Verb Tables

by Roger K.W. Hui

language design mathematics and statistics

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 91

The verb tables produced by f/ in J (see Reference [1]; f/ is ∘.f in APL) are generalizations of the familiar addition and multiplication tables of arithmetic. The utility verbs over and by (defined and analyzed in the Appendix) are used to annotate verb tables. For example:

   a by b over (a=.2 3 5)+/b=.i.5

The table of a+b, with row heads 2 3 5 and column heads 0 1 2 3 4

Verb tables facilitate systematic experimentation, and can reveal new patterns and new meanings even for familiar and seemingly quotidian verbs.

Cross (~) is an adverb such that f~y is y f y and x f~ y is y f x, and is useful in generating verb tables of a list against itself. Thus:

   y by y over y */ y=.2-~i.5        y by y over %/~ y

The times and divide tables of _2 _1 0 1 2 against itself; the divide table shows infinities for division by 0

   2 1 0 </2 1 3 0 1 2         *./~0 1        +./~0 1
0 0 1 0 0 0                  0 0            0 1
1 0 1 0 0 1                  0 1            1 1
1 1 1 0 1 1

Note the pattern of signs in both the times (*) and divide (%) tables, the result of 0%0, and the infinite possibilities of dividing by 0 [3,4]. The signs of a real table t can be displayed using the expression ' +-'{~*t, using the signum (*) of t to select from ({) ' +-'. Tables of boolean verbs (truth tables) such as less than (<), and (*.), and or (+.) offer useful classifications.

Out of (!) produces binomial coefficients; !/ produces Pascal’s triangle. 5&|@* is multiplication modulo 5:

   y by y over !/~y=.4-~i.9          y by y over 5&|@*/~y

The out-of table of _4 to 4 against itself, showing Pascal’s triangle extended to negative arguments, and the table of multiplication modulo 5

a =. -: 4-~i.9
b =. -: 3-~i.7

   a by b over a ^/ b

The power table of a (from _2 to 2 by halves) against b (from _1.5 to 1.5 by halves), with complex results for negative bases and infinities for zero raised to negative powers

The power (^) table contains many subtle patterns. For example, examine the column headed by 0, and the rows headed by _1 and 0. The reader may wish to attempt the same table on other systems; the results may prove interesting.

The verb j. produces a complex number from its real and imaginary parts; thus 3 j. 4 is 3j4. The expression (|.y) <.@j.~/ y=.5%~i.5 produces a table of the verb <.@j.~ on the reverse (|.) of y against y itself; i.e., floor (<.) on the lower left corner of the first quadrant of the complex plane. The pattern is made more evident by mapping the result to single glyphs, using the symbols s=.' @.+&-$*=%'.

   (|.y) by y over (|.y) <.@j.~/ y =. 5%~i.5

The complex-floor table on 0 0.2 0.4 0.6 0.8, with values 0, 0j1 and 1

   s =. ' @.+&-$*=%'
   map =. {&s @ (~.@, i. ])
   map (|.y) <.@j.~/ y=.10%~i.20

The mapped table: twenty rows of the glyphs @ . + & - $ * forming sloping diagonal bands

Thus, the floor of the complex plane is tiled by "sloping bricks" — unit-area rectangles inclined at minus 45°, whose long sides are the diagonal of the unit square [2]. map is a composition of two verbs, {&s (from bonded with s) atop the fork ~.@, i. ]; it works by computing the indices of the floor values in the nubbed ravel (~.@,) of such values, which are then used to select from s. Fork and atop are defined by the following diagrams:

 (f g h) y      x (f g h) y        f@g y       x f@g y
     g               g               f            f
    / \             / \              |            |
   f   h           f   h             g            g
   |   |          / \ / \            |           / \
   y   y          x y x y            y          x   y

Appendix: over and by¶

rowhead =. ' '&; @ ,.
colhead =. {. ; }.
over    =. colhead @ ": @ ,
by      =. rowhead@[ ,. ]

$The development of over and by, step by step: a and b, the power table a^/b, x =. b, a^/b and its shape 4 4, y =. ": x and its shape 4 26, {. y and }. y, ({.y) ; }.y, colhead y, and colhead ": b , a^/b$

Continuing: over applied to b and a^/b, 2 1 $ b over a^/b, ,.a and its shape 3 1, ' ' ; ,.a, rowhead a and 2 1 $ rowhead a, (rowhead a) ,. c=.b over a^/b, the table 1 2 3 ,. 9 10 11 and its shape 3 2, (rowhead a[c) ,. a]c, and 1 2['abc' and 1 2]'abc'

Finally, by =. rowhead@[ ,. ] applied as a by c, and a by b over a^/b

References¶

[1] Iverson, K.E., ISI Dictionary of J, Version 4, 1991 12.

[2] McDonnell, E.E., Complex Floor, APL73 Conference Proceedings.

[3] McDonnell, E.E., Zero Divided by Zero, APL76 Conference Proceedings.

[4] McDonnell, E.E., and Shallit, J.O., Extending APL to Infinity, APL 80, North-Holland Publishing Company, 1980 6.