Leading axis agreement: Difference between revisions

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


<source lang=j>
<syntaxhighlight lang=j>
   ]x =: 2 3 $ 10
   ]x =: 2 3 $ 10
10 10 10
10 10 10
Line 17: Line 17:
10 11 12
10 11 12
13 14 15
13 14 15
</source>
</syntaxhighlight>
{{Works in|[[J]]}}
{{Works in|[[J]]}}


as well as when one is a [[scalar]]:
as well as when one is a [[scalar]]:


<source lang=j>
<syntaxhighlight lang=j>
   ]x =: 10
   ]x =: 10
10
10
Line 31: Line 31:
10 11 12
10 11 12
13 14 15
13 14 15
</source>
</syntaxhighlight>
{{Works in|[[J]]}}
{{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:
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:


<source lang=j>
<syntaxhighlight lang=j>
   ]x =: 10 20
   ]x =: 10 20
10 20
10 20
Line 45: Line 45:
10 11 12
10 11 12
23 24 25
23 24 25
</source>
</syntaxhighlight>
{{Works in|[[J]]}}
{{Works in|[[J]]}}


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


== Model ==
== Model ==


In dialects that do not feature leading axis agreement, it can nevertheless be utilised by the introduction of an explicit operator:
In dialects that do not feature leading axis agreement, it can nevertheless be utilised by the introduction of an explicit operator:
<source lang=apl>
<syntaxhighlight lang=apl>
       _LA←{⍺ ⍺⍺⍤(-⍺⌊⍥(≢⍴)⍵)⊢⍵}
       _LA←{⍺ ⍺⍺⍤(-⍺⌊⍥(≢⍴)⍵)⊢⍵}
       ⊢x ← 10 20
       ⊢x ← 10 20
Line 63: Line 63:
10 11 12
10 11 12
23 24 25
23 24 25
</source>
</syntaxhighlight>
{{Works in|Dyalog APL}}
{{Works in|Dyalog APL}}


== Aligning axes using the Rank operator ==
== Aligning axes using the Rank operator ==


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 [[axis|axes]] to be matched.
When using the [[Rank (operator)|Rank operator]] for dyadic functions as in <syntaxhighlight lang=apl inline>X (f⍤m n) Y</syntaxhighlight>, the [[Frame|frames]] of <syntaxhighlight lang=apl inline>X</syntaxhighlight> and <syntaxhighlight lang=apl inline>Y</syntaxhighlight> are checked for conformability. Combined with leading axis agreement, the Rank operator can be used to align the [[axis|axes]] to be matched.


<source lang=j>
<syntaxhighlight lang=j>
   NB. $x        : 2|3
   NB. $x        : 2|3
   NB. $y        :  |3 2
   NB. $y        :  |3 2
Line 90: Line 90:
53 54
53 54
65 66
65 66
</source>
</syntaxhighlight>


{{Works in|[[J]]}}
{{Works in|[[J]]}}
[[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 22:17, 10 September 2022

Leading axis agreement, sometimes called prefix agreement, is a conformability rule designed for leading axis theory and used by J and BQN. 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.

Model

In dialects that do not feature leading axis agreement, it can nevertheless be utilised by the introduction of an explicit operator:

      _LA←{⍺ ⍺⍺⍤(-⍺⌊⍥(≢⍴)⍵)⊢⍵}
      ⊢x ← 10 20
10 20
      ⊢y ← 2 3 ⍴ ⍳ 6
0 1 2
3 4 5
      x +_LA y
10 11 12
23 24 25
Works in: Dyalog APL

Aligning axes using the Rank operator

When using the Rank operator for dyadic functions as in X (f⍤m 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
APL features [edit]
Built-ins Primitives (functions, operators) ∙ Quad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespaceFunction array
Concepts and paradigms Conformability (Scalar extension, Leading axis agreement) ∙ Scalar function (Pervasion) ∙ Identity elementComplex floorArray ordering (Total) ∙ Tacit programming (Function composition, Close composition) ∙ GlyphLeading axis theoryMajor cell search
Errors LIMIT ERRORRANK ERRORSYNTAX ERRORDOMAIN ERRORLENGTH ERRORINDEX ERRORVALUE ERROREVOLUTION ERROR