Frame agreement

Frame agreement is a conformability rule designed for leading axis theory and used by J. It states that a dyadic function can be applied between two arrays only if one of their frames, with frame length based on the function's dyadic function rank, is a prefix of the other. The leading shape of the result is that of the argument with 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), which may be empty. With respect to frame agreement, the generic term "cells" denotes the l-cells (for the left argument) or the r-cells (for the right argument). If the frames are identical, each cell of x is paired with the corresponding cell of y in a 1-to-1 pairing. If 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. This 1-to-n pairing can be viewed as extending the shorter frame to match the longer frame. The collective results of the individual applications of f are framed by the longer frame.

Examples

   x=: i.2
   y=: i.2 3 2
   x+"0 1 y
 0  1
 2  3
 4  5

 7  8
 9 10
11 12

Works in: J

The table below shows the pairing of cells from the above example. Here the notation [shape] denotes the cell shape, and | denotes the division between the common frame and the remaining trailing axes.

Argument	Step 1: Frames / Cells	Step 2: Common frame—cells paired
x	2 []	2|[]
y	2 3 [2]	2|3 [2]

The same example, but without considering cell shape:

Argument	Step 1: Frame / Cells	Step 2: Common frame
x	2 C	2|C
y	2 3 C	2|3 C

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 atom of x is paired with the corresponding 3 rows of y.

The expanded example below uses APL to model frame agreement.

      x←⍳2
      y←2 3 2⍴⍳12
      l r←0 1                    ⍝ the left and right ranks of the function +⍤0 1
      ⊢lf rf←(-l r)↓¨x,⍥(⊂∘⍴)y   ⍝ frames
┌─┬───┐
│2│2 3│
└─┴───┘
      ⊢cf←lf(⊃{⍺⍵}⌷⍨(,⍳⌊)⍥≢)rf   ⍝ common (shorter) frame
2
      x{⍺⍵}⍤(-≢cf)⊢y             ⍝ 1st step: 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│
└─┴─────┘
      x{⍺⍵}⍤l r⍤(-≢cf)⊢y         ⍝ 2nd step: the (-≢cf)-cells in each pairing are split into l- and r-cells, and these are paired 1-to-n (or 1-to-1 if the frames are identical)
┌─┬─────┐
│0│0 1  │
├─┼─────┤
│0│2 3  │
├─┼─────┤
│0│4 5  │
└─┴─────┘
┌─┬─────┐
│1│6 7  │
├─┼─────┤
│1│8 9  │
├─┼─────┤
│1│10 11│
└─┴─────┘
      x+⍤0 1⊢y                   ⍝ n-to-m pairing in dialects without frame agreement; invalid
RANK ERROR
      x+⍤l r⍤(-≢cf)⊢y            ⍝ matches the result of J's frame agreement
 0  1
 2  3
 4  5

 7  8
 9 10
11 12

Works in: Dyalog APL