A Fractal Verb in J

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 131

At Christmas I sent a copy of J to a professor at Cornell College in Iowa where I went to school. I also sent this verb. It prints one line at a time like Norman Thomson’s program in Vector Vol.11 No.3 page 17.

Primitives like + and programs like F are verbs. F makes a square fractal.

   F 16
┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼
┼ ┼ ┼ ┼ ┼ ┼ ┼ ┼
┼┼  ┼┼  ┼┼  ┼┼
┼   ┼   ┼   ┼
┼┼┼┼    ┼┼┼┼
┼ ┼     ┼ ┼
┼┼      ┼┼
┼       ┼
┼┼┼┼┼┼┼┼
┼ ┼ ┼ ┼
┼┼  ┼┼
┼   ┼
┼┼┼┼
┼ ┼
┼┼
┼

Consider the sentence "She walks fast.". The adverb "fast" changes the meaning of "walk". English doesn’t need a "walkfast" verb. Consider the composition (+/). The Insert adverb (/) sticks the Plus verb (+) between each item of the argument. J doesn’t need a SUM built-in function.

   +/ 1 2 3   NB. +/ changes 1 2 3 to 1+2+3
6

J programs can be composed of nothing but verbs. You don’t have to refer to application data. Iverson calls this "tacit definition". F is a tacit program. The execution sequence is mapped by Track. In a track map each composed subverb is pegged with a horizontal line.

   F=. Kill@Draw@({.&1)
   Kill=. (1.0 0)"_
   Draw=. Next@NEIP^:#
   Next=. [ Screen
   Screen=. {&' ┼' (1!:2) 2:
   NEIP=. ~:/\

![Track'Next Screen NEIP': the track maps of Next ( Screen), Screen ({& ' ┼' 1!:2 2:) and NEIP (~:/ )

A verb applies to a noun on each side (like +) or to a noun on the right (like F). An adverb applies to an argument on the left (like Insert). The argument may be verb or noun. A conjunction applies to an argument on each side. The result of an adverb or conjunction is a verb. The result of a verb is a noun. Verbs, adverbs, conjunctions and nouns can all be specified with the copula (=.).

Atop (@) and Bond (&) are conjunctions. In F, Atop runs the (Kill@Draw) verb after or "atop" the ({.&1) verb. The latter is composed of Take ({.), Bond, and 1. Names of primitives are constructed from one or two characters. If two, the second is period (.) or colon. This is what ({.&1) does.

   4 {. 1 5 0 3 2 3    NB. No composition, two arguments for {.
1 5 0 3
   4 {. 1              NB. J supplies zeroes if necessary
1 0 0 0
   {.&1 (4)            NB. Composed verb ({.&1) applied to 4
1 0 0 0

The argument of F is the argument of ({.&1). If the argument is 4 the result of ({.&1) is (1 0 0 0) and (1 0 0 0) is the argument of (Kill@Draw). In Draw the Power (^:) conjunction runs (Next@NEIP) # (Tally) times. The argument of # is (1 0 0 0) as is the initial argument of (Next@NEIP).

   # 1 0 0 0
4

NEIP (~:/\) makes the fractal. Not Equal (~:) is a verb. Insert (/) and Prefix (\) are adverbs. NEIP applies Not Equal Insert (~:/) to each prefix. Prefix structures its argument as a bunch of prefixes: the first item, the first two, the first three, etc.

   <\ 1 0 0 0          NB. Box (<) each prefix (\)
┌─┬───┬─────┬───────┐
│1│1 0│1 0 0│1 0 0 0│
└─┴───┴─────┴───────┘

This is what J does when Not Equal Insert is applied to the last prefix (1 0 0 0) in place of Box.

The result is also 1 when Not Equal Insert is applied to any other prefix. The result of NEIP on (1 0 0 0) is (1 1 1 1). In Next, Screen displays '┼┼┼┼' for (1 1 1 1), but Left ([) returns (1 1 1 1) when applied to (1 1 1 1) and '┼┼┼┼'.

   NEIP 1 0 0 0
1 1 1 1
   Next@NEIP 1 0 0 0
┼┼┼┼
1 1 1 1

   NEIP NEIP 1 0 0 0
1 0 1 0
   Next@NEIP Next@NEIP 1 0 0 0
┼┼┼┼
┼ ┼
1 0 1 0
   Next@NEIP ^: # 1 0 0 0
┼┼┼┼
┼ ┼
┼┼
┼
1 0 0 0
   F 4
┼┼┼┼
┼ ┼
┼┼
┼

A scalar verb like Minus (-) is applied to each item of a list. An underbar (_) touching a number is a negative sign, not a verb. Scalar technique is extended to all verbs with the Rank (") conjunction. Rank applies a non-scalar verb to each part of its argument.

   1 - 6 1.2 4
_5 _0.2 _3
   - 6 1.2 4
_6 _1.2 _4
   F"0 (2 0 4)
┼┼
┼
┼┼┼┼
┼ ┼
┼┼
┼

Tacit definition sharpens thought. Also, it facilitates the kind of pre-chip manipulation that is increasingly required for parallel and non-parallel hardware. The Fix (f.) adverb substitutes definitions for names. When Track is applied to a single name the linear definition appears below the map.

   Fractal=. F f.
   Track'Fractal'

(1.0 0)"_@(([ ({&' ┼' (1!:2) 2:))@(~:/\)^:#)@({.&1)

The right argument of many conjunctions is executed before the left. In Fractal, ([ ——) is Next and (—— —— 2:) is Screen. Two isolated verbs are a "hook" and three are a "fork". A fork applies the first and last verbs to its argument, and then applies the middle verb to their results. A hook applies the last verb to its argument, and then applies the first verb to its argument and to the result of the last verb.

Vocabulary¶

Two English words like A and Ad usually have nothing in common. Two J words like { and {. frequently do. From selects any. Take selects from the top or bottom. Rank is (verb"noun). Constant is (noun"noun). Foreign links the operating system. Write is 1!:2. Boxed representation is 5!:2. Underbar (_) is infinity only when isolated.