Leading axis theory: Difference between revisions

'''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]].

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 <source lang=apl inline>,</source> and <source lang=apl inline>⍪</source>, with <source lang=j inline>,</source> as [[Ravel]] and [[Catenate First]] and <source lang=j inline>,.</source> for second- (rather than last-) axis forms of these functions; <source lang=j inline>,.</source> is identical to <source lang=j inline>,"_1</source> (which transliterates to APL <source lang=apl inline>,⍤¯1</source>) in both valences.

Because APL was designed before the leading axis model was developed, many APL primitives do not naturally adhere to the theory and some are not compatible with it at all. The following table describes how functions and operators that act on specific axes of their arguments (omitting for example [[Enclose]] and [[Match]] which apply to entire arrays) interact with leading axis theory.

| Already compatible        || [[Grade]] (<source lang=apl inline>⍋</source>, <source lang=apl inline>⍒</source>), [[Decode]] (<source lang=apl inline>⊥</source>), [[Encode]] (<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>), [[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>)

| Unclear                    || [[Find]] (<source lang=apl inline>⍷</source>)

| Designed for leading axes  || [[Rank operator]] (<source lang=apl inline>⍤</source>), [[Tally]] (<source lang=apl inline>≢</source>), [[Interval Index]] (<source lang=apl inline>⍸</source>), [[Key]] (<source lang=apl inline>⌸</source>), [[Raze]] (<source lang=apl inline>⊃</source>)

Backwards-compatible language changes are only effective for functions which are extendible to leading axes. The following extensions have been made in order to support leading axis theory:

! Functions                                !! [[SHARP APL]]      !! [[Dyalog APL]]      !!style="min-width:5em"| [[A+]]  !!style="min-width:5em"| [[J]]

| [[Take]], [[Drop]]                      || {{Yes|19.0}}        || {{Yes|13.0}}        || {{Yes}}                        || {{Yes}}

| [[Indexing]] function                   || {{Yes|19.0}}        || {{Yes|13.0}}        || {{Yes}}                        || {{Yes}}

| [[Bracket indexing]]                    || {{No}}              || {{No}}              || {{Yes}}                        ||style="text-align:center;"| N/A

| [[Scalar dyadic]]s                      || {{Yes}}             || {{No}}              || {{No}}                        || {{Yes}}

| [[Index Of]]                            || {{No|Incompatible}} || {{Yes|14.0}}        || {{Yes}}                        || {{Yes}}

| [[Membership]]                          || {{No|Incompatible}} || {{No|Incompatible}} || {{Yes}}                        || {{Yes}}

| [[Unique]]                              || {{Yes}}            || {{Yes|17.0}}        ||style="text-align:center;"| N/A || {{Yes}}

| [[Union]], [[Intersection]], [[Without]] || {{No}}              || {{No}}              ||style="text-align:center;"| N/A || {{Yes}}

[[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]].

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. 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.

{{APL features}}