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The leading axis theory as a complete model of programming is best exemplified by [[J]], since it was designed entirely using the theory from the start. While various APLs have developed or adopted features of the leading axis model, [[backwards compatibility]] requirements can prevent them from making certain changes to align with the theory. | The leading axis theory as a complete model of programming is best exemplified by [[J]], since it was designed entirely using the theory from the start. While various APLs have developed or adopted features of the leading axis model, [[backwards compatibility]] requirements can prevent them from making certain changes to align with the theory. | ||
J defines all one-dimensional functions to work on the first [[axis]], that is, to manipulate [[major cell]]s of their arguments. In this way it unifies APL's pairs of first- and last-axis functions and operators including [[Reverse]], [[Rotate]], [[Replicate]], [[Expand]], [[Reduce]], and [[Scan]]. J retains two functions corresponding to < | J defines all one-dimensional functions to work on the first [[axis]], that is, to manipulate [[major cell]]s of their arguments. In this way it unifies APL's pairs of first- and last-axis functions and operators including [[Reverse]], [[Rotate]], [[Replicate]], [[Expand]], [[Reduce]], and [[Scan]]. J retains two functions corresponding to <syntaxhighlight lang=apl inline>,</syntaxhighlight> and <syntaxhighlight lang=apl inline>⍪</syntaxhighlight>, with <syntaxhighlight lang=j inline>,</syntaxhighlight> as [[Ravel]] and [[Catenate First]] and <syntaxhighlight lang=j inline>,.</syntaxhighlight> for second- (rather than last-) axis forms of these functions; <syntaxhighlight lang=j inline>,.</syntaxhighlight> is identical to <syntaxhighlight lang=j inline>,"_1</syntaxhighlight> (which transliterates to APL <syntaxhighlight lang=apl inline>,⍤¯1</syntaxhighlight>) in both valences. | ||
J also extends [[Rotate]] so that it can work on multiple leading axes rather than a single axis: additional values in the left argument apply to leading axes of the right in order. This aligns Rotate with the [[SHARP APL]] extensions to [[Take]], [[Drop]], and [[Squad]] allowing short left arguments: in each case the left argument is a [[vector]] corresponding to axes of the right argument starting at the first. | J also extends [[Rotate]] so that it can work on multiple leading axes rather than a single axis: additional values in the left argument apply to leading axes of the right in order. This aligns Rotate with the [[SHARP APL]] extensions to [[Take]], [[Drop]], and [[Squad]] allowing short left arguments: in each case the left argument is a [[vector]] corresponding to axes of the right argument starting at the first. | ||
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! Compatibility !! Functions | ! Compatibility !! Functions | ||
|- | |- | ||
| Already compatible || [[Grade]] (< | | Already compatible || [[Grade]] (<syntaxhighlight lang=apl inline>⍋</syntaxhighlight>, <syntaxhighlight lang=apl inline>⍒</syntaxhighlight>), [[Decode]] (<syntaxhighlight lang=apl inline>⊥</syntaxhighlight>), [[Encode]] (<syntaxhighlight lang=apl inline>⊤</syntaxhighlight>) | ||
|- | |- | ||
| Use first axis form only || [[Reverse]], [[Rotate]] (< | | Use first axis form only || [[Reverse]], [[Rotate]] (<syntaxhighlight lang=apl inline>⊖</syntaxhighlight>), [[Replicate]], [[Reduce]] (<syntaxhighlight lang=apl inline>⌿</syntaxhighlight>), [[Expand]], [[Scan]] (<syntaxhighlight lang=apl inline>⍀</syntaxhighlight>), [[Catenate]] (<syntaxhighlight lang=apl inline>⍪</syntaxhighlight>) | ||
|- | |- | ||
| Extendible to leading axes || [[Take]] (< | | Extendible to leading axes || [[Take]] (<syntaxhighlight lang=apl inline>↑</syntaxhighlight>), [[Drop]] (<syntaxhighlight lang=apl inline>↓</syntaxhighlight>), [[Squad Indexing]] (<syntaxhighlight lang=apl inline>⌷</syntaxhighlight>), [[scalar dyadic]]s, [[Unique]] (<syntaxhighlight lang=apl inline>∪</syntaxhighlight> or <syntaxhighlight lang=apl inline>↑</syntaxhighlight>) and most [[set function]]s (<syntaxhighlight lang=apl inline>⍳∪∩~</syntaxhighlight>) | ||
|- | |- | ||
| Incompatible || [[Split]] (< | | Incompatible || [[Split]] (<syntaxhighlight lang=apl inline>↓</syntaxhighlight>), [[First]] (<syntaxhighlight lang=apl inline>↑</syntaxhighlight> or <syntaxhighlight lang=apl inline>⊃</syntaxhighlight>), [[Membership]] (<syntaxhighlight lang=apl inline>∊</syntaxhighlight>), [[Partition]] (<syntaxhighlight lang=apl inline>⊂</syntaxhighlight> and <syntaxhighlight lang=apl inline>⊆</syntaxhighlight>) | ||
|- | |- | ||
| Unclear || [[Find]] (< | | Unclear || [[Find]] (<syntaxhighlight lang=apl inline>⍷</syntaxhighlight>) | ||
|- | |- | ||
| Designed for leading axes || [[Rank operator]] (< | | Designed for leading axes || [[Rank operator]] (<syntaxhighlight lang=apl inline>⍤</syntaxhighlight>), [[Tally]] (<syntaxhighlight lang=apl inline>≢</syntaxhighlight>), [[Interval Index]] (<syntaxhighlight lang=apl inline>⍸</syntaxhighlight>), [[Key]] (<syntaxhighlight lang=apl inline>⌸</syntaxhighlight>), [[Raze]] (<syntaxhighlight lang=apl inline>⊃</syntaxhighlight>) | ||
|} | |} | ||
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|} | |} | ||
[[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 < | [[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 <syntaxhighlight lang=apl inline>1 0</syntaxhighlight> 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). | ||
== History == | == History == | ||
Leading axis theory was first developed by | Leading axis theory was first developed by employees of [[I. P. Sharp Associates]] including [[Ken Iverson]], [[Arthur Whitney]], and [[Bob Bernecky]] in the early 1980s: the [[Rank operator]] itself is attributed to Whitney, who invented it while travelling to the [[APL82]] conference.<ref>[[Roger Hui]] and [[Morten Kromberg]]. [https://dl.acm.org/doi/abs/10.1145/3386319 ''APL since 1978'']. ACM [[HOPL]] IV. 2020-06.</ref> It was further developed by Iverson and [[Roger Hui]] when creating the [[J]] language in the 1990s and 2000s; the leading axis model and its various incompatibilities with APL had been a major reason to break with APL and create a new language. | ||
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. | ||
== References == | == References == | ||
<references/> | <references /> | ||
{{APL features}}[[Category:Leading axis theory| ]] | {{APL features}}[[Category:Leading axis theory| ]] | ||