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'''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.
'''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.


== Description ==
== Description ==
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.
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>.


== Examples ==
== Examples ==
<syntaxhighlight lang=j>
<syntaxhighlight lang=j>
      (i.2) +"0 1] (i.2 3 2)
  a=: 3 4
</syntaxhighlight>
  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
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>.
  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
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.
12 13
 
14 15
{|class=wikitable
  'A B'=: (<"lr]a),&<(<"rr]b)     NB. frame each argument in terms of lr- or rr-cells
! Argument          !!        Step 1: Frame / Cell shape          !! Step 2: Common frame—<syntaxhighlight lang=j inline>(-#cf)</syntaxhighlight>-cells paired
   A                              NB. a framed as lr-cells
|-
┌─┬─┐
| <syntaxhighlight lang=j inline>x</syntaxhighlight>    || <syntaxhighlight lang=j inline>2 []</syntaxhighlight>                    || <syntaxhighlight lang=j inline>2|[]</syntaxhighlight>
│3│4│
|-
└─┴─┘
| <syntaxhighlight lang=j inline>y</syntaxhighlight>     || <syntaxhighlight lang=j inline>2 3 [2]</syntaxhighlight>                  || <syntaxhighlight lang=j inline>2|3 [2]</syntaxhighlight>
   $A
|}
 
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)}.$x          NB. x's frame
2
2
   ]yf=: (-r)}.$y          NB. y's frame
   B                              NB. b framed as rr-cells
┌───┬───┬─────┐
│0 1│2 3│4 5  │
├───┼───┼─────┤
│6 7│8 9│10 11│
└───┴───┴─────┘
  $B
2 3
2 3
   ]cf=: xf                  NB. the common frame (always the shorter of the two frames)
   ]pairs=: A([;'+';])&>_1]B      NB. each cell of sa is paired with each cell among the corresponding group of (*/sf) cells of la
2
┌─┬─┬─────┐
      ]pairs_1=:x;"(-#cf)]y    NB.  first the (-#cf)-cells are paired 1-to-1 between x and y
│3│+│0 1 │
┌─┬─────┐
├─┼─┼─────┤
│0│0 1  │
│3│+│2 3  │
│ │2 3  │
├─┼─┼─────┤
│ │4 5  │
│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


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


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.
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.


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

Revision as of 20:57, 8 July 2023

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.

APL features [edit]
Built-ins Primitives (functions, operators) ∙ Quad name
Array model Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index (Indexing) ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ Box ∙ Namespace ∙ Function array
Concepts and paradigms Conformability (Scalar extension, Leading axis agreement) ∙ Scalar function (Pervasion) ∙ Identity element ∙ Complex floor ∙ Array ordering (Total) ∙ Tacit programming (Function composition, Close composition) ∙ Glyph ∙ Leading axis theory ∙ Major cell search ∙ First-class function
Errors LIMIT ERROR ∙ RANK ERROR ∙ SYNTAX ERROR ∙ DOMAIN ERROR ∙ LENGTH ERROR ∙ INDEX ERROR ∙ VALUE ERROR ∙ EVOLUTION ERROR
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