Difference between revisions of "Leading axis agreement"

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#REDIRECT [[Leading axis theory]]
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'''Leading axis agreement''', sometimes called '''prefix agreement''', is a [[conformability]] rule designed for [[leading axis theory]]. It states that a [[dyadic]] [[scalar function]] can be applied between two arrays only if one of their [[Shape|shapes]] is a prefix of the other. The shape of the result is that of the argument with higher rank.
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== Examples ==
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The following examples use [[J]] for demonstration purposes.
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A scalar dyadic function works when the two arrays have the same shape:
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<source lang=j>
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  ]x =: 2 3 $ 10
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10 10 10
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10 10 10
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  ]y =: 2 3 $ i.6
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0 1 2
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3 4 5
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  x + y
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10 11 12
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13 14 15
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</source>
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{{Works in|[[J]]}}
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as well as when one is a scalar:
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<source lang=j>
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  ]x =: 10
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10
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  ]y =: 2 3 $ i.6
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0 1 2
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3 4 5
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  x + y
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10 11 12
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13 14 15
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</source>
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{{Works in|[[J]]}}
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The two cases above are already supported in other APLs in the form of [[scalar extension]]. J goes one step further, allowing the lower-rank array argument to have nonzero rank, as long as the leading dimensions match:
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<source lang=j>
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  ]x =: 10 20
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10 20
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  ]y =: 2 3 $ i.6
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0 1 2
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3 4 5
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  x + y
 +
10 11 12
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23 24 25
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</source>
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{{Works in|[[J]]}}
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In this case, <source lang=j inline>x</source> has shape <source lang=j inline>2</source> and <source lang=j inline>y</source> has shape <source lang=j inline>2 3</source>. Since the leading axes agree and the rank difference is 1, each atom (or 0-[[cell]]) of <source lang=j inline>x</source> is matched with each row (or 1-cell) of <source lang=j inline>y</source>, and the two rows in the result are the results of <source lang=j inline>10 + 0 1 2</source> and <source lang=j inline>20 + 3 4 5</source>, respectively.
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== Aligning axes using the Rank operator ==
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When using the [[Rank (operator)|Rank operator]] for dyadic functions as in <source lang=apl inline>X (f⍤m n) Y</source>, the [[Frame|frames]] of <source lang=apl inline>X</source> and <source lang=apl inline>Y</source> are checked for conformability. Combined with leading axis agreement, the Rank operator can be used to align the axes to be matched.
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<source lang=j>
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  NB. $x        : 2|3
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  NB. $y        :  |3 2
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  NB. ------------------
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  NB. $x +"1 2 y : 2 3 2
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  ]x =: 2 3 $ 10 20 30 40 50 60
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10 20 30
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40 50 60
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  ]y =: 3 2 $ 1 2 3 4 5 6
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1 2
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3 4
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5 6
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  x +"1 2 y
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11 12
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23 24
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35 36
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41 42
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53 54
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65 66
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</source>
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{{Works in|[[J]]}}

Revision as of 03:04, 18 February 2021

Leading axis agreement, sometimes called prefix agreement, is a conformability rule designed for leading axis theory. It states that a dyadic scalar function can be applied between two arrays only if one of their shapes is a prefix of the other. The shape of the result is that of the argument with higher rank.

Examples

The following examples use J for demonstration purposes.

A scalar dyadic function works when the two arrays have the same shape:

   ]x =: 2 3 $ 10
10 10 10
10 10 10
   ]y =: 2 3 $ i.6
0 1 2
3 4 5
   x + y
10 11 12
13 14 15
Works in: J

as well as when one is a scalar:

   ]x =: 10
10
   ]y =: 2 3 $ i.6
0 1 2
3 4 5
   x + y
10 11 12
13 14 15
Works in: J

The two cases above are already supported in other APLs in the form of scalar extension. J goes one step further, allowing the lower-rank array argument to have nonzero rank, as long as the leading dimensions match:

   ]x =: 10 20
10 20
   ]y =: 2 3 $ i.6
0 1 2
3 4 5
   x + y
10 11 12
23 24 25
Works in: J

In this case, x has shape 2 and y has shape 2 3. Since the leading axes agree and the rank difference is 1, each atom (or 0-cell) of x is matched with each row (or 1-cell) of y, and the two rows in the result are the results of 10 + 0 1 2 and 20 + 3 4 5, respectively.

Aligning axes using the Rank operator

When using the Rank operator for dyadic functions as in X (fm n) Y, the frames of X and Y are checked for conformability. Combined with leading axis agreement, the Rank operator can be used to align the axes to be matched.

   NB. $x         : 2|3
   NB. $y         :  |3 2
   NB. ------------------
   NB. $x +"1 2 y : 2 3 2
   ]x =: 2 3 $ 10 20 30 40 50 60
10 20 30
40 50 60
   ]y =: 3 2 $ 1 2 3 4 5 6
1 2
3 4
5 6
   x +"1 2 y
11 12
23 24
35 36

41 42
53 54
65 66
Works in: J