Defined function (traditional)

A user-defined function (or tradfn, for "traditional function", in Dyalog APL) is a function defined using a header that includes the function's name. Introduced in APL\360, function definition was universally supported by APL dialects for much of the language's history, and is still commonly used in mainstream APLs. Since the 1990s other ways to describe functions have appeared, with J and K rejecting function definition in favor of anonymous function description. Beginning in the 2010s Dyalog-based APL dialects including ngn/apl, dzaima/APL, and APL\iv have removed function definition in favor of dfns.

In many dialects the function header syntax of defined functions is adapted to allow user-defined operators as well.

Examples
Ambivalent function with optional left argument and one local variable (Dyalog APL):

GNU APL allows functions and operators accept an axis argument:

Representations
The canonical representation form is equivalent to what the user would type into the line editor to define the function. An alternative representation consists of the entire session log transcript after having such a definition has been made. A tradfn operator can also be called a tradop (pronounced "trad op").

Properties
Tradfns allow both functional and procedural programming, and are essential for object-oriented programming. They support a full set of keywords for flow control and object declarations.

One of the most noticeable differences from dfns is that tradfns must declare their locals, because assignments are global by default. The declaration is done in a header line which also determines the function's name and calling syntax. Dyalog APL adds two extensions to this:

Tradfns cannot be nested but can contain dfns. A tradfn can dynamically create another tradfn by "fixing" its source, and if the function thus created has a name which has been localised, the inner function will only exist the scope of the outer function. Nested functions are less necessary due to tradfns using dynamic scoping as opposed to the lexical scoping of dfns. In other words, a tradfn can "see" locals of its caller.
 * 1) Dynamic localisation of names while a tradfn is running.
 * 2) The declaration of additional local names on separate lines following the header line.

A tradfn can be niladic which causes it to behave syntactically like an array. However, every time its name is referenced, it will run to create a result (if any). Such methods are often used to return a cache or as an entry point for the user.

Dyalog APL extensions
In addition to the two above-mentioned extensions for localising names, Dyalog APL adds a few features to tradfn.

The the header syntax can explicitly specify that a function is ambivalent by enclosing the left argument name in curly braces (for example ). Others dialects treat all functions that declare a left argument name as ambivalent and leave it up to the programmer to check for the presence of a value.

A tradfn can return a function value as result. The returned function will replace the function and its arguments (if any) in the calling expression: A tradfn can be declared as shy, that is, it exhibits the same behaviour as an assignment, in that by default, no result is printed when the function terminates, but attempting to use the result still succeeds. This declaration is done by putting curly braces around the result name (for example ).

The right argument and the result can be a name list instead of single name. The interpreter will unpack the right argument when the function is called, and collect the result when the function returns.

A+
A+ uses a reworked style of function and operator definition than maintains the principle of a header that matches the way the function will be used, but differs in many details:
 * The result name is not included in the header; instead, the result of the last executed statement is returned (and so functions that do not return a result cannot be defined).
 * The header is separated from the body with a colon, and the body of a multi-line function is enclosed in curly braces.
 * Functions have lexical scope. Variables assigned are local by default, and can be made global by enclosing their names in parentheses when assigning.

Comparison to dfns
Wikipedia has a comparison of dfns versus tradfns.

Tutorials

 * APL Cultivation: tradfns
 * Bernard Legrand. Mastering Dyalog APL (page 176). Dyalog Ltd. November 2009.

Documentation

 * Dyalog – Introduction – Functions
 * Dyalog – Defined Functions & Operators
 * APLX