Frame agreement

Frame agreement is a conformability rule designed for leading axis theory and used by J. It is more general than Prefix agreement, which applies only to scalar functions. It states that a dyadic function can be applied between two arrays only if one of their frames is a prefix of the other. The results of the individual applications of the function are collectively framed by the longer frame.

Description

A dyadic function f with left and right ranks l and r, respectively, splits its left argument x into l-cells, and splits its right argument y into r-cells. Each argument's shape is thus split into a frame and a cell shape. Given that one frame must be a prefix of the other, the shorter frame is called the common frame (denoted here as cf). The term "cells" will denote the l-cells (for the left argument x) or the r-cells (for the right argument y). If the frames are identical, each cell of x is paired with the corresponding cell of y in a 1-to-1 pairing. For the case in which the frame lengths differ, each cell of the shorter-framed argument is paired with each among the corresponding group of cells of the longer-framed argument. The collective results of the individual applications of f are framed by the longer frame.

Examples

      (i.2) +"0 1] (i.2 3 2)

Based on the ranks l r of the function +"0 1, x's frame is ,2 and its cell shape is empty (⍬); y's frame is 2 3 and its cell shape is ,2. The shorter of the frames, and thus the common frame, is ,2. Thus, relative to the common frame, each of the two atoms of x is paired with each of the corresponding cells of y.

Below, the notation [shape] denotes cell shape (l- or r-cells), and a | denotes the division between the common frame and the remaining trailing axes.

Argument	Step 1: Frame / Cell shape	Step 2: Common frame—(-#cf)-cells paired
x	2 []	2|[]
y	2 3 [2]	2|3 [2]

So each atom of x is paired with each corresponding group of 3 vectors of y.

The same example, but without considering cell shape:

Argument	Step 1: Frame / Cell shape	Step 2: Common frame—(-#cf)-cells paired
x	2 C	2|C
y	2 3 C	2|3 C

   x=: i.2
   y=: i.2 3 2
   v=:+"0 1
   ]'l r'=: }.v b.0       NB. l and r denote the left and right ranks of the function +"0 1
0 1
   ]xf=: (-l)}.$x           NB. x's frame
2
   ]yf=: (-r)}.$y           NB. y's frame
2 3
   ]cf=: xf                  NB. the common frame (always the shorter of the two frames)
2
      ]pairs_1=:x;"(-#cf)]y     NB.  first the (-#cf)-cells are paired 1-to-1 between x and y
┌─┬─────┐
│0│0 1  │
│ │2 3  │
│ │4 5  │
├─┼─────┤
│1│ 6  7│
│ │ 8  9│
│ │10 11│
└─┴─────┘
   ]pairs_2=: <@([,' + ',])&":"(l,r)&>/"1]pairs_1     NB.  next, within each (-#cf)-cell pairing, the arguments are split into l- and r-cells, and these are paired 1-to-n (or 1-to-1 if the l- and r-frames are identical). we can see here that the results are framed by the longer frame, yf.
┌───────┬───────┬─────────┐
│0 + 0 1│0 + 2 3│0 + 4 5  │
├───────┼───────┼─────────┤
│1 + 6 7│1 + 8 9│1 + 10 11│
└───────┴───────┴─────────┘
   ]pairs=: x<@([,' + ',])&":"(l,r)"(-#cf)]y      NB. same thing but both rank applications done back-to-back on the original x and y
┌───────┬───────┬─────────┐
│0 + 0 1│0 + 2 3│0 + 4 5  │
├───────┼───────┼─────────┤
│1 + 6 7│1 + 8 9│1 + 10 11│
└───────┴───────┴─────────┘
   x +"0 1]y
 0  1
 2  3
 4  5

 7  8
 9 10
11 12

Works in: J

In APL and BQN, frame agreement can only be modeled for functions modified by the Rank operator, since these languages do not formalize function rank in general, and thus an arbitrary function not modified by the Rank operator does not necessarily create frames for its arguments at all.