Leading axis theory
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 cells and mapping functions over leading 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 choices of axis.
Features
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 onedimensional functions to work on the first axis, that is, to manipulate major cells of their arguments. In this way it unifies APL's pairs of first and lastaxis functions and operators including Reverse, Rotate, Replicate, Expand, Reduce, and Scan. J retains two functions corresponding to ,
and ⍪
, with ,
as Ravel and Catenate First and ,.
for second (rather than last) axis forms of these functions; ,.
is identical to ,"_1
(which transliterates to APL ,⍤¯1
) 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.
The Rank operator is present in every language influenced by leading axis theory. By mapping over leading axes of the arguments, it allows a left operand which works with leading axes of its own arguments to be applied on axes other than the first. The Rank operator is the reason to define functions on the leading axes: by applying Rank to a leadingaxis function, the function can be made to work on any axis or contiguous sequence of axes in the argument.
In J and SHARP APL, every function has a rank defined by the language. For example, scalar functions inherently have rank 0 as they apply only to scalars. J and SHARP APL also define close compositions, which compose two functions while retaining the rank of the first one applied. While the concept of applying with rank makes these features possible, it's unclear whether they are part of leading axis theory or simply a decision made by two languages with a shared heritage. J introduced nonclose compositions, and Dyalog APL has added only nonclose compositions, avoiding introducing function rank or close compositions despite otherwise adhering to leading axis theory.
Adoption in APL
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.
Compatibility  Functions 

Already compatible  Grade (⍋ , ⍒ ), Decode (⊥ ), Encode (⊤ )

Use first axis form only  Reverse, Rotate (⊖ ), Replicate, Reduce (⌿ ), Expand, Scan (⍀ ), Catenate (⍪ )

Extendible to leading axes  Take (↑ ), Drop (↓ ), Squad Indexing (⌷ ), scalar dyadics, Unique (∪ or ↑ ) and most set functions (⍳∪∩~ )

Incompatible  Split (↓ ), First (↑ or ⊃ ), Membership (∊ ), Partition (⊂ and ⊆ )

Unclear  Find (⍷ )

Designed for leading axes  Rank operator (⍤ ), Tally (≢ ), Interval Index (⍸ ), Key (⌸ ), Raze (⊃ )

In addition to these, the Rank operator and scalar dyadics can be extended with leading axis agreement, so that arguments with differentlength frames can match as long as one frame is a prefix of the other. The original introduction of Rank had only an analogue of scalar extension, where an argument with an empty frame is repeated to pair it with an argument with a nonempty frame. The following extensions have been made in order to support leading axis theory:
Functions  SHARP APL  Dyalog APL  A+  J  BQN 

Take, Drop  19.0  13.0  Yes  Yes  Yes 
Indexing function  19.0  13.0  Yes  Yes  Yes 
Bracket indexing  No  No  Yes  N/A  N/A 
Scalar dyadics  No  No  No  Yes  Yes 
Index Of  Incompatible  14.0  Yes  Yes  Yes 
Membership  Incompatible  Incompatible  Yes  Yes  Yes 
Unique  Yes  17.0  N/A  Yes  Yes 
Union, Intersection, Without  No  No  N/A  Yes  N/A 
Index Of in SHARP APL was not extended to apply to left argument major cells as in J and Dyalog; instead it was given rank 1 0
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
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.
Leading axis theory was brought to 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 [edit]  

Builtins  Primitives (functions, operators) ∙ Quad name 
Array model  Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index (Indexing) ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype 
Data types  Number (Boolean, Complex number) ∙ Character (String) ∙ Box ∙ Namespace 
Concepts and paradigms  Leading axis theory ∙ Scalar extension ∙ Conformability ∙ Leading axis agreement ∙ Scalar function ∙ Pervasion ∙ Glyph ∙ Identity element ∙ Complex floor ∙ Total array ordering 
Errors  LIMIT ERROR ∙ RANK ERROR ∙ SYNTAX ERROR ∙ DOMAIN ERROR ∙ LENGTH ERROR ∙ INDEX ERROR ∙ VALUE ERROR 