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{{Built-in|Partitioned Enclose|⊂}} is a [[dyadic function]] which splits its right argument into differently sized pieces as determined by the non-negative integer<ref group=note name=max>Most dialects restrict this number to a maximum of 1.</ref> left argument.
:''In the [[APL2]] family and many related dialects, <syntaxhighlight lang=apl inline>⊂</syntaxhighlight> indicates [[Partition]].''


On a vector right argument, the corresponding element in the left argument indicates how many<ref group=note name=max></ref> '''divisions''' to begin there:
{{Built-in|Partitioned Enclose|⊂}} is a [[dyadic function]] which splits its right argument into differently sized pieces as determined by the left argument. It was introduced by [[NARS]] and appears in [[Dyalog APL]] and some newer dialects influenced by it.
<source lang=apl>
== Description ==
In the simplest case, and on a vector right argument, the corresponding element in the left argument indicates where '''divisions''' begin:
<syntaxhighlight lang=apl>
       1 0 1 0 0 0 0⊂'HiEarth'
       1 0 1 0 0 0 0⊂'HiEarth'
┌──┬─────┐
┌──┬─────┐
│Hi│Earth│
│Hi│Earth│
└──┴─────┘
└──┴─────┘
</source>
</syntaxhighlight>
{{Works in|[[Dyalog APL]]}}
{{Works in|[[Dyalog APL]], [[Extended Dyalog APL]]}}
Empty divisions can be inserted by beginning more than 1 division at a particular index<ref group=note name=not>Most dialects do not support this.</ref>:
=== Non-Boolean left argument ===
<source lang=apl>
Older dialects restrict the left argument to provide only the above functionality. However, the left argument can be interpreted as a count of how many partitions begin with a particular position:
<syntaxhighlight lang=apl>
       2 0 3 0 0 0 0⊂'HiEarth'
       2 0 3 0 0 0 0⊂'HiEarth'
┌┬──┬┬┬─────┐
┌┬──┬┬┬─────┐
││Hi│││Earth│
││Hi│││Earth│
└┴──┴┴┴─────┘
└┴──┴┴┴─────┘
</source>
</syntaxhighlight>
Additional trailing empty divisions are created by adding an additional division count corresponding to the index beyond the end of the right argument<ref group=note name=not></ref>:
=== Short left argument ===
<source lang=apl>
The dialects that support this extension also allow omission of trailing zeros, which is useful if the partitioning vector is generated by [[indices#Inverse|where's inverse]]:
<syntaxhighlight lang=apl>
      ⍸⍣¯1⊢1 1 3 3 3
2 0 3
      2 0 3⊂'HiEarth'
┌┬──┬┬┬─────┐
││Hi│││Earth│
└┴──┴┴┴─────┘
</syntaxhighlight>
=== Long left argument ===
Additional trailing empty divisions are created by adding an additional division count corresponding to the position beyond the end of the right argument:
<syntaxhighlight lang=apl>
       1 0 1 0 0 0 0 1⊂'HiEarth'
       1 0 1 0 0 0 0 1⊂'HiEarth'
┌──┬─────┬┐
┌──┬─────┬┐
│Hi│Earth││
│Hi│Earth││
└──┴─────┴┘
└──┴─────┴┘
</source>
</syntaxhighlight>
Partition allows a simple definition<ref group=note>In dialects that implement the full functionality of Partitioned Enclose.</ref> of a split-into-lengths function:
 
== Split into lengths ==
The above extensions allow a simple definition of a split-into-lengths function:


[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/qLdP/VHf1GBXvZRHbROqNQxtH3UufNS7S/NRx/JHvWvAnC1AvhVQEVABkJWSVqJg8Ki7BSKQkllcABSshQgAjdEAUuHB6uquZal5JerqGurqwanFxZn5eQFFmSABBXX1FHV1zUddi4EKnSOBvLS8YqCA@v/ggpzMEpAjHvVuVXjU1fSod4XCo94dj3oXH1pvqKAdowCkHrVNNtQB2ff/v7GCsYKJAliTgrpHfrljUWpkfqm9OgA Try it online!]
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/qLdP/VHf1GBXvZRHbROqNQxtH3UufNS7S/NRx/JHvWvAnC1AvhVQEVABkJWSVqJg8Ki7BSKQkllcABSshQgAjdEAUuHB6uquZal5JerqGurqwanFxZn5eQFFmSABBXX1FHV1zUddi4EKnSOBvLS8YqCA@v/ggpzMEpAjHvVuVXjU1fSod4XCo94dj3oXH1pvqKAdowCkHrVNNtQB2ff/v7GCsYKJAliTgrpHfrljUWpkfqm9OgA Try it online!]
<source lang=apl>
<syntaxhighlight lang=apl>
       Split←{⍵ ⊂⍨ ⍸⍣¯1 +\ ¯1↓1,⍺}
       Split←{⍵ ⊂⍨ ⍸⍣¯1 +\ ¯1↓1,⍺}
       3 3 4 Split 'HowAreYou?'
       3 3 4 Split 'HowAreYou?'
Line 32: Line 48:
│How│Are│You?│
│How│Are│You?│
└───┴───┴────┘
└───┴───┴────┘
</source>
</syntaxhighlight>
{{Works in|[[Extended Dyalog APL]]}}
 
==Notes==
== Extension support ==
<references group="note"/>
The following table shows which of the extensions to <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> introduced by [[Dyalog APL 18.0]] are supported in various dialects. [[dzaima/APL]] doesn't support extended lengths but instead requires a short <syntaxhighlight lang=apl inline>⍺</syntaxhighlight>, assuming the first value is 1. This means it can't drop elements as the original Partitioned Enclose typically does, but also means it can't produce leading or trailing empty partitions.
{| class=wikitable
! Languages                !! Non-boolean !! Left argument length
|-
| [[NARS]]                  || {{No}}      || <syntaxhighlight lang=apl inline>=≢⍵</syntaxhighlight>
|-
| [[Dyalog APL]], [[April]] || {{Yes}}     || <syntaxhighlight lang=apl inline>≤1+≢⍵</syntaxhighlight>
|-
| [[dzaima/APL]]            || {{Yes}}    || <syntaxhighlight lang=apl inline>=¯1+≢⍵</syntaxhighlight> (first entry assumed 1)
|-
| [[Kap]] (<syntaxhighlight lang=apl inline>⊆</syntaxhighlight>) || {{Yes}}          || <syntaxhighlight lang=apl inline>=≢⍵</syntaxhighlight>
|}
 
== See also ==
* [[Partition]]
* [[Cut (K)]]
* [[Partition representations]]
 
==External links==
==External links==
===Tutorials===
===Tutorials===
Line 41: Line 74:
===Documentation===
===Documentation===
* [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Partitioned%20Enclose.htm Dyalog]
* [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Partitioned%20Enclose.htm Dyalog]
{{APL built-ins}}
{{APL built-ins}}[[Category:Primitive functions]]

Latest revision as of 21:33, 28 January 2024

In the APL2 family and many related dialects, ⊂ indicates Partition.
⊂

Partitioned Enclose (⊂) is a dyadic function which splits its right argument into differently sized pieces as determined by the left argument. It was introduced by NARS and appears in Dyalog APL and some newer dialects influenced by it.

Description

In the simplest case, and on a vector right argument, the corresponding element in the left argument indicates where divisions begin: 

      1 0 1 0 0 0 0⊂'HiEarth'
┌──┬─────┐
│Hi│Earth│
└──┴─────┘
Works in: Dyalog APL, Extended Dyalog APL

Non-Boolean left argument

Older dialects restrict the left argument to provide only the above functionality. However, the left argument can be interpreted as a count of how many partitions begin with a particular position: 

      2 0 3 0 0 0 0⊂'HiEarth'
┌┬──┬┬┬─────┐
││Hi│││Earth│
└┴──┴┴┴─────┘

Short left argument

The dialects that support this extension also allow omission of trailing zeros, which is useful if the partitioning vector is generated by where's inverse: 

      ⍸⍣¯1⊢1 1 3 3 3
2 0 3
      2 0 3⊂'HiEarth'
┌┬──┬┬┬─────┐
││Hi│││Earth│
└┴──┴┴┴─────┘

Long left argument

Additional trailing empty divisions are created by adding an additional division count corresponding to the position beyond the end of the right argument: 

      1 0 1 0 0 0 0 1⊂'HiEarth'
┌──┬─────┬┐
│Hi│Earth││
└──┴─────┴┘

Split into lengths

The above extensions allow a simple definition of a split-into-lengths function:

Try it online! 

      Split←{⍵ ⊂⍨ ⍸⍣¯1 +\ ¯1↓1,⍺}
      3 3 4 Split 'HowAreYou?'
┌───┬───┬────┐
│How│Are│You?│
└───┴───┴────┘

Extension support

The following table shows which of the extensions to ⍺ introduced by Dyalog APL 18.0 are supported in various dialects. dzaima/APL doesn't support extended lengths but instead requires a short ⍺, assuming the first value is 1. This means it can't drop elements as the original Partitioned Enclose typically does, but also means it can't produce leading or trailing empty partitions.

Languages Non-boolean Left argument length
NARS No =≢⍵
Dyalog APL, April Yes ≤1+≢⍵
dzaima/APL Yes =¯1+≢⍵ (first entry assumed 1)
Kap (⊆) Yes =≢⍵

See also

External links

Tutorials

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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