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Leading axis theory: Difference between revisions
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'''Leading axis theory''', or the '''leading axis model''', is an approach to array language design and use that emphasizes working with arrays by manipulating their [[cell]]s and mapping functions over leading [[Axis|axes]] implicitly using [[function rank]] or explicitly using the [[Rank operator]]. It was initially developed in [[SHARP APL]] in the early 1980s and is now a major feature of [[J]] and [[Dyalog APL]], as well as languages influenced by these. The name "leading axis" comes from the [[frame]], which consists of leading axes of an array, the related concept of [[leading axis agreement]], which extends [[scalar]] [[conformability]], and the emphasis on first axis forms of functions while deprecating or discarding other [[function axis|choices of axis]]. | '''Leading axis theory''', or the '''leading axis model''', is an approach to array language design and use that emphasizes working with arrays by manipulating their [[cell]]s and mapping functions over leading [[Axis|axes]] implicitly using [[function rank]] or explicitly using the [[Rank (operator)|Rank operator]]. It was initially developed in [[SHARP APL]] in the early 1980s and is now a major feature of [[J]] and [[Dyalog APL]], as well as languages influenced by these. The name "leading axis" comes from the [[frame]], which consists of leading axes of an array, the related concept of [[leading axis agreement]], which extends [[scalar]] [[conformability]], and the emphasis on first axis forms of functions while deprecating or discarding other [[function axis|choices of axis]]. | ||
== Features == | == Features == | ||
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| Use first axis form only || [[Reverse]], [[Rotate]] (<source lang=apl inline>⊖</source>), [[Replicate]], [[Reduce]] (<source lang=apl inline>⌿</source>), [[Expand]], [[Scan]] (<source lang=apl inline>⍀</source>), [[Catenate]] (<source lang=apl inline>⍪</source>) | | Use first axis form only || [[Reverse]], [[Rotate]] (<source lang=apl inline>⊖</source>), [[Replicate]], [[Reduce]] (<source lang=apl inline>⌿</source>), [[Expand]], [[Scan]] (<source lang=apl inline>⍀</source>), [[Catenate]] (<source lang=apl inline>⍪</source>) | ||
|- | |- | ||
| Extendible to leading axes || [[Take]] (<source lang=apl inline>↑</source>), [[Drop]] (<source lang=apl inline>↓</source>), [[ | | Extendible to leading axes || [[Take]] (<source lang=apl inline>↑</source>), [[Drop]] (<source lang=apl inline>↓</source>), [[Squad Indexing]] (<source lang=apl inline>⌷</source>), [[scalar dyadic]]s, [[Unique]] (<source lang=apl inline>∪</source> or <source lang=apl inline>↑</source>) and most [[set function]]s (<source lang=apl inline>⍳∪∩~</source>) | ||
|- | |- | ||
| Incompatible || [[Split]] (<source lang=apl inline>↓</source>), [[First]] (<source lang=apl inline>↑</source> or <source lang=apl inline>⊃</source>), [[Membership]] (<source lang=apl inline>∊</source>), [[Partition]] (<source lang=apl inline>⊂</source> and <source lang=apl inline>⊆</source>) | | Incompatible || [[Split]] (<source lang=apl inline>↓</source>), [[First]] (<source lang=apl inline>↑</source> or <source lang=apl inline>⊃</source>), [[Membership]] (<source lang=apl inline>∊</source>), [[Partition]] (<source lang=apl inline>⊂</source> and <source lang=apl inline>⊆</source>) | ||
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{|class=wikitable | {|class=wikitable | ||
! Functions !! [[SHARP APL]] !!style="min-width:5em"| [[A+]] !! [[ | ! Functions !! [[SHARP APL]] !! [[Dyalog APL]] !!style="min-width:5em"| [[A+]] !!style="min-width:5em"| [[J]] !!style="min-width:5em"| [[BQN]] | ||
|- | |- | ||
| [[Take]], [[Drop]] || {{Yes|19.0}} | | [[Take]], [[Drop]] || {{Yes|19.0}} || {{Yes|13.0}} || {{Yes}} || {{Yes}} || {{Yes}} | ||
|- | |- | ||
| [[Indexing]] function | | [[Squad Indexing|Indexing]] function || {{Yes|19.0}} || {{Yes|13.0}} || {{Yes}} || {{Yes}} || {{Yes}} | ||
|- | |- | ||
| [[Bracket indexing]] || {{No}} | | [[Bracket indexing]] || {{No}} || {{No}} || {{Yes}} ||style="text-align:center;"| N/A ||style="text-align:center;"| N/A | ||
|- | |- | ||
| [[Scalar dyadic]]s || {{Yes}} | | [[Scalar dyadic]]s || {{Yes}} || {{No}} || {{No}} || {{Yes}} || {{Yes}} | ||
|- | |- | ||
| [[ | | [[Index Of]] || {{No|Incompatible}} || {{Yes|14.0}} || {{Yes}} || {{Yes}} || {{Yes}} | ||
|- | |- | ||
| [[ | | [[Membership]] || {{No|Incompatible}} || {{No|Incompatible}} || {{Yes}} || {{Yes}} || {{Yes}} | ||
|- | |- | ||
| [[Union]], [[Intersection]], [[Without]] || {{No}} | | [[Unique]] || {{Yes}} || {{Yes|17.0}} ||style="text-align:center;"| N/A || {{Yes}} || {{Yes}} | ||
|- | |||
| [[Union]], [[Intersection]], [[Without]] || {{No}} || {{No}} ||style="text-align:center;"| N/A || {{Yes}} ||style="text-align:center;"| N/A | |||
|} | |} | ||
[[Index Of]] in | [[Index Of]] in SHARP APL was not extended to apply to left argument [[major cell]]s as in J and Dyalog; instead it was given rank <source lang=apl inline>1 0</source> making such a change impossible. In A+ not only [[Index Of]] but also [[Membership]] was changed to follow the leading axis model, breaking compatibility with other APLs. A+ also restricts [[Take]] and [[Drop]] to allow only a [[singleton]] left argument (but any right argument rank is permitted). The leading-axis extension of scalar dyadics in SHARP is a direct consequence of giving them a function rank 0, as SHARP's concept of function rank includes [[prefix agreement]]. | ||
== History == | == History == | ||
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Leading axis theory was brought to [[Nested array theory|nested]] APLs by [[Dyalog APL]] in the 2010s after [[Dyalog Ltd.]] employed Hui. Working with [[Jay Foad]] and [[Morten Kromberg]], Hui designed and implemented versions of Rank and other J functionality compatible with Dyalog's nested arrays. | Leading axis theory was brought to [[Nested array theory|nested]] APLs by [[Dyalog APL]] in the 2010s after [[Dyalog Ltd.]] employed Hui. Working with [[Jay Foad]] and [[Morten Kromberg]], Hui designed and implemented versions of Rank and other J functionality compatible with Dyalog's nested arrays. | ||
{{APL features}} | {{APL features}}[[Category:Leading axis theory| ]] | ||