{"history":[],"article":{"id":"9c0ac194-5bd6-475b-afad-c08a13bd4a87","title":"APL syntax and symbols","slug":"apl-syntax-and-symbols","content":":::summary\n:::\n\n# \nThe programming language [APL](https://bliptext.com/articles/apl-programming-language) is distinctive in being symbolic rather than lexical: its primitives are denoted by symbols, not words.\nThese symbols were originally devised as a [mathematical notation](https://bliptext.com/articles/mathematical-notation) to describe algorithms.\nAPL programmers often assign informal names when discussing functions and operators (for example, \"product\" for ×/) but the core functions and operators provided by the language are denoted by non-textual symbols.\n\n# Monadic and dyadic functions\nMost symbols denote functions or operators.\nA monadic function takes as its argument the result of evaluating everything to its right.\n(Moderated in the usual way by parentheses.) A dyadic function has another argument, the first item of data on its left.\nMany symbols denote both monadic and dyadic functions, interpreted according to use.\nFor example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments.\n\n# Functions and operators\nAPL uses the term operator in [Heaviside](https://bliptext.com/articles/oliver-heaviside)’s sense as a moderator of a function as opposed to some other programming language's use of the same term as something that operates on data, ref.\n[relational operator](https://bliptext.com/articles/relational-operator) and [operators generally](https://bliptext.com/articles/operator-computer-programming).\nOther programming languages also sometimes use this term interchangeably with function, however both terms are used in APL more precisely.\nEarly definitions of APL symbols were very specific about how symbols were categorized.\nFor example, the operator reduce is denoted by a forward slash and reduces an array along one axis by interposing its function operand.\nAn example of [reduce](https://bliptext.com/articles/fold-higher-order-function):\nIn the above case, the **reduce** or **slash** operator moderates the multiply function.\nThe expression **×/2 3 4** evaluates to a scalar (1 element only) result through **reducing** an array by multiplication.\nThe above case is simplified, imagine multiplying (adding, subtracting or dividing) more than just a few numbers together.\n(From a vector, **×/** returns the product of all its elements.)\nThe above dyadic functions examples [left and right examples] (using the same **/** symbol, right example) demonstrate how [Boolean](https://bliptext.com/articles/boolean-algebra) values (0s and 1s) can be used as left arguments for the **\\ expand** and **/** replicate functions to produce exactly opposite results.\nOn the left side, the **2-element** [vector](https://bliptext.com/articles/array-data-structure) {45 67} is **expanded** where Boolean 0s occur to result in a **3-element** [vector](https://bliptext.com/articles/array-data-structure) {45 0 67} — note how APL inserted a 0 into the [vector](https://bliptext.com/articles/array-data-structure).\nConversely, the exact opposite occurs on the right side — where a 3-element vector becomes just 2-elements; Boolean 0s delete items using the dyadic **/ slash** function.\nAPL symbols also operate on [lists](https://bliptext.com/articles/enumeration) (vector) of items using data types other than just numeric, for example a 2-element vector of character strings {\"Apples\" \"Oranges\"} could be substituted for numeric vector {45 67} above.\n\n# Syntax rules\nIn APL the [precedence hierarchy](https://bliptext.com/articles/order-of-operations) for functions or operators is strictly positional: expressions are evaluated right-to-left.\nAPL does not follow the usual [operator precedence](https://bliptext.com/articles/operator-precedence) of other programming languages; for example, does not bind its operands any more \"tightly\" than.\nInstead of operator precedence, APL defines a notion of scope.\nThe scope of a [function](https://bliptext.com/articles/first-class-function) determines its [arguments](https://bliptext.com/articles/parameter-computer-programming).\nFunctions have long right scope: that is, they take as right arguments everything to their right.\nA dyadic function has short left scope: it takes as its left arguments the first piece of data to its left.\nFor example, (leftmost column below is actual [program code](https://bliptext.com/articles/source-code) from an APL [user session](https://bliptext.com/articles/session-computer-science), indented = actual [user input](https://bliptext.com/articles/user-interface), not-indented = result returned by [APL interpreter](https://bliptext.com/articles/apl-programming-language)):\n<< First note there are no parentheses and\nAPL is going to execute from right-to-left.\nStep 1{of topmost APL code entered at left}) 4-5 = -1.\nStep 2) 3 times -1 = -3.\nStep 3) Take the **floor** or **lower** of 2 and -3 = -3.\nStep 4) Divide 1 by -3 = -0.3333333333 = final result.\nAn operator may have function or data [operands](https://bliptext.com/articles/operand) and evaluate to a dyadic or monadic function.\nOperators have long left scope.\nAn operator takes as its left operand the longest function to its left.\nFor example:\n∘.=/⍳¨3 3 1 0 0 0 1 0 0 0 1\nAPL [atomic](https://bliptext.com/articles/linearizability) or piecemeal sub-analysis (**full explanation**):\nBeginning rightmost: ⍳¨3 3 produces a 2-element nested APL vector { {1 2 3} {1 2 3} } where each element is itself a vector {1 2 3}.\n**Iota ⍳3** by itself would produce {1 2 3}.\nThe **diaeresis ¨** or mini double-dot means repeat or **over each** or perform each separately so **iota repeats** (in human i.e., reversed terms, the APL interpreter reads 3 3 **over each** use iota), concisely: **iota for each 3**.\nThe left operand for the **over-each** operator is the **index ⍳** function.\nThe derived function is used monadically and takes as its right operand the vector.\nThe left scope of **each** is terminated by the **reduce** operator, denoted by the forward **slash**.\nIts left operand is the function expression to its left: the **outer product** of the **equals** function.\nThe result of ∘.=/ is a monadic function.\nWith a function's usual long right scope, it takes as its right argument the result of ⍳¨3 3.\nThus\n**(⍳3)(⍳3)** 1 2 3 1 2 3 **(⍳3)∘.=⍳3** 1 0 0 0 1 0 0 0 1 ⍳¨3 3 1 2 3 1 2 3 **∘.=/⍳¨3 3** 1 0 0 0 1 0 0 0 1\nEquivalent results in APL: and << Rightmost expression is more concise.\nThe matrix of 1s and 0s similarly produced by and is called an [identity matrix](https://bliptext.com/articles/identity-matrix).\nIdentity matrices are useful in solving [matrix determinants](https://bliptext.com/articles/determinant), groups of [linear equations](https://bliptext.com/articles/system-of-linear-equations) and [multiple regression](https://bliptext.com/articles/regression-analysis).\nSome APL interpreters support the **[compose](https://bliptext.com/articles/composition-operator) operator ∘** and the **[commute](https://bliptext.com/articles/commuting-matrices) operator ⍨**.\nThe former **∘ glues** functions together so that **[foo∘bar](https://bliptext.com/articles/foobar)**, for example, could be a hypothetical function that applies defined function foo to the result of defined function bar; foo and bar can represent any existing function.\nIn cases where a dyadic function is moderated by **commute** and then used monadically, its right argument is taken as its left argument as well.\nThus, a **derived** or **composed** function (named im at left) is used in the APL user session to return a 9-[element](https://bliptext.com/articles/element-mathematics) identity matrix using its right [argument](https://bliptext.com/articles/argument-of-a-function), [parameter](https://bliptext.com/articles/parameter-computer-programming) or operand = 3.\nExample using APL to **index ⍳** or **find** (or not **find**) elements in a character vector :\nFirst, variable **Letters** is assigned a vector of 5-elements, in this case - letters of the alphabet.\nThe **shape ⍴** or character vector-length of **Letters** is 5.\nVariable **FindIt** is assigned what to **search for** in **Letters** and its length is 4 characters.\n1 2 3 4 5 << vector positions or index #'s in **Letters**\nABCDE\nAt left, dyadic function **iota searches** through its left argument(Letters) for the search string (iota's right argument, FindIt).\nIota finds letter \"C\" at position 3 in Letters, it finds \"A\" at position 1, and \"B\" at position 2.\n**Iota** does **not find** letter \"S\"\nanywhere in variable Letters so it returns the number 6 which is **1 greater than the length** of Letters.\n**Iota** found letters\n\"CAB\" (3 1 2).\n**Iota** correctly did **not find** \"S\" (6).\n\n# Monadic functions\n\n# Dyadic functions\n\n# Operators and axis indicator\n**Notes:** The reduce and scan operators expect a dyadic function on their left, forming a monadic composite function applied to the vector on its right.\nThe product operator \".\"\nexpects a dyadic function on both its left and right, forming a dyadic composite function applied to the vectors on its left and right.\nIf the function to the left of the dot is \"∘\" (signifying null) then the composite function is an outer product, otherwise it is an inner product.\nAn inner product intended for conventional matrix multiplication uses the + and × functions, replacing these with other dyadic functions can result in useful alternative operations.\nSome functions can be followed by an axis indicator in (square) brackets, i.e., this appears between a function and an array and should not be confused with array subscripts written after an array.\nFor example, given the ⌽ (reversal) function and a two-dimensional array, the function by default operates along the last axis but this can be changed using an axis indicator:\n4 rows by 3 cols matrix created, using **rho ⍴** and **iota ⍳**.\nThe 4 x 3 matrix is then stored in a [variable](https://bliptext.com/articles/variable-computer-science) named **A**.\n**A** is now reflected or flipped along its vertical axis as **symbol ⌽** visually indicates.\n**A** is now reflected using the **[1] axis indicator** or **first dimension modifier**.\nThe result is that variable A has been reflected across the horizontal axis, instead of vertically.\n**A** is now reflected both **vertically ⊖** and **horizontally ⌽**.\n**A** is **⍉ transposed** to a 3 row by 4 col matrix such that rows-cols become exchanged, as **symbol ⍉** visually portrays.\nCompare the result here to the original matrix stored in A, topmost matrix.\nThese types of data transformations are useful in [time series](https://bliptext.com/articles/time-series) analysis and [spatial coordinates](https://bliptext.com/articles/rotation-mathematics), just two examples, [more exist](https://bliptext.com/articles/data-transformation).\nAs a particular case, if the dyadic **catenate** **\",\"** function is followed by an axis indicator (or axis modifier to a symbol/function), it can be used to laminate (interpose) two arrays depending on whether the axis indicator is less than or greater than the [index origin](https://bliptext.com/articles/zero-based-numbering) ([index origin](https://bliptext.com/articles/zero-based-numbering) = 1 in illustration below):\nAt left, variable 'B' is first assigned a vector of 4 consecutive integers (e.g., **⍳4**).\nVar **C** is then assigned 4 more consecutive integers (such as **4+⍳4**).\n'B' and **C** are then **concatenated** or **raveled** together for illustration purposes, resulting in a single vector (**⍳8**).\nIn the particular case at left, if the dyadic **catenate \",\"** function is followed by an **axis indicator** (**[0.5]** which is less than 1), it can be used to **laminate** (**interpose**) two arrays (vectors in this case) depending on whether the **axis indicator** is less than or greater than the index origin(1).\nThe first result (of **B,[0.5]**C****) is a 2 row by 4 column matrix, vertically joining 'B' and **C** row-wise.\nThe second result (of **B,[1.5]C** which is greater than 1) is a 4 row by 2 column matrix.\n\n# Nested arrays\n[Arrays](https://bliptext.com/articles/array-data-structure) are structures which have elements grouped linearly as [vectors](https://bliptext.com/articles/vector-mathematics-and-physics) or in table form as [matrices](https://bliptext.com/articles/matrix-mathematics)—and higher dimensions (3D or cubed, [4D or cubed over time](https://bliptext.com/articles/four-dimensional-space), etc.).\nArrays containing both characters and numbers are termed mixed arrays.\nArray structures containing elements which are also arrays are called nested arrays.\n\n# Flow control\nA user may define custom functions which, like variables, are identified by name rather than by a non-textual symbol.\nThe function header defines whether a custom function is niladic (no arguments), monadic (one right argument) or dyadic (left and right arguments), the local name of the result (to the left of the ← assign arrow), and whether it has any local variables (each separated by semicolon ';').\nWhether functions with the same identifier but different [adicity](https://bliptext.com/articles/adicity) are distinct is implementation-defined.\nIf allowed, then a function CURVEAREA could be defined twice to replace both monadic CIRCLEAREA and dyadic SEGMENTAREA above, with the monadic or dyadic function being selected by the context in which it was referenced.\nCustom dyadic functions may usually be applied to parameters with the same conventions as built-in functions, i.e., arrays should either have the same number of elements or one of them should have a single element which is extended.\nThere are exceptions to this, for example a function to convert pre-decimal UK currency to dollars would expect to take a parameter with precisely three elements representing pounds, shillings and pence.\nInside a program or a custom function, control may be conditionally transferred to a statement identified by a line number or explicit label; if the target is 0 (zero) this terminates the program or returns to a function's caller.\nThe most common form uses the APL compression function, as in the template (condition)/target which has the effect of evaluating the condition to 0 (false) or 1 (true) and then using that to mask the target (if the condition is false it is ignored, if true it is left alone so control is transferred).\nHence function SEGMENTAREA may be modified to abort (just below), returning zero if the parameters (DEGREES and RADIUS below) are of different sign:\nThe above function SEGMENTAREA works as expected if the parameters are scalars or single-element **arrays**, but **not** if they are multiple-element **arrays** since the condition ends up being based on a single element of the SIGN array - on the other hand, the user function could be modified to correctly handle vectorized arguments.\nOperation can sometimes be unpredictable since APL defines that computers with vector-processing capabilities should parallelise and may reorder array operations as far as possible - thus, **test and debug** user functions particularly if they will be used with vector or even matrix arguments.\nThis affects not only explicit application of a custom function to arrays, but also its use anywhere that a dyadic function may reasonably be used such as in generation of a table of results:\nA more concise way and sometimes better way - to formulate a function is to avoid explicit transfers of control, instead using expressions which evaluate correctly in all or the expected conditions.\nSometimes it is correct to let a function fail when one or both **input** arguments are **incorrect** - precisely to let user know that one or both arguments used were **incorrect**.\nThe following is more concise than the above SEGMENTAREA function.\nThe below importantly **correctly** handles vectorized arguments:\nAvoiding explicit transfers of control also called branching, if not reviewed or carefully controlled - can promote use of excessively complex one liners, veritably \"misunderstood and complex idioms\" and a \"write-only\" style, which has done little to endear APL to influential commentators such as [Edsger Dijkstra](https://bliptext.com/articles/edsger-dijkstra).\nConversely however APL idioms can be fun, educational and useful - if used with helpful **comments ⍝**, for example including source and intended meaning and function of the idiom(s).\nHere is an [APL idioms list](https://bliptext.com/articles/undefined), an [IBM APL2 idioms list here](https://bliptext.com/articles/undefined) and [Finnish APL idiom library here](https://bliptext.com/articles/undefined).\n\n# Miscellaneous\nMost APL implementations support a number of system variables and functions, usually preceded by the **⎕ (quad)** and/or **\")\"** (**hook**=close parenthesis) character. Note that the quad character is not the same as the [Unicode missing character symbol](https://bliptext.com/articles/specials-unicode-block). Particularly important and widely implemented is the ⎕IO ([Index Origin](https://bliptext.com/articles/zero-based-numbering)) variable, since while the original IBM APL based its arrays on 1 some newer variants base them on zero:\nThere are also system functions available to users for saving the current workspace e.g., **)SAVE** and terminating the APL environment, e.g., **)OFF** - sometimes called hook commands or functions due to the use of a leading right parenthesis or hook.\nThere is some standardization of these quad and hook functions.\n\n# Fonts\nThe Unicode [Basic Multilingual Plane](https://bliptext.com/articles/mapping-of-unicode-character-planes) includes the APL symbols in the [Miscellaneous Technical](https://bliptext.com/articles/miscellaneous-technical) block, which are thus usually rendered accurately from the larger Unicode fonts installed with most modern operating systems.\nThese fonts are rarely designed by typographers familiar with APL glyphs.\nSo, while accurate, the glyphs may look unfamiliar to APL programmers or be difficult to distinguish from one another.\nSome Unicode fonts have been designed to display APL well: APLX Upright, APL385 Unicode, and SimPL.\nBefore Unicode, APL interpreters were supplied with fonts in which APL characters were mapped to less commonly used positions in the ASCII character sets, usually in the upper 128 code points.\nThese mappings (and their national variations) were sometimes unique to each APL vendor's interpreter, which made the display of APL programs on the Web, in text files and manuals - frequently problematic.\n\n# APL2 keyboard function to symbol mapping\nNote the APL On/Off Key - topmost-rightmost key, just below.\nAlso note the keyboard had some 55 unique (68 listed per tables above, including comparative symbols but several symbols appear in both monadic and dyadic tables) APL symbol keys (55 APL functions (operators) are listed in IBM's 5110 APL Reference Manual), thus with the use of alt, shift and ctrl keys - it would theoretically have allowed a maximum of some **59** (keys) ***4** (with 2-key pressing) ***3** (with tri-key pressing, e.g., ctrl-alt-del) or some 472 different maximum key combinations, approaching the 512 [EBCDIC](https://bliptext.com/articles/ebcdic) character max (256 chars times 2 codes for each keys-combination).\nAgain, in theory the keyboard pictured here would have allowed for about 472 different APL symbols/functions to be keyboard-input, actively used.\nIn practice, early versions were only using something roughly equivalent to 55 APL special symbols (excluding letters, numbers, punctuation, etc. keys).\nThus, early APL was then only using about 11% (55/472) of a symbolic language's at-that-time utilization potential, based on keyboard # keys limits, again excluding numbers, letters, punctuation, etc. In another sense keyboard symbols utilization was closer to 100%, highly efficient, since EBCDIC only allowed 256 distinct chars, and [ASCII](https://bliptext.com/articles/ascii) only 128.\n# [APL2 Keyboard | upload.wikimedia.org/wikipedia/commons/6/6d/APL2-nappaimisto.png]\n\n# Solving puzzles\nAPL has proved to be extremely useful in solving mathematical puzzles, several of which are described below.\n\n## Pascal's triangle\nTake [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), which is a triangular array of numbers in which those at the ends of the rows are 1 and each of the other numbers is the sum of the nearest two numbers in the row just above it (the apex, 1, being at the top).\nThe following is an APL one-liner function to visually depict Pascal's triangle:\n\n## Prime numbers, contra proof via factors\nDetermine the number of [prime numbers](https://bliptext.com/articles/list-of-prime-numbers) (prime # is a natural number greater than 1 that has no positive divisors other than 1 and itself) up to some number N. [Ken Iverson](https://bliptext.com/articles/kenneth-e-iverson) is credited with the following one-liner APL solution to the problem:\nExamining the converse or opposite of a mathematical solution is frequently needed ([integer factors of a number](https://bliptext.com/articles/integer-factorization)): Prove for the subset of integers from 1 through 15 that they are **non-prime** by listing their decomposition factors.\nWhat are their non-one factors (#'s divisible by, except 1)?\n\n## Fibonacci sequence\nGenerate a [Fibonacci number](https://bliptext.com/articles/fibonacci-number) sequence, where each subsequent number in the sequence is the sum of the prior two:\n\n## Generic online tutorials\n\n## Syntax rules","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T06:56:26.112Z","updated_at":"2025-01-26T06:56:26.112Z","search_vector":"'apl':1 'symbol':4 'syntax':2","search_content":" \n \n\n# \nThe programming language [APL](https://bliptext.com/articles/apl-programming-language) is distinctive in being symbolic rather than lexical: its primitives are denoted by symbols, not words.\nThese symbols were originally devised as a [mathematical notation](https://bliptext.com/articles/mathematical-notation) to describe algorithms.\nAPL programmers often assign informal names when discussing functions and operators (for example, \"product\" for ×/) but the core functions and operators provided by the language are denoted by non-textual symbols.\n\n# Monadic and dyadic functions\nMost symbols denote functions or operators.\nA monadic function takes as its argument the result of evaluating everything to its right.\n(Moderated in the usual way by parentheses.) A dyadic function has another argument, the first item of data on its left.\nMany symbols denote both monadic and dyadic functions, interpreted according to use.\nFor example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments.\n\n# Functions and operators\nAPL uses the term operator in [Heaviside](https://bliptext.com/articles/oliver-heaviside)’s sense as a moderator of a function as opposed to some other programming language's use of the same term as something that operates on data, ref.\n[relational operator](https://bliptext.com/articles/relational-operator) and [operators generally](https://bliptext.com/articles/operator-computer-programming).\nOther programming languages also sometimes use this term interchangeably with function, however both terms are used in APL more precisely.\nEarly definitions of APL symbols were very specific about how symbols were categorized.\nFor example, the operator reduce is denoted by a forward slash and reduces an array along one axis by interposing its function operand.\nAn example of [reduce](https://bliptext.com/articles/fold-higher-order-function):\nIn the above case, the **reduce** or **slash** operator moderates the multiply function.\nThe expression **×/2 3 4** evaluates to a scalar (1 element only) result through **reducing** an array by multiplication.\nThe above case is simplified, imagine multiplying (adding, subtracting or dividing) more than just a few numbers together.\n(From a vector, **×/** returns the product of all its elements.)\nThe above dyadic functions examples [left and right examples] (using the same **/** symbol, right example) demonstrate how [Boolean](https://bliptext.com/articles/boolean-algebra) values (0s and 1s) can be used as left arguments for the **\\ expand** and **/** replicate functions to produce exactly opposite results.\nOn the left side, the **2-element** [vector](https://bliptext.com/articles/array-data-structure) {45 67} is **expanded** where Boolean 0s occur to result in a **3-element** [vector](https://bliptext.com/articles/array-data-structure) {45 0 67} — note how APL inserted a 0 into the [vector](https://bliptext.com/articles/array-data-structure).\nConversely, the exact opposite occurs on the right side — where a 3-element vector becomes just 2-elements; Boolean 0s delete items using the dyadic **/ slash** function.\nAPL symbols also operate on [lists](https://bliptext.com/articles/enumeration) (vector) of items using data types other than just numeric, for example a 2-element vector of character strings {\"Apples\" \"Oranges\"} could be substituted for numeric vector {45 67} above.\n\n# Syntax rules\nIn APL the [precedence hierarchy](https://bliptext.com/articles/order-of-operations) for functions or operators is strictly positional: expressions are evaluated right-to-left.\nAPL does not follow the usual [operator precedence](https://bliptext.com/articles/operator-precedence) of other programming languages; for example, does not bind its operands any more \"tightly\" than.\nInstead of operator precedence, APL defines a notion of scope.\nThe scope of a [function](https://bliptext.com/articles/first-class-function) determines its [arguments](https://bliptext.com/articles/parameter-computer-programming).\nFunctions have long right scope: that is, they take as right arguments everything to their right.\nA dyadic function has short left scope: it takes as its left arguments the first piece of data to its left.\nFor example, (leftmost column below is actual [program code](https://bliptext.com/articles/source-code) from an APL [user session](https://bliptext.com/articles/session-computer-science), indented = actual [user input](https://bliptext.com/articles/user-interface), not-indented = result returned by [APL interpreter](https://bliptext.com/articles/apl-programming-language)):\n<< First note there are no parentheses and\nAPL is going to execute from right-to-left.\nStep 1{of topmost APL code entered at left}) 4-5 = -1.\nStep 2) 3 times -1 = -3.\nStep 3) Take the **floor** or **lower** of 2 and -3 = -3.\nStep 4) Divide 1 by -3 = -0.3333333333 = final result.\nAn operator may have function or data [operands](https://bliptext.com/articles/operand) and evaluate to a dyadic or monadic function.\nOperators have long left scope.\nAn operator takes as its left operand the longest function to its left.\nFor example:\n∘.=/⍳¨3 3 1 0 0 0 1 0 0 0 1\nAPL [atomic](https://bliptext.com/articles/linearizability) or piecemeal sub-analysis (**full explanation**):\nBeginning rightmost: ⍳¨3 3 produces a 2-element nested APL vector { {1 2 3} {1 2 3} } where each element is itself a vector {1 2 3}.\n**Iota ⍳3** by itself would produce {1 2 3}.\nThe **diaeresis ¨** or mini double-dot means repeat or **over each** or perform each separately so **iota repeats** (in human i.e., reversed terms, the APL interpreter reads 3 3 **over each** use iota), concisely: **iota for each 3**.\nThe left operand for the **over-each** operator is the **index ⍳** function.\nThe derived function is used monadically and takes as its right operand the vector.\nThe left scope of **each** is terminated by the **reduce** operator, denoted by the forward **slash**.\nIts left operand is the function expression to its left: the **outer product** of the **equals** function.\nThe result of ∘.=/ is a monadic function.\nWith a function's usual long right scope, it takes as its right argument the result of ⍳¨3 3.\nThus\n**(⍳3)(⍳3)** 1 2 3 1 2 3 **(⍳3)∘.=⍳3** 1 0 0 0 1 0 0 0 1 ⍳¨3 3 1 2 3 1 2 3 **∘.=/⍳¨3 3** 1 0 0 0 1 0 0 0 1\nEquivalent results in APL: and << Rightmost expression is more concise.\nThe matrix of 1s and 0s similarly produced by and is called an [identity matrix](https://bliptext.com/articles/identity-matrix).\nIdentity matrices are useful in solving [matrix determinants](https://bliptext.com/articles/determinant), groups of [linear equations](https://bliptext.com/articles/system-of-linear-equations) and [multiple regression](https://bliptext.com/articles/regression-analysis).\nSome APL interpreters support the **[compose](https://bliptext.com/articles/composition-operator) operator ∘** and the **[commute](https://bliptext.com/articles/commuting-matrices) operator ⍨**.\nThe former **∘ glues** functions together so that **[foo∘bar](https://bliptext.com/articles/foobar)**, for example, could be a hypothetical function that applies defined function foo to the result of defined function bar; foo and bar can represent any existing function.\nIn cases where a dyadic function is moderated by **commute** and then used monadically, its right argument is taken as its left argument as well.\nThus, a **derived** or **composed** function (named im at left) is used in the APL user session to return a 9-[element](https://bliptext.com/articles/element-mathematics) identity matrix using its right [argument](https://bliptext.com/articles/argument-of-a-function), [parameter](https://bliptext.com/articles/parameter-computer-programming) or operand = 3.\nExample using APL to **index ⍳** or **find** (or not **find**) elements in a character vector :\nFirst, variable **Letters** is assigned a vector of 5-elements, in this case - letters of the alphabet.\nThe **shape ⍴** or character vector-length of **Letters** is 5.\nVariable **FindIt** is assigned what to **search for** in **Letters** and its length is 4 characters.\n1 2 3 4 5 << vector positions or index #'s in **Letters**\nABCDE\nAt left, dyadic function **iota searches** through its left argument(Letters) for the search string (iota's right argument, FindIt).\nIota finds letter \"C\" at position 3 in Letters, it finds \"A\" at position 1, and \"B\" at position 2.\n**Iota** does **not find** letter \"S\"\nanywhere in variable Letters so it returns the number 6 which is **1 greater than the length** of Letters.\n**Iota** found letters\n\"CAB\" (3 1 2).\n**Iota** correctly did **not find** \"S\" (6).\n\n# Monadic functions\n\n# Dyadic functions\n\n# Operators and axis indicator\n**Notes:** The reduce and scan operators expect a dyadic function on their left, forming a monadic composite function applied to the vector on its right.\nThe product operator \".\"\nexpects a dyadic function on both its left and right, forming a dyadic composite function applied to the vectors on its left and right.\nIf the function to the left of the dot is \"∘\" (signifying null) then the composite function is an outer product, otherwise it is an inner product.\nAn inner product intended for conventional matrix multiplication uses the + and × functions, replacing these with other dyadic functions can result in useful alternative operations.\nSome functions can be followed by an axis indicator in (square) brackets, i.e., this appears between a function and an array and should not be confused with array subscripts written after an array.\nFor example, given the ⌽ (reversal) function and a two-dimensional array, the function by default operates along the last axis but this can be changed using an axis indicator:\n4 rows by 3 cols matrix created, using **rho ⍴** and **iota ⍳**.\nThe 4 x 3 matrix is then stored in a [variable](https://bliptext.com/articles/variable-computer-science) named **A**.\n**A** is now reflected or flipped along its vertical axis as **symbol ⌽** visually indicates.\n**A** is now reflected using the **[1] axis indicator** or **first dimension modifier**.\nThe result is that variable A has been reflected across the horizontal axis, instead of vertically.\n**A** is now reflected both **vertically ⊖** and **horizontally ⌽**.\n**A** is **⍉ transposed** to a 3 row by 4 col matrix such that rows-cols become exchanged, as **symbol ⍉** visually portrays.\nCompare the result here to the original matrix stored in A, topmost matrix.\nThese types of data transformations are useful in [time series](https://bliptext.com/articles/time-series) analysis and [spatial coordinates](https://bliptext.com/articles/rotation-mathematics), just two examples, [more exist](https://bliptext.com/articles/data-transformation).\nAs a particular case, if the dyadic **catenate** **\",\"** function is followed by an axis indicator (or axis modifier to a symbol/function), it can be used to laminate (interpose) two arrays depending on whether the axis indicator is less than or greater than the [index origin](https://bliptext.com/articles/zero-based-numbering) ([index origin](https://bliptext.com/articles/zero-based-numbering) = 1 in illustration below):\nAt left, variable 'B' is first assigned a vector of 4 consecutive integers (e.g., **⍳4**).\nVar **C** is then assigned 4 more consecutive integers (such as **4+⍳4**).\n'B' and **C** are then **concatenated** or **raveled** together for illustration purposes, resulting in a single vector (**⍳8**).\nIn the particular case at left, if the dyadic **catenate \",\"** function is followed by an **axis indicator** (**[0.5]** which is less than 1), it can be used to **laminate** (**interpose**) two arrays (vectors in this case) depending on whether the **axis indicator** is less than or greater than the index origin(1).\nThe first result (of **B,[0.5]**C****) is a 2 row by 4 column matrix, vertically joining 'B' and **C** row-wise.\nThe second result (of **B,[1.5]C** which is greater than 1) is a 4 row by 2 column matrix.\n\n# Nested arrays\n[Arrays](https://bliptext.com/articles/array-data-structure) are structures which have elements grouped linearly as [vectors](https://bliptext.com/articles/vector-mathematics-and-physics) or in table form as [matrices](https://bliptext.com/articles/matrix-mathematics)—and higher dimensions (3D or cubed, [4D or cubed over time](https://bliptext.com/articles/four-dimensional-space), etc.).\nArrays containing both characters and numbers are termed mixed arrays.\nArray structures containing elements which are also arrays are called nested arrays.\n\n# Flow control\nA user may define custom functions which, like variables, are identified by name rather than by a non-textual symbol.\nThe function header defines whether a custom function is niladic (no arguments), monadic (one right argument) or dyadic (left and right arguments), the local name of the result (to the left of the ← assign arrow), and whether it has any local variables (each separated by semicolon ';').\nWhether functions with the same identifier but different [adicity](https://bliptext.com/articles/adicity) are distinct is implementation-defined.\nIf allowed, then a function CURVEAREA could be defined twice to replace both monadic CIRCLEAREA and dyadic SEGMENTAREA above, with the monadic or dyadic function being selected by the context in which it was referenced.\nCustom dyadic functions may usually be applied to parameters with the same conventions as built-in functions, i.e., arrays should either have the same number of elements or one of them should have a single element which is extended.\nThere are exceptions to this, for example a function to convert pre-decimal UK currency to dollars would expect to take a parameter with precisely three elements representing pounds, shillings and pence.\nInside a program or a custom function, control may be conditionally transferred to a statement identified by a line number or explicit label; if the target is 0 (zero) this terminates the program or returns to a function's caller.\nThe most common form uses the APL compression function, as in the template (condition)/target which has the effect of evaluating the condition to 0 (false) or 1 (true) and then using that to mask the target (if the condition is false it is ignored, if true it is left alone so control is transferred).\nHence function SEGMENTAREA may be modified to abort (just below), returning zero if the parameters (DEGREES and RADIUS below) are of different sign:\nThe above function SEGMENTAREA works as expected if the parameters are scalars or single-element **arrays**, but **not** if they are multiple-element **arrays** since the condition ends up being based on a single element of the SIGN array - on the other hand, the user function could be modified to correctly handle vectorized arguments.\nOperation can sometimes be unpredictable since APL defines that computers with vector-processing capabilities should parallelise and may reorder array operations as far as possible - thus, **test and debug** user functions particularly if they will be used with vector or even matrix arguments.\nThis affects not only explicit application of a custom function to arrays, but also its use anywhere that a dyadic function may reasonably be used such as in generation of a table of results:\nA more concise way and sometimes better way - to formulate a function is to avoid explicit transfers of control, instead using expressions which evaluate correctly in all or the expected conditions.\nSometimes it is correct to let a function fail when one or both **input** arguments are **incorrect** - precisely to let user know that one or both arguments used were **incorrect**.\nThe following is more concise than the above SEGMENTAREA function.\nThe below importantly **correctly** handles vectorized arguments:\nAvoiding explicit transfers of control also called branching, if not reviewed or carefully controlled - can promote use of excessively complex one liners, veritably \"misunderstood and complex idioms\" and a \"write-only\" style, which has done little to endear APL to influential commentators such as [Edsger Dijkstra](https://bliptext.com/articles/edsger-dijkstra).\nConversely however APL idioms can be fun, educational and useful - if used with helpful **comments ⍝**, for example including source and intended meaning and function of the idiom(s).\nHere is an [APL idioms list](https://bliptext.com/articles/undefined), an [IBM APL2 idioms list here](https://bliptext.com/articles/undefined) and [Finnish APL idiom library here](https://bliptext.com/articles/undefined).\n\n# Miscellaneous\nMost APL implementations support a number of system variables and functions, usually preceded by the **⎕ (quad)** and/or **\")\"** (**hook**=close parenthesis) character. Note that the quad character is not the same as the [Unicode missing character symbol](https://bliptext.com/articles/specials-unicode-block). Particularly important and widely implemented is the ⎕IO ([Index Origin](https://bliptext.com/articles/zero-based-numbering)) variable, since while the original IBM APL based its arrays on 1 some newer variants base them on zero:\nThere are also system functions available to users for saving the current workspace e.g., **)SAVE** and terminating the APL environment, e.g., **)OFF** - sometimes called hook commands or functions due to the use of a leading right parenthesis or hook.\nThere is some standardization of these quad and hook functions.\n\n# Fonts\nThe Unicode [Basic Multilingual Plane](https://bliptext.com/articles/mapping-of-unicode-character-planes) includes the APL symbols in the [Miscellaneous Technical](https://bliptext.com/articles/miscellaneous-technical) block, which are thus usually rendered accurately from the larger Unicode fonts installed with most modern operating systems.\nThese fonts are rarely designed by typographers familiar with APL glyphs.\nSo, while accurate, the glyphs may look unfamiliar to APL programmers or be difficult to distinguish from one another.\nSome Unicode fonts have been designed to display APL well: APLX Upright, APL385 Unicode, and SimPL.\nBefore Unicode, APL interpreters were supplied with fonts in which APL characters were mapped to less commonly used positions in the ASCII character sets, usually in the upper 128 code points.\nThese mappings (and their national variations) were sometimes unique to each APL vendor's interpreter, which made the display of APL programs on the Web, in text files and manuals - frequently problematic.\n\n# APL2 keyboard function to symbol mapping\nNote the APL On/Off Key - topmost-rightmost key, just below.\nAlso note the keyboard had some 55 unique (68 listed per tables above, including comparative symbols but several symbols appear in both monadic and dyadic tables) APL symbol keys (55 APL functions (operators) are listed in IBM's 5110 APL Reference Manual), thus with the use of alt, shift and ctrl keys - it would theoretically have allowed a maximum of some **59** (keys) ***4** (with 2-key pressing) ***3** (with tri-key pressing, e.g., ctrl-alt-del) or some 472 different maximum key combinations, approaching the 512 [EBCDIC](https://bliptext.com/articles/ebcdic) character max (256 chars times 2 codes for each keys-combination).\nAgain, in theory the keyboard pictured here would have allowed for about 472 different APL symbols/functions to be keyboard-input, actively used.\nIn practice, early versions were only using something roughly equivalent to 55 APL special symbols (excluding letters, numbers, punctuation, etc. keys).\nThus, early APL was then only using about 11% (55/472) of a symbolic language's at-that-time utilization potential, based on keyboard # keys limits, again excluding numbers, letters, punctuation, etc. In another sense keyboard symbols utilization was closer to 100%, highly efficient, since EBCDIC only allowed 256 distinct chars, and [ASCII](https://bliptext.com/articles/ascii) only 128.\n# [APL2 Keyboard | upload.wikimedia.org/wikipedia/commons/6/6d/APL2-nappaimisto.png]\n\n# Solving puzzles\nAPL has proved to be extremely useful in solving mathematical puzzles, several of which are described below.\n\n## Pascal's triangle\nTake [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), which is a triangular array of numbers in which those at the ends of the rows are 1 and each of the other numbers is the sum of the nearest two numbers in the row just above it (the apex, 1, being at the top).\nThe following is an APL one-liner function to visually depict Pascal's triangle:\n\n## Prime numbers, contra proof via factors\nDetermine the number of [prime numbers](https://bliptext.com/articles/list-of-prime-numbers) (prime # is a natural number greater than 1 that has no positive divisors other than 1 and itself) up to some number N. [Ken Iverson](https://bliptext.com/articles/kenneth-e-iverson) is credited with the following one-liner APL solution to the problem:\nExamining the converse or opposite of a mathematical solution is frequently needed ([integer factors of a number](https://bliptext.com/articles/integer-factorization)): Prove for the subset of integers from 1 through 15 that they are **non-prime** by listing their decomposition factors.\nWhat are their non-one factors (#'s divisible by, except 1)?\n\n## Fibonacci sequence\nGenerate a [Fibonacci number](https://bliptext.com/articles/fibonacci-number) sequence, where each subsequent number in the sequence is the sum of the prior two:\n\n## Generic online tutorials\n\n## Syntax rules","revisionCount":0},"hasMore":false,"nextCursor":null}

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