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 lr and rr, respectively, splits its left argument x into lf-cells, and splits its right argument y into rf-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 prefix (i.e. shorter) frame is called the common frame (denoted here as cf), and the suffix of the longer frame relative to the common frame is called the surplus frame (denoted here as sf. "Cells" will be used here to denote the lr-cells (for the left argument a) or the rr-cells (for the right argument b). 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, we'll denote the shorter-framed argument as sa and the longer-framed argument as la. Each cell of sa is paired with each among the corresponding group of */sf cells of la. The collective results among all applications of f are framed by the longer frame lf.

Examples

   a=: 3 4
   b=: i.2 3 2
   'lr rr'=: }.+"0 1 b.0           NB. let lv and rv denote the left and right ranks of the verb +"0 1
   lr;rr
┌─┬─┐
│0│1│
└─┴─┘
   ]af=: (-lr)}.$a                 NB. a's frame
2
   ]bf=: (-rr)}.$b                 NB. b's frame
2 3
   af =/@(<.&# {.&> ;) bf          NB. one frame prefixes the other
1
   ]cf=: af {.@(<.&#{.&>;~) bf     NB. the common frame (always the shorter of the two frames)
2
   ]lf=: af >@(-.@>&# { ;) bf      NB. the longer of the two frames
2 3
   ]sf=: lf }.~ #cf                NB. the surplus frame results from removing the common frame from the left side of longer frame 
3
   a +"0 1]b
 3  4
 5  6
 7  8

10 11
12 13
14 15
   'A B'=: (<"lr]a),&<(<"rr]b)     NB. frame each argument in terms of lr- or rr-cells
   A                               NB. a framed as lr-cells
┌─┬─┐
│3│4│
└─┴─┘
   $A
2
   B                               NB. b framed as rr-cells
┌───┬───┬─────┐
│0 1│2 3│4 5  │
├───┼───┼─────┤
│6 7│8 9│10 11│
└───┴───┴─────┘
   $B
2 3
   ]pairs=: A([;'+';])&>_1]B       NB. each cell of sa is paired with each cell among the corresponding group of (*/sf) cells of la 
┌─┬─┬─────┐
│3│+│0 1  │
├─┼─┼─────┤
│3│+│2 3  │
├─┼─┼─────┤
│3│+│4 5  │
└─┴─┴─────┘

┌─┬─┬─────┐
│4│+│6 7  │
├─┼─┼─────┤
│4│+│8 9  │
├─┼─┼─────┤
│4│+│10 11│
└─┴─┴─────┘
   A+&>"_1]B                       NB. the results of all applications of + are framed by lf
 3  4
 5  6
 7  8

10 11
12 13
14 15
   pairs -: a ([;'+';])"_1"(-#cf)]b
1

Works in: J

In the example above, based on the ranks 0 1 of the verb +"0 1, a's frame is ,2 and its cell shape is empty (0$0); b's frame is 2 3 and its cell shape is ,2. The shorter of these, and thus the common frame, is ,2, so each of the two 0$0-shaped cells (atoms) of a is paired with each of the corresponding 6 (i.e. */sf) corresponding b cells.