements in the two planes.  Twelve different sums are performed,
and the result is an array that is 3 by 4.  The action happens
over the first dimension of the array.

But we might wish to add up numbers in rows.  We can do it this way:

    Sum "1 data
 450 490 530
 850 890 930

And to add up numbers in each column:

    Sum "2 data
 363 366 369 372
 663 666 669 672

The expressions "1 and "2 are read as 'rank 1' and 'rank 2'.  The rank
1 objects of 'data' are the rows, the rank 2 objects are the planes,
and the rank 3 object is the whole object itself.  So, when one asks
for  Sum"2 data ,  one asks for the action (Sum) to happen over the
rank two arrays, and the action happens over the first dimension of
these arrays.

Finally, the expression  Sum"3 data  has the same meaning as Sum data .

Now, recall the definition of Average.

    Average =. Sum % #

Apply the proverb Average to the pronoun data, to get:

    Average data
 161 162 163 164
 171 172 173 174
 181 182 183 184

The result is the average of corresponding numbers in the two planes.
That is, the average is calculated over the objects along the first
dimension of data.  If we would like to know the average of the
numbers in the rows, we type:

    Average "1 data
 112.5 122.5 132.5
 212.5 222.5 232.5

and finally, averaging the columns of the two planes:

    Average "2 data
 121 122 123 124
 221 222 223 224


Again, compare the sizes of these results with the sizes of
the results above where we were asking simply about sums.
The verb $ (shape of) will help us here:

    Sum "1 data
 450 490 530
 850 890 930

    $ Sum "1 data
 2 3
    $ Average "1 data
 2 3

    $ Sum "2 data
 2 4
    $ Average "2 data
 2 4


What the author finds exciting about this example is not just how
easily the nouns and verbs were built, but how every verb acts in a
uniform and predictable way on the data.  Rank one action is the same
kind of action whether one is summing, averaging, or whatever.  It
has been said that the concept of rank is one of the top ten computer
inventions of all time.  [Monardo, 1991]

    UseNet Article in the news group comp.lang.apl, by
    Pat Monardo of Cold Spring Harbor Laboratory,
    Message-id: <[e-mail]>.

    ...  i also believe that the rank operator is a wonderful
    invention, up there in the top ten computer inventions of all time.

Conclusion:

In this article, only a few of the many verbs in J have been used, but
the reader has had a glimpse some of the power of J, as seen in hooks
and forks, in the implicit definition of verbs, and in the uniform way
that J acts on sub-arrays by means of the rank operator.  J is powerful
because one can get results quickly.  It is a carefully thought out and
beautifully consistent programming language for people who need to
write programs for themselves, but who don't want to spend inordinately
large amounts of time programming to get those results.  If you have to
write programs for others, this could still be a language for you,
especially if you get paid by the job, and not by time.


Availability:

J is available by anonymous ftp from watserv1.waterloo.edu in the
directory "~/languages/apl/j" and is available from certain other servers.
At the Waterloo site, versions of J are available that run on several
kinds of computers.  J is a product of Iverson Software Inc., 33 Major
Street, Toronto, Ontario, Canada  M5S 2K9.  [phone], and when
purchased from them, comes with a dictionary of J and other aids
for learning J.


Learning J:

Because J is a relatively new language, there are not yet many books or
articles about it.  At the Waterloo site, there is a tutorial that has
much material concentrated into a small space.   With each release
of J to date, there has been a status file that tells which features
have been implemented.  Several status files are available.

The reader who chooses to learn J from these materials might like to
print the 47 tutorial files and study them one at a time, and to have
at hand a copy of a status file listing the names of the verbs.  For
each example given in the tutorial, the reader might wish to experiment
with J by inventing nouns, verbs, adverbs, and conjunctions to test
ideas related to those presented in the tutorial and thereby to let J
itself be the guide and the interpreter (pun intended).


The author of J:

The inventor-developer of J is Kenneth E. Iverson, the same man who
invented APL, a pioneer of computing, and recipient of the prestigious
ACM Turing Award.  Some years ago, he retired from IBM and went to work
for I.P.Sharp Associates, a Toronto company that developed the world's
first private packet switching network, the finest APL available, and
was at one time the world's largest time sharing service which just
happened to be devoted entirely to APL.  The descendent of this
company, Reuter:file, still has offices in many of the world's major
cities.  Now Iverson is retired (again) and lives in Toronto, where he
spends his time developing and promoting J.

The implementor for J is Roger Hui.  Roger hails from Edmonton, and has
lived in Toronto since he joined I.P.Sharp Associates in 1975.  The
source code for J, written in C, is worthy of some comment, because it
makes extensive use of macro expansion.  Its style, influenced by the
work of Arthur Whitney of Morgan Stanley in New York, reflects Roger's
strong background in APL.  Today, Roger spends much of his time
developing J and does occasional contract work in APL.


Copyright (c) 1991,1992 by Leroy J. Dickey
Permission is granted to use this for
nonprofit and educational purposes.
It may be given away, but it may not be sold.


Revised 1992-03-21.