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...ph]] <syntaxhighlight lang=apl inline>~</syntaxhighlight> with the monadic function [[Not]]. Others can easily define such a function:

...[[set function]] which computes the [[wikipedia:intersection (set theory)|set intersection]] of the two [[vector]] [[argument|arguments]]. Others can easily define such a function:

{{Main|Comparison function}} :''For the set function, see [[Without]].''

...yadic]] [[set function]] which computes the [[wikipedia:union (set theory)|set union]] of the two [[vector]] [[argument|arguments]]. Others can easily define such a function:

A '''primitive function''' is a [[function]] defined by the language. Outside of the array community, such functions m ...of the argument or its subarrays without regard to their values, and [[set function]]s, which return a result or manipulate arrays based on which [[cell]]s or

...lished by [[Ken Iverson]] in 1978, collecting a set of [[operator]]s and [[function]]s he had developed over the preceding years. Released two years before Ive ...ighlight lang=apl inline>~</syntaxhighlight> || || [[Set Difference]]

...in a similar manner because ordinary assignment has the form of a [[dyadic function]] while [[modified assignment]] works like a [[monadic operator]] applied d The following glyphs may be subject to function-operator overloading:

:''This article is about the items of an array. For the set function, see [[Membership]]. ...ther array, and an element of an array can be retrieved using the [[Pick]] function. [[Stranding]] forms a [[vector]] directly from its elements.

| function styles = [[defined function]]s, [[direct definition (operator)]] ...chard More]], and development manager [[Bob Smith]], NARS introduced a new set of functions for working with arrays that contain arrays, as well as many n

...apply inside [[boxes]]. Functions which are not scalar, are called [[mixed function]]s. Only a particular [[valence]] of a function is labelled "scalar". The scalar monad [[Not]] usually shares the glyph <sy

.../10.1145/390007.805334 10.1145/390007.805334]</ref><ref>Berry, Michael. "A set of APL functions to play new Eleusis". IJCAI'81: Proceedings of the 7th int ...unction is not revealed to the Students, nor is an explanation of what the function does.

...''' applies to [[function]] or [[array]] [[operand]]s to yield a [[derived function]] result. Operators bind with higher [[precedence]] than functions and are ...king no operands, because this would be indistinguishable from a [[niladic function]].

| function styles = [[defined function]] ...ted from right to left, much as if the semicolon was the Left [[identity]] function (<syntaxhighlight lang=apl inline>⊣</syntaxhighlight>).

...represents, and usually has a different name: for example, the [[dyadic]] function [[Take]] is represented with the glyph [[Up Arrow]]. ...[wikipedia:ASCII|ASCII]], now considered a "standard" or "basic" character set, but all of them are included in [[Unicode]], as one goal of Unicode was to

...brary for [[SHARP APL]] that translated a direct definition to a [[defined function]]. The argument naming and use of <code>:</code> for a condition later appe ...nymous function]]s. For example, the following expression would define the function <syntaxhighlight lang=apl inline>plus</syntaxhighlight> to add its argument

...name of a 256-[[character]] [[vector]] which contains the entire character set of older APL implementations that did not support [[Unicode]], but instead ...fined character vector containing every member of the ''required character set'' exactly once".<ref>BS ISO/IEC 13751:2001: Information technology. Program

| function styles = [[defined function]] ...0: it is a 16-bit machine rather than 32-bit, with a different instruction set, and has no native floating-point support: instead, Hirondo Kuki's 32-bit f

...nline>⍺</syntaxhighlight> being the character vector representation of the function (<syntaxhighlight lang=apl inline>'{∊⍵⍺⍵⍺,4/⍵}'</syntaxhighligh ...]. This is precisely what the function does. Given the representation of a function <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> (itself in this case

A vector differs from a [[set]] or [[multiset]] in that its elements are ordered (additionally, sets do n ...line>/</syntaxhighlight> in [[J]]) passes entire cells, and performs fewer function calls.

'''Sort Up''' and '''Sort Down''' are [[monadic function]]s that order the [[major cell]]s of an array in ascending or descending or ...xhighlight>) or From function, which for example allows the ascending sort function on any rank to be written as the [[train]] <syntaxhighlight lang=apl inline

| function styles = [[dfn]] ...not from Miscellaneous Technical block, are instead borrowed from Julia's set<ref>https://docs.juliahub.com/CalculusWithJulia/AZHbv/0.0.5/misc/unicode.ht

| function styles = [[tacit]] ...ntended to give Jelly an advantage in [[code golf]], e.g. divmod, the hash function, and string compression. Jelly uses different terminology from normal APL u

...]]s is allowed and actually performs a lamination. In APLs which allow a [[function axis]] for Catenate, Laminate can be invoked by giving a fractional (non-in ...oving the ability to specify an axis, and add a new dedicated [[Laminate]] function.

...primitive operator|primitive]] [[monadic operator]] which takes a [[dyadic function]] [[operand]] where specified keys group the indices or major cells of an a ...and applies the [[function]] operand once for each unique major cell. The function is applied with the unique major cell as left argument, while the right arg

...perator while the behavior of functions with axis must be defined for each function separately. However, [[GNU APL]] implements the syntax and extends it so it The following [[Monadic function|monads]] may allow an axis:

...s play a crucial role for Dyalog APL developers. Dyalog comes with a rich set of user commands, but independent parties also develop and maintain user co Dyalog installs its own set of user commands into <syntaxhighlight lang=apl inline>[DYALOG]/SALT/spice<

...rse]]/[[Rotate]] (<syntaxhighlight lang=apl inline>⌽</syntaxhighlight>), [[Function axis]] | 1974 || [[APL/700]] || [[Union]], [[Intersection]], [[Set Difference]] (but these functions only return unique elements), [[modified

...f>) and '''Fit''' in [[J]], is a special [[dyadic operator]] which takes a function as its left [[operand]] and modifies its behaviour as specified by the arra ...his time in the form of a dyadic operator that would customise [[primitive function]]s.

| function styles = [[tradfn]], [[dfn]], [[tacit]] * All set functions support arguments of any rank.

...pth, a number associated with every array. For the operator that applies a function at a given depth, see [[Depth (operator)]].'' {{Built-in|Depth|≡}} is a [[monadic]] [[primitive function]] that returns an array's depth. In the APL [[array model]], the '''depth''

...e First" or an equivalent. Replicate is a widely-accepted extension of the function '''Compress''', which requires the number of copies to be [[Boolean]]: each ...etimes resolved by making Replicate an operator itself, and sometimes by [[function-operator overloading]] allowing both syntactic elements to coexist.

| [[Replicate]] [[function]] | [[Expand]] [[function]]

...as the [[Commute]] and [[Rank (operator)|Rank]] operators, the [[Identity function]]s, and [[GCD]] and [[LCM]] extending Or and And. ...hile others, such as [[APL+Win]] and [[Dyalog]], have a slightly different set of [[quad name]]s and provide the [[migration level]] functionality to impr

...ntax, are always considered the most important to maintain, while [[system function]]s are often handled more loosely and [[I-beam]]s or [[user command]]s may ...ce to appear. The traditional way to do so was to insert a line with one a function on it. The [[FinnAPL idiom library]] includes

...ilt-in|Nub Sieve|≠}}, '''Nubsieve''', or '''Unique Mask''', is a [[monadic function]] that returns a [[Boolean]] [[vector]] indicating which [[major cells]] of ...Unique Mask can also be derived from the [[self-classify]] or index-in-nub function <syntaxhighlight lang=apl inline>(∪⍳1/⊢)</syntaxhighlight>. The follo

{{Built-ins|Unique|∪|↑}}, or '''Nub''', is a [[monadic]] [[set function]] which removes duplicate [[major cells]] from an array. It returns only th ...[Ken Iverson|Iverson]]'s 1978 [[Operators and Functions]] along with other set functions including [[Union]], [[Intersection]], and [[Difference]]. There

| function styles = [[defined function]] ...\3000 followed the [[APL.SV]] model closely, and featured exactly the same set of primitives. However, it featured the new implementation techniques of a

:''This page is about the array theory concept. See [[Index (function)]], [[Indexing]], [[Indices]], [[Index Generator]], [[Index of]], and [[Int ...the ordering within each axis. One definition of an array is simply a map (function) which specifies an element for each index, where the possible indices are

{{Built-in|Indices|⍸}}, or '''Where''', is a [[monadic]] [[primitive function]] which returns the [[Index|indices]] of all ones in a [[Boolean]] array. M ...the argument by the number of times it appears. It is identical to the APL function:

Some modern versions of APL include a set of keywords for [[wikipedia:Control flow|controlling flow]] of execution. S ...control structures defined by these keywords is only allowed in [[defined function]]s.

{{Built-in|Format|⍕}} is an [[ambivalent]] [[primitive function]] which formats the right [[argument]] into a [[simple]] [[character]] arra ...pports '''Format by example''', which uses string left argument for a rich set of format specifications:

The relevant document for the APL character set is the ''APL Character Repertoire'' (ACR)<ref>ISO-IEC/JTC1/SC22/WG3. [http: ...</ref> was that in some implementations the symbol for the logical [[And]] function was U+005E only, in some implementations it was U+2227 only, and in some bo

...namespace, namespaces serve as a way to combine the context for a set of [[function]]s or other operations without interfering with other parts of a program th ...as a form of name scope (IBM used the term "namescope" for a workspace or function scope) that could be manipulated directly by the programmer.<ref name="swai

...PL implementations dropped the closing symbol for consistency with monadic function syntax. ...nion]] are written with <math>\cap</math> and <math>\cup</math>, and the [[Set Difference]] with <math>\Delta</math>.

...view of all forms; in particular [[dfn]]s are considered a type of defined function in [[Dyalog APL]] and [[GNU APL]].'' ...with [[J]] and [[K]] rejecting function definition in favor of [[anonymous function]] description.

<syntaxhighlight lang=apl inline>FixSimpleXaml</syntaxhighlight> is a function used to execute the XAML and return the root element as a .Net object. All ...button1''' (line 22) are attached automatically to the root element by the function <syntaxhighlight lang=apl inline>FixSimpleXaml</syntaxhighlight>:

...those that result from [[Bracket indexing|indexing]] that array by a full set of [[index]] arrays each of which is either a [[scalar]] or a [[vector]] of The [[Find]] (<syntaxhighlight lang=apl inline>⍷</syntaxhighlight>) function searches for occurrences of the left argument as a subarray of the right ar

| function styles = [[Defined function|defined]] ...soft Windows, Linux, and macOS. Though APLX keeps APL's extended character set, APLX is a bit more verbose, due to the prevalence of system functions with

[[File:20190910-02-ML.jpg|thumb|right|[[Boolean function|Boolean algorithm]]s at [[Dyalog '19]].]] ...its deficiencies. Uses Unicode [[glyph]]s, but has a different character set than APL.

...nar]] on 20th September 2020, [[Ray Polivka]] described a programming test set in the 1950s in the course "Programs1" in the [[wikipedia:ILLIAC|ILLIAC]] w ...ntaxhighlight lang=apl inline>Shirley</syntaxhighlight> is a [[traditional function]] which one might have [[APL\360|written in 1970]] (except for the [[Mix]]

...s on first axis forms of functions while deprecating or discarding other [[function axis|choices of axis]]. ...ions on the leading axes: by applying Rank to a leading-axis function, the function can be made to work on any axis or contiguous sequence of axes in the argum

...tml emacs APL tutorial] provides easy-to-follow instructions on how to get set up using GNU APL or Dyalog APL.

...tion you want, but they tend to use 1 by default. To switch convention, we set the variable <syntaxhighlight lang=apl inline>⎕IO</syntaxhighlight>: ...or|generates indices]] until is has made ''N'' [[Index|indices]]. Since we set <syntaxhighlight lang=apl inline>⎕IO</syntaxhighlight> to 0, we count fro

...axhighlight lang=apl inline>⌊</syntaxhighlight> with the dyadic arithmetic function [[Minimum]]. [[Comparison_with_traditional_mathematics#Prefix|Traditional m ⎕pp←34 ⍝ set to max as we are using strings, the execute and format primitives round the

The simplest way to set up an APL keyboard on Linux is with the following <syntaxhighlight lang=con X11 and Wayland use XKB rules to determine their layout. To set a layout create a file called <syntaxhighlight lang=console inline>00-keybo

| <syntaxhighlight lang=apl inline>)FNS</syntaxhighlight> || Lists global function names. | <syntaxhighlight lang=apl inline>)wsid {Y}</syntaxhighlight> || Query / Set the identity of the active workspace.

* [https://code.jsoftware.com/wiki/Essays/Mertens_Function Mertens Function] * [https://code.jsoftware.com/wiki/Essays/Power_Set Power Set]

[[File:Function compositions.png|thumb|right|Diagrams of [[function composition]]s, an important part of tacit programming.]] ...]]s following a small set of rules. This allows creating complex [[derived function]]s without specifying any arguments explicitly.

...al argument an infinite list of all the results obtained by iterating that function (as with the [[Power operator]]) on the argument. [[Eugene McDonnell]] cite ...ly in every program — the primitive functions form a harmonious and useful set.<ref name="lyrical">Perlis, Alan. [http://www.jsoftware.com/papers/perlis77

| function styles = [[tradfn]], [[dfn]], [[tacit]] ...ot derived from Dyalog such as [[GNU APL]] and [[NARS2000]], [[dfn]]-style function syntax has become common, and Dyalog has also popularised [[SHARP APL]] and

...for consistency. The latter representation matches [[K]]'s [[Cut (K)|Cut]] function, which produces a complete partition exactly when the left argument contain This style of partition function is slightly harder to understand and implement, but does have advantages:

| function styles = [[tradfn]], [[dfn]], [[tacit]] ...conversion]] as a <syntaxhighlight lang=apl inline>⎕C</syntaxhighlight> [[function]] rather than the proposed [[operator]]. Dyalog is also planning to add [[R

...lled by the name of its [[glyph]] '''Iota''', is a [[monadic]] [[primitive function]] which returns an array of [[index|indices]] with [[shape]] given by the r ...that is, it matches Iota of its [[bound]], [[reshape]]d. A+ refers to this function as '''Interval''' while J calls it '''Integers'''.

| function styles = block (like [[dfn]]s), [[tacit]] ...with [[wikipedia:first-class function|first-class function]]s, uses a new set of [[glyph]]s with different primitive pairings, and implements [[character

...generation of a population "evolves" from the previous one according to a set of predefined rules. The Game of Life is defined on an infinite [[Boolean]] ...me, so that the function makes little use of APL's array capabilities. The function was followed by a 9-line implementation in February 1972<ref>Bruce A. Beebe

* The song [[APL Blossom Time]] includes numerous references to [[primitive function|primitives]]. ...tually speaks about forcing function/operator [[hybrid]]s into acting as [[function]]s by either giving them a (constant) array left-[[argument]] or letting th

...ighlight> was added for better integration with .NET. Another new [[system function]], <syntaxhighlight lang=apl inline>⎕NLOCK</syntaxhighlight>, was added f ...the work of creating a hash table for that argument every time the search function was called after the first. The APLScript mechanism, allowing source files

...and Mesh is similar to that between [[Compress]] and [[Expand]]. A related function also named Mesh is implemented in [[NARS]] and [[NARS2000]] as a case of [[ ...Here the expressions which use [[indexing]] must have the [[index origin]] set to 0.

# <syntaxhighlight lang=apl inline>⎕LX</syntaxhighlight> is set to run function <syntaxhighlight lang=apl inline>Run</syntaxhighlight> # This function calls 3 sub-functions: <syntaxhighlight lang=apl inline>Initial</syntaxhigh

...w [[Dyalog APL Vision]], changing some proposals. Notable are the [[system function]]s for creating, updating, and unpacking [[namespace]]s, because this part ...log/2018/09/enhanced-debugging-with-function-keys/ Enhanced Debugging with Function Keys]

...ritten <syntaxhighlight lang=apl inline>></syntaxhighlight>, and given a [[function rank]] of 0, which causes it to behave as [[Mix]]. [[First]] is not a primi == Function definition ==

...rnate implementations, in his [[dfns workspace]]. The workspace includes a function which differs in a few small details. In this page we will break down the function <syntaxhighlight lang=apl inline>Life</syntaxhighlight> to see how each par

* [[APL84]]: [https://doi.org/10.1145/800058.801080 Function assignment and arrays of functions]. ...PL93]]: [https://doi.org/10.1145/166197.166203 Extending the APL character set]. With Brent Hawks and Ray Trimble.

| function styles = [[Anonymous function|explicit]], [[tacit]] ...use of terms from English grammar to describe the language. For example, [[function]]s are called "verbs" while arrays are called "nouns".

<syntaxhighlight lang=apl inline>@</syntaxhighlight> applies a function or substitutes elements ''[[at]]'' specific locations. ...nadically, it can be called ''[[self]]ie'', in that it applies the operand function to the argument with the argument itself as left argument too.

* [https://code.jsoftware.com/wiki/Essays/Ackermann%27s_Function "Ackermann's Function"] ...at APL is its use of an expanded set of operations relative to the typical set used in computer science research. It's true that APL dedicates much of its

...tement (namely [[function axis]]/[[bracket indexing]] and [[precedence]]/[[function train]]s). Thus, while <syntaxhighlight lang=apl inline>2 ⟦3⟧</syntaxhi The overloading of square brackets, currently in use only for [[function axis]] and [[bracket indexing]], to mean a higher-rank array, poses a probl

...00]] processor. Zilog wanted an APL for the Z8000 because of a requirement set by the Swedish Ministry of Defence for Unix suppliers. ...hired to begin work on a graphical user interface, supporting the [[system function]] <syntaxhighlight lang=apl inline>⎕WC</syntaxhighlight>.

|rowspan=2| 175. || Gamma function ||style="text-align: right;"|<syntaxhighlight lang=apl inline>X←D0</synta |rowspan=2| 177. || Beta function ||style="text-align: right;"|<syntaxhighlight lang=apl inline>X←D0; Y←D

* '''A Set of APL Programs for Use in Network Theory'''. Paul Penfield, Jr. ([[wikiped * '''The variety of definitions of a simple function'''. [[Eugene McDonnell|E. E. McDonnell]].

* '''D06: Function Trains for Dyalog APL''', [[John Scholes]] * '''D09: Performance Improvements in Set Operations''', Karta Kooner ([https://www.dyalog.com/uploads/conference/dya