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'''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 [[array]]s only if one of their [[frame]]s, 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.
'''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 [[array]]s only if one of their [[frame]]s is a [[prefix]] of the other. The results of the individual applications of the function are collectively framed by the longer frame.


== Description ==
== Description ==
A dyadic function <syntaxhighlight lang=j inline>f</syntaxhighlight> with left and right ranks <syntaxhighlight lang=j inline>lr</syntaxhighlight> and <syntaxhighlight lang=j inline>rr</syntaxhighlight>, respectively, splits its left argument <syntaxhighlight lang=j inline>x</syntaxhighlight> into <syntaxhighlight lang=j inline>lf</syntaxhighlight>-[[cell]]s, and splits its right argument <syntaxhighlight lang=j inline>y</syntaxhighlight> into <syntaxhighlight lang=j inline>rf</syntaxhighlight>-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 <syntaxhighlight lang=j inline>cf</syntaxhighlight>), and the suffix of the longer frame relative to the common frame is called the surplus frame (denoted here as <syntaxhighlight lang=j inline>sf</syntaxhighlight>. "Cells" will be used here to denote the <syntaxhighlight lang=j inline>lr</syntaxhighlight>-cells (for the left argument <syntaxhighlight lang=j inline>a</syntaxhighlight>) or the <syntaxhighlight lang=j inline>rr</syntaxhighlight>-cells (for the right argument <syntaxhighlight lang=j inline>b</syntaxhighlight>). If the frames are identical, each cell of <syntaxhighlight lang=j inline>x</syntaxhighlight> is paired with the corresponding cell of <syntaxhighlight lang=j inline>y</syntaxhighlight> in a 1-to-1 pairing. For the case in which the frame lengths differ, we'll denote the shorter-framed argument as <syntaxhighlight lang=j inline>sa</syntaxhighlight> and the longer-framed argument as <syntaxhighlight lang=j inline>la</syntaxhighlight>. Each cell of <syntaxhighlight lang=j inline>sa</syntaxhighlight> is paired with each among the corresponding group of <syntaxhighlight lang=j inline>*/sf</syntaxhighlight> cells of <syntaxhighlight lang=j inline>la</syntaxhighlight>. The collective results among all applications of <syntaxhighlight lang=j inline>f</syntaxhighlight> are framed by the longer frame <syntaxhighlight lang=j inline>lf</syntaxhighlight>.
A dyadic function <syntaxhighlight lang=j inline>f</syntaxhighlight> with left and right ranks <syntaxhighlight lang=j inline>l</syntaxhighlight> and <syntaxhighlight lang=j inline>r</syntaxhighlight>, respectively, splits its left argument <syntaxhighlight lang=j inline>x</syntaxhighlight> into <syntaxhighlight lang=j inline>l</syntaxhighlight>-cells, and splits its right argument <syntaxhighlight lang=j inline>y</syntaxhighlight> into <syntaxhighlight lang=j inline>r</syntaxhighlight>-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 <syntaxhighlight lang=j inline>cf</syntaxhighlight>). The term "cells" will denote the <syntaxhighlight lang=j inline>l</syntaxhighlight>-cells (for the left argument <syntaxhighlight lang=j inline>x</syntaxhighlight>) or the <syntaxhighlight lang=j inline>r</syntaxhighlight>-cells (for the right argument <syntaxhighlight lang=j inline>y</syntaxhighlight>). If the frames are identical, each cell of <syntaxhighlight lang=j inline>x</syntaxhighlight> is paired with the corresponding cell of <syntaxhighlight lang=j inline>y</syntaxhighlight> 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 <syntaxhighlight lang=j inline>f</syntaxhighlight> are framed by the longer frame.


== Examples ==
== Examples ==
<syntaxhighlight lang=j>
<syntaxhighlight lang=j>
  a=: 3 4
      (i.2) +"0 1] (i.2 3 2)
  b=: i.2 3 2
</syntaxhighlight>
  'lr rr'=: }.+"0 1 b.0          NB. let lv and rv denote the left and right ranks of the verb +"0 1
 
  lr;rr
Based on the ranks <syntaxhighlight lang=apl inline>l r</syntaxhighlight> of the function <syntaxhighlight lang=j inline>+"0 1</syntaxhighlight>, <syntaxhighlight lang=j inline>x</syntaxhighlight>'s frame is <syntaxhighlight lang=j inline>,2</syntaxhighlight> and its cell shape is [https://aplwiki.com/wiki/Zilde empty] (<syntaxhighlight lang=j inline>⍬</syntaxhighlight>); y's frame is <syntaxhighlight lang=j inline>2 3</syntaxhighlight> and its cell shape is <syntaxhighlight lang=j inline>,2</syntaxhighlight>. The shorter of the frames, and thus the common frame, is <syntaxhighlight lang=j inline>,2</syntaxhighlight>. Thus, relative to the common frame, each of the two atoms of <syntaxhighlight lang=j inline>x</syntaxhighlight> is paired with each of the corresponding cells of <syntaxhighlight lang=j inline>y</syntaxhighlight>.
┌─┬─┐
 
│0│1│
 
└─┴─┘
Below, the notation <syntaxhighlight lang=j inline>[shape]</syntaxhighlight> denotes cell shape (l- or r-cells), and a <syntaxhighlight lang=j inline>|</syntaxhighlight> denotes the division between the common frame and the remaining trailing axes.
  ]af=: (-lr)}.$a                NB. a's frame
 
2
{|class=wikitable
  ]bf=: (-rr)}.$b                NB. b's frame
! Argument          !!        Step 1: Frame / Cell shape          !! Step 2: Common frame—<syntaxhighlight lang=j inline>(-#cf)</syntaxhighlight>-cells paired
2 3
|-
   af =/@(<.&# {.&> ;) bf          NB. one frame prefixes the other
| <syntaxhighlight lang=j inline>x</syntaxhighlight>    || <syntaxhighlight lang=j inline>2 []</syntaxhighlight>                    || <syntaxhighlight lang=j inline>2|[]</syntaxhighlight>
1
|-
   ]cf=: af {.@(<.&#{.&>;~) bf    NB. the common frame (always the shorter of the two frames)
| <syntaxhighlight lang=j inline>y</syntaxhighlight>    || <syntaxhighlight lang=j inline>2 3 [2]</syntaxhighlight>                  || <syntaxhighlight lang=j inline>2|3 [2]</syntaxhighlight>
|}
 
So each atom of <syntaxhighlight lang=j inline>x</syntaxhighlight> is paired with each corresponding group of 3 vectors of <syntaxhighlight lang=j inline>y</syntaxhighlight>.
 
The same example, but without considering cell shape:
{|class=wikitable
! Argument          !!        Step 1: Frame / Cell shape          !! Step 2: Common frame—<syntaxhighlight lang=j inline>(-#cf)</syntaxhighlight>-cells paired
|-
| <syntaxhighlight lang=j inline>x</syntaxhighlight>    || <syntaxhighlight lang=j inline>2 C</syntaxhighlight>                    || <syntaxhighlight lang=j inline>2|C</syntaxhighlight>
|-
| <syntaxhighlight lang=j inline>y</syntaxhighlight>    || <syntaxhighlight lang=j inline>2 3 C</syntaxhighlight>                  || <syntaxhighlight lang=j inline>2|3 C</syntaxhighlight>
|}
 
 
<syntaxhighlight lang=j>
  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)}.$x          NB. x's frame
2
2
   ]lf=: af >@(-.@>&# { ;) bf      NB. the longer of the two frames
   ]yf=: (-r)}.$y          NB. y's frame
2 3
2 3
   ]sf=: lf }.~ #cf                NB. the surplus frame results from removing the common frame from the left side of longer frame
   ]cf=: xf                  NB. the common frame (always the shorter of the two frames)
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
2
  B                              NB. b framed as rr-cells
      ]pairs_1=:x;"(-#cf)]y    NB. first the (-#cf)-cells are paired 1-to-1 between x and y
┌───┬───┬─────┐
┌─┬─────┐
│0 1│2 3│4 5  │
│0│0 1  │
├───┼───┼─────┤
│ │2 3  │
│6 7│8 9│10 11│
│ │4 5  │
└───┴───┴─────┘
├─┼─────┤
  $B
│1│ 6  7│
2 3
│ │ 8  9│
   ]pairs=: A([;'+';])&>_1]B      NB. each cell of sa is paired with each cell among the corresponding group of (*/sf) cells of la
│ │10 11│
┌─┬─┬─────┐
└─┴─────┘
│3│+│0 1 │
   ]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.
├─┼─┼─────┤
┌───────┬───────┬─────────┐
│3│+│2 3 │
│0 + 0 1│0 + 2 3│0 + 4 5 │
├─┼─┼─────┤
├───────┼───────┼─────────┤
│3│+│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


┌─┬─┬─────┐
│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
  7  8
 
9 10
10 11
11 12
12 13
14 15
  pairs -: a ([;'+';])"_1"(-#cf)]b
1
</syntaxhighlight>
</syntaxhighlight>
{{Works in|[[J]]}}
{{Works in|[[J]]}}


In the example above, based on the ranks <syntaxhighlight lang=j inline>0 1</syntaxhighlight> of the verb <syntaxhighlight lang=j inline>+"0 1</syntaxhighlight>, <syntaxhighlight lang=j inline>a</syntaxhighlight>'s frame is <syntaxhighlight lang=j inline>,2</syntaxhighlight> and its cell shape is [[empty array|empty]] (<syntaxhighlight lang=j inline>0$0</syntaxhighlight>); b's frame is <syntaxhighlight lang=j inline>2 3</syntaxhighlight> and its cell shape is <syntaxhighlight lang=j inline>,2</syntaxhighlight>. The shorter of these, and thus the common frame, is <syntaxhighlight lang=j inline>,2</syntaxhighlight>, so each of the two <syntaxhighlight lang=j inline>0$0</syntaxhighlight>-shaped cells (atoms) of <syntaxhighlight lang=j inline>a</syntaxhighlight> is paired with each of the corresponding <syntaxhighlight lang=j inline>6</syntaxhighlight> (i.e. <syntaxhighlight lang=j inline>*/sf</syntaxhighlight>) corresponding <syntaxhighlight lang=j inline>b</syntaxhighlight> cells.
In [[APL]] and [[BQN]], frame agreement can only be modeled for functions modified by the [https://aplwiki.com/wiki/Rank_(operator) 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.


[[Category:Leading axis theory]][[Category:Function characteristics]][[Category:Conformability]]{{APL features}}
[[Category:Leading axis theory]][[Category:Function characteristics]][[Category:Conformability]]{{APL features}}
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