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:''This page is about the class of primitive functions. For the result of an empty reduction, see [[Identity element]].'' | :''This page is about the class of primitive functions. For the result of an empty reduction, see [[Identity element]].'' | ||
An '''Identity function''', or '''tack function''', is one of the three [[primitive function]]s which returns one of its [[argument]]s with no modification: | An '''Identity function''', or '''tack function''', is one of the three [[primitive function]]s which returns one of its [[argument]]s with no modification: | ||
* '''Identity''', '''Same''', or '''Pass''' (<syntaxhighlight lang=apl inline>⊣</ | * '''Identity''', '''Same''', or '''Pass''' (<syntaxhighlight lang=apl inline>⊣</syntaxhighlight> or <syntaxhighlight lang=apl inline>⊢</syntaxhighlight>) is [[monadic]] and returns its only argument. | ||
* '''Left Identity''', or '''Left''' (<syntaxhighlight lang=apl inline>⊣</ | * '''Left Identity''', or '''Left''' (<syntaxhighlight lang=apl inline>⊣</syntaxhighlight>) is [[dyadic]] and returns its left argument. | ||
* '''Right Identity''', or '''Right''' (<syntaxhighlight lang=apl inline>⊢</ | * '''Right Identity''', or '''Right''' (<syntaxhighlight lang=apl inline>⊢</syntaxhighlight>) is [[dyadic]] and returns its right argument. | ||
The ''right tack'' glyph <syntaxhighlight lang=apl inline>⊢</ | The ''right tack'' glyph <syntaxhighlight lang=apl inline>⊢</syntaxhighlight>, when used for Right, is almost paired with Identity for the monadic case. ''Left tack'', <syntaxhighlight lang=apl inline>⊣</syntaxhighlight> is usually used for Identity as well, but may be given a different meaning, such as [[Stop]] (which returns the constant <syntaxhighlight lang=apl inline>0 0⍴0</syntaxhighlight>) in [[SHARP APL]] and [[APLX]], or [[Hide]] (which returns the constant <syntaxhighlight lang=apl inline>0</syntaxhighlight> as a [[shy]] result) in [[GNU APL]]. | ||
Identity functions (Identity in particular) may be used like elements of [[APL syntax|syntax]] to break up [[stranding]], or to force a [[shy]] result to be shown. They can also be combined with an array-oriented [[operator]] to perform structural manipulations on arrays. Identity functions are a central feature of [[tacit programming]], in which functions and operators rather than names are used to direct the flow of arguments. The pairing of both Left and Right with monadic Identity makes it easier to design [[ambivalent]] functions which usefully work with one or two arguments. | Identity functions (Identity in particular) may be used like elements of [[APL syntax|syntax]] to break up [[stranding]], or to force a [[shy]] result to be shown. They can also be combined with an array-oriented [[operator]] to perform structural manipulations on arrays. Identity functions are a central feature of [[tacit programming]], in which functions and operators rather than names are used to direct the flow of arguments. The pairing of both Left and Right with monadic Identity makes it easier to design [[ambivalent]] functions which usefully work with one or two arguments. | ||
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⊢pi←3.14 | ⊢pi←3.14 | ||
3.14 | 3.14 | ||
</ | </syntaxhighlight> | ||
The result of an identity function is never [[shy]], even if the argument is. Thus the result of the second expression above is displayed, although the assignment <syntaxhighlight lang=apl inline>pi←3.14</ | The result of an identity function is never [[shy]], even if the argument is. Thus the result of the second expression above is displayed, although the assignment <syntaxhighlight lang=apl inline>pi←3.14</syntaxhighlight> on its own would not produce any display. | ||
Left and Right return the left and right arguments, respectively, when called [[dyad]]ically. | Left and Right return the left and right arguments, respectively, when called [[dyad]]ically. | ||
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'left' ⊢ 'right' | 'left' ⊢ 'right' | ||
right | right | ||
</ | </syntaxhighlight> | ||
== Uses == | == Uses == | ||
| Line 39: | Line 39: | ||
2 -⍤⊢ 7 | 2 -⍤⊢ 7 | ||
¯7 | ¯7 | ||
</ | </syntaxhighlight> | ||
In both cases the derived function, when called monadically, simply acts on the right argument, as there is no left argument to ignore. | In both cases the derived function, when called monadically, simply acts on the right argument, as there is no left argument to ignore. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
| Line 46: | Line 46: | ||
-⍤⊢ 7 | -⍤⊢ 7 | ||
¯7 | ¯7 | ||
</ | </syntaxhighlight> | ||
The mirror image—using only the left argument while ignoring the right—is attained by using [[Atop]] (either the operator, or a 2-train) with Left as the right operand. | The mirror image—using only the left argument while ignoring the right—is attained by using [[Atop]] (either the operator, or a 2-train) with Left as the right operand. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
| Line 53: | Line 53: | ||
-⍤⊣ 2 | -⍤⊣ 2 | ||
¯2 | ¯2 | ||
</ | </syntaxhighlight> | ||
Within a [[function train]] (as an "argument" function, that is, the rightmost function, or one an even number of steps away), Right indicates the right argument to the train, and Left indicates the left argument. The 3-train <syntaxhighlight lang=apl inline>≠⊆⊢</ | Within a [[function train]] (as an "argument" function, that is, the rightmost function, or one an even number of steps away), Right indicates the right argument to the train, and Left indicates the left argument. The 3-train <syntaxhighlight lang=apl inline>≠⊆⊢</syntaxhighlight> thus applies [[Partition]] to the result of [[Not Equal to]] on both arguments and the right argument. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
' ' (≠⊆⊢) 'split on spaces' | ' ' (≠⊆⊢) 'split on spaces' | ||
| Line 61: | Line 61: | ||
│split│on│spaces│ | │split│on│spaces│ | ||
└─────┴──┴──────┘ | └─────┴──┴──────┘ | ||
</ | </syntaxhighlight> | ||
=== Structural manipulation === | === Structural manipulation === | ||
With [[Reduce]], Left selects the first [[element]]s along the reduction axis, and Right selects the last. For example, <syntaxhighlight lang=apl inline>⊣⌿</ | With [[Reduce]], Left selects the first [[element]]s along the reduction axis, and Right selects the last. For example, <syntaxhighlight lang=apl inline>⊣⌿</syntaxhighlight> gives the first [[major cell]] of an array while <syntaxhighlight lang=apl inline>⊣/</syntaxhighlight> gives the first element along each row, for example the first column of a [[matrix]]. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
⊢A ← 3 4⍴⍳12 | ⊢A ← 3 4⍴⍳12 | ||
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⊢⌿ A ⍝ Last column | ⊢⌿ A ⍝ Last column | ||
9 10 11 12 | 9 10 11 12 | ||
</ | </syntaxhighlight> | ||
The combinations <syntaxhighlight lang=apl inline>⊣/</ | The combinations <syntaxhighlight lang=apl inline>⊣/</syntaxhighlight> <syntaxhighlight lang=apl inline>⊣⌿</syntaxhighlight> <syntaxhighlight lang=apl inline>⊢/</syntaxhighlight> <syntaxhighlight lang=apl inline>⊢⌿</syntaxhighlight> are [[Idiom recognition|recognized idioms]] in [[Dyalog APL]]. | ||
A [[Scan]] using Left extends the first element along each axis to the whole axis, while retaining the argument's shape. This is because a scan reduces on [[prefix]]es, and the first element of a prefix is the first element of the entire array. On the other hand, Right Scan doesn't change the argument, since the last element from each prefix gives the entire array. | A [[Scan]] using Left extends the first element along each axis to the whole axis, while retaining the argument's shape. This is because a scan reduces on [[prefix]]es, and the first element of a prefix is the first element of the entire array. On the other hand, Right Scan doesn't change the argument, since the last element from each prefix gives the entire array. | ||
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⊢\ 'vector' | ⊢\ 'vector' | ||
vector | vector | ||
</ | </syntaxhighlight> | ||
Right [[Each]] checks the arguments for [[conformability]] and returns the right argument, possibly applying [[scalar extension]] or [[singleton extension]]; Left Each does the same for the left argument. | Right [[Each]] checks the arguments for [[conformability]] and returns the right argument, possibly applying [[scalar extension]] or [[singleton extension]]; Left Each does the same for the left argument. | ||
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A⊢¨1 2 3 | A⊢¨1 2 3 | ||
∧ | ∧ | ||
</ | </syntaxhighlight> | ||
The [[outer product]] with Left adds the axes from the right argument to the left argument, while the outer product with Right adds the axes from the left argument to the right argument. In each case the resulting array is constant along any of the added axes. In the case of Right outer product, the result is composed of [[cell]]s matching the right argument, and can also be obtained by [[Reshape|reshaping]] the right argument. | The [[outer product]] with Left adds the axes from the right argument to the left argument, while the outer product with Right adds the axes from the left argument to the right argument. In each case the resulting array is constant along any of the added axes. In the case of Right outer product, the result is composed of [[cell]]s matching the right argument, and can also be obtained by [[Reshape|reshaping]] the right argument. | ||
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1 2 3 4 5 6 | 1 2 3 4 5 6 | ||
1 2 3 4 5 6 | 1 2 3 4 5 6 | ||
</ | </syntaxhighlight> | ||
== External links == | == External links == | ||