At Play with J: MIMD Machines

by Eugene McDonnell

I had a request recently from someone who wanted to apply a verb a different number of times to a list of arguments. What was wanted was a simpler way of writing:

     (f a),(f f b),(f f f c)

My initial response was to say that J did not as yet have a way of describing Multiple Instruction-Multiple Data machine architectures (MIMD), although such a mechanism had been described [Be91]. I pointed out that a collapsing transpose could solve the problem, but my questioner would have none of that, as it implied a great deal of useless computation. There the matter rested for a while. After several months I had another request from the same person who wanted to know if I had made any progress on the problem. Actually, I hadn’t thought about it at all in the interim, but since my questioner seemed to be a determined type, I gave it a few minutes more thought, and found what I think is a neat use of one of J’s more interesting differences from APL, the way scan is defined: that is, the verb applied is monadic, not dyadic.

For example, whereas in APL one writes +\1 2 3 to obtain the continued sum of the values in the argument, in J one would have to write +/\1 2 3 to obtain the same result.

     +\1 2 3
1 0 0
1 2 0
1 2 3

Here the monadic verb conjugate, denoted by +, is being applied, first to 1, next to 1 2, and last to 1 2 3; since these are real numbers, their conjugates are the same as the arguments, and since J reshapes results so that they conform, and then appends them, we get the zero fills at the right of the top two rows. Compare this with

     +/\1 2 3
1 3 6

which is the analog to APL’s +\1 2 3.

Finally, here is the solution to the MIMD problem.

First I define three variables, a, b, and c:

     'abc'=.3 4 5

Next, I define a verb f to be the natural logarithm (^.).

      f=.^.

and apply it once, twice, and thrice, to a, b, and c, respectively:

     f a
1.09861
     f f b
0.326634
     f f f c
_0.742579

This is the desired result, but done the hard way. Now for the easy way:

Define a verb g which in which the verb f is applied (@) to the tail ({:) of its argument a number of times (^:) equal to the length (#) of its argument:

     g=.f@{:^:#

For example, g 3 1 4 1 5 9 applies f six times to 9:

     f f f f f f 9
0.854804j1.01575
     g 3 1 4 1 5 9
0.854804j1.01575

Perhaps you already see how this will end. We apply the prefix scan (\) adverb to g, and apply this derived verb to a,b,c:

     (g\)a,b,c  NB. apply g to successively longer prefixes
1.09861 0.326634 _0.742579
     NB. q.e.f.

Showing once again that where there’s a will there’s a way. Note that because of the way prefix scan is defined, it is easy to visualize how, in a multiprocessor environment, the applications of g to all three arguments can be carried out simultaneously.

     [Be91]     Bernecky, Robert, and Roger Hui, ‘Gerunds and Representations’, APL Quote Quad 21, 4, August 1991, Stanford, California 1991, pp 39-45