# APL syntax and symbols

The programming language APL is distinctive in being symbolic rather than lexical: its primitives are denoted by symbols, not words. These symbols were originally devised as a mathematical notation to describe algorithms. APL 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.

## Monadic and dyadic functions

Most symbols denote functions or operators. A monadic function takes as its argument the result of evaluating everything to its right. (Moderated in the usual way by parentheses.) A dyadic function has another argument, the first item of data on its left. Many symbols denote both monadic and dyadic functions, interpreted according to use. For example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments.

## Functions and operators

APL uses the term operator in 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. relational operator and operators generally. Other programming languages also sometimes use this term interchangeably with function, however both terms are used in APL more precisely. Early definitions of APL symbols were very specific about how symbols were categorized. For example, the operator reduce is denoted by a forward slash and reduces an array along one axis by interposing its function operand. An example of reduce:
| ×/2 3 4 24 | << Equivalent results in APL >> << Reduce operator / used at left | 2×3×4 24 |
In the above case, the reduce or slash operator moderates the multiply function. The expression ×/2 3 4 evaluates to a scalar (1 element only) result through reducing an array by multiplication. The above case is simplified, imagine multiplying (adding, subtracting or dividing) more than just a few numbers together. (From a vector, ×/ returns the product of all its elements.)
| 1 0 1\45 67 45 0 67 | << Opposite results in APL >> << Expand dyadic function \ used at left Replicate dyadic function / used at right >> | 1 0 1/45 0 67 45 67 |
The above dyadic functions examples [left and right examples] (using the same / symbol, right example) demonstrate how Boolean values (0s and 1s) can be used as left arguments for the \ expand and / replicate functions to produce exactly opposite results. On the left side, the 2-element vector {45 67} is expanded where Boolean 0s occur to result in a 3-element vector {45 0 67} — note how APL inserted a 0 into the vector. Conversely, 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. APL symbols also operate on lists (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.

## Syntax rules

In APL the precedence hierarchy for functions or operators is strictly positional: expressions are evaluated right-to-left. APL does not follow the usual operator precedence of other programming languages; for example, × does not bind its operands any more "tightly" than +. Instead of operator precedence, APL defines a notion of scope.
The scope of a function determines its arguments. Functions have long right scope: that is, they take as right arguments everything to their right. A dyadic function has short left scope: it takes as its left arguments the first piece of data to its left. For example, (leftmost column below is actual program code from an APL user session, indented = actual user input, not-indented = result returned by APL interpreter):
An operator may have function or data operands and evaluate to a dyadic or monadic function. Operators have long left scope. An operator takes as its left operand the longest function to its left. For example:
The left operand for the over-each operator ¨ is the index ⍳ function. The derived function ⍳¨ is used monadically and takes as its right operand the vector 3 3. The left scope of each is terminated by the reduce operator, denoted by the forward slash. Its left operand is the function expression to its left: the outer product of the equals function. The result of ∘.=/ is a monadic function. With a function's usual long right scope, it takes as its right argument the result of ⍳¨3 3. Thus

## Monadic functions

| Name(s) | Notation | Meaning | Unicode code point |
| Roll | ?B | One integer selected randomly from the first B integers | U+003F ? QUESTION MARK |
| Ceiling | ⌈B | Least integer greater than or equal to B | U+2308 ⌈ LEFT CEILING |
| Floor | ⌊B | Greatest integer less than or equal to B | U+230A ⌊ LEFT FLOOR |
| Shape, Rho | ⍴B | Number of components in each dimension of B | U+2374 ⍴ APL FUNCTIONAL SYMBOL RHO |
| Not, Tilde | ∼B | Logical: ∼1 is 0, ∼0 is 1 | U+223C ∼ TILDE OPERATOR |
| Absolute value | ∣B | Magnitude of B | U+2223 ∣ DIVIDES |
| Index generator, Iota | ⍳B | Vector of the first B integers | U+2373 ⍳ APL FUNCTIONAL SYMBOL IOTA |
| Exponential | ⋆B | e to the B power | U+22C6 ⋆ STAR OPERATOR |
| Negation | −B | Changes sign of B | U+2212 − MINUS SIGN |
| Conjugate | +B | The complex conjugate of B (real numbers are returned unchanged) | U+002B + PLUS SIGN |
| Signum | ×B | ¯1 if B<0; 0 if B=0; 1 if B>0 | U+00D7 × MULTIPLICATION SIGN |
| Reciprocal | ÷B | 1 divided by B | U+00F7 ÷ DIVISION SIGN |
| Ravel, Catenate, Laminate | ,B | Reshapes B into a vector | U+002C , COMMA |
| Matrix inverse, Monadic Quad Divide | ⌹B | Inverse of matrix B | U+2339 ⌹ APL FUNCTIONAL SYMBOL QUAD DIVIDE |
| Pi times | ○B | Multiply by π | U+25CB ○ WHITE CIRCLE |
| Logarithm | ⍟B | Natural logarithm of B | U+235F ⍟ APL FUNCTIONAL SYMBOL CIRCLE STAR |
| Reversal | ⌽B | Reverse elements of B along last axis | U+233D ⌽ APL FUNCTIONAL SYMBOL CIRCLE STILE |
| Reversal | ⊖B | Reverse elements of B along first axis | U+2296 ⊖ CIRCLED MINUS |
| Grade up | ⍋B | Indices of B which will arrange B in ascending order | U+234B ⍋ APL FUNCTIONAL SYMBOL DELTA STILE |
| Grade down | ⍒B | Indices of B which will arrange B in descending order | U+2352 ⍒ APL FUNCTIONAL SYMBOL DEL STILE |
| Execute | ⍎B | Execute an APL expression | U+234E ⍎ APL FUNCTIONAL SYMBOL DOWN TACK JOT |
| Monadic format | ⍕B | A character representation of B | U+2355 ⍕ APL FUNCTIONAL SYMBOL UP TACK JOT |
| Monadic transpose | ⍉B | Reverse the axes of B | U+2349 ⍉ APL FUNCTIONAL SYMBOL CIRCLE BACKSLASH |
| Factorial | !B | Product of integers 1 to B | U+0021 ! EXCLAMATION MARK |
| Depth | ≡B | Nesting depth: 1 for un-nested array | U+2261 ≡ IDENTICAL TO |
| Table | ⍪B | Makes B into a table, a 2-dimensional array. | U+236A ⍪ APL FUNCTIONAL SYMBOL COMMA BAR |

## Dyadic functions

| Name(s) | Notation | Meaning | Unicode code point |
| Add | A+B | Sum of A and B | U+002B + PLUS SIGN |
| Subtract | A−B | A minus B | U+2212 − MINUS SIGN |
| Multiply | A×B | A multiplied by B | U+00D7 × MULTIPLICATION SIGN |
| Divide | A÷B | A divided by B | U+00F7 ÷ DIVISION SIGN |
| Exponentiation | A⋆B | A raised to the B power | U+22C6 ⋆ STAR OPERATOR |
| Circle | A○B | Trigonometric functions of B selected by A A=1: sin(B) A=5: sinh(B) A=2: cos(B) A=6: cosh(B) A=3: tan(B) A=7: tanh(B) Negatives produce the inverse of the respective functions | U+25CB ○ WHITE CIRCLE |
| Deal | A?B | A distinct integers selected randomly from the first B integers | U+003F ? QUESTION MARK |
| Membership, Epsilon | A∈B | 1 for elements of A present in B; 0 where not. | U+2208 ∈ ELEMENT OF |
| Find, Epsilon Underbar | A⍷B | 1 for starting point of multi-item array A present in B; 0 where not. | U+2377 ⍷ APL FUNCTIONAL SYMBOL EPSILON UNDERBAR |
| Maximum, Ceiling | A⌈B | The greater value of A or B | U+2308 ⌈ LEFT CEILING |
| Minimum, Floor | A⌊B | The smaller value of A or B | U+230A ⌊ LEFT FLOOR |
| Reshape, Dyadic Rho | A⍴B | Array of shape A with data B | U+2374 ⍴ APL FUNCTIONAL SYMBOL RHO |
| Take | A↑B | Select the first (or last) A elements of B according to ×A | U+2191 ↑ UPWARDS ARROW |
| Drop | A↓B | Remove the first (or last) A elements of B according to ×A | U+2193 ↓ DOWNWARDS ARROW |
| Decode | A⊥B | Value of a polynomial whose coefficients are B at A | U+22A5 ⊥ UP TACK |
| Encode | A⊤B | Base-A representation of the value of B | U+22A4 ⊤ DOWN TACK |
| Residue | A∣B | B modulo A | U+2223 ∣ DIVIDES |
| Catenation | A,B | Elements of B appended to the elements of A | U+002C , COMMA |
| Expansion, Dyadic Backslash | A\B | Insert zeros (or blanks) in B corresponding to zeros in A | U+005C \ REVERSE SOLIDUS |
| Compression, Dyadic Slash | A/B | Select elements in B corresponding to ones in A | U+002F / SOLIDUS |
| Index of, Dyadic Iota | A⍳B | The location (index) of B in A; 1+⍴A if not found | U+2373 ⍳ APL FUNCTIONAL SYMBOL IOTA |
| Matrix divide, Dyadic Quad Divide | A⌹B | Solution to system of linear equations, multiple regression Ax = B | U+2339 ⌹ APL FUNCTIONAL SYMBOL QUAD DIVIDE |
| Rotation | A⌽B | The elements of B are rotated A positions | U+233D ⌽ APL FUNCTIONAL SYMBOL CIRCLE STILE |
| Rotation | A⊖B | The elements of B are rotated A positions along the first axis | U+2296 ⊖ CIRCLED MINUS |
| Logarithm | A⍟B | Logarithm of B to base A | U+235F ⍟ APL FUNCTIONAL SYMBOL CIRCLE STAR |
| Dyadic format | A⍕B | Format B into a character matrix according to A | U+2355 ⍕ APL FUNCTIONAL SYMBOL UP TACK JOT |
| General transpose | A⍉B | The axes of B are ordered by A | U+2349 ⍉ APL FUNCTIONAL SYMBOL CIRCLE BACKSLASH |
| Combinations | A!B | Number of combinations of B taken A at a time | U+0021 ! EXCLAMATION MARK |
| Diaeresis, Dieresis, Double-Dot | A¨B | Over each, or perform each separately; B = on these; A = operation to perform or using (e.g., iota) | U+00A8 ¨ DIAERESIS |
| Less than | A<B | Comparison: 1 if true, 0 if false | U+003C < LESS-THAN SIGN |
| Less than or equal | A≤B | Comparison: 1 if true, 0 if false | U+2264 ≤ LESS-THAN OR EQUAL TO |
| Equal | A=B | Comparison: 1 if true, 0 if false | U+003D = EQUALS SIGN |
| Greater than or equal | A≥B | Comparison: 1 if true, 0 if false | U+2265 ≥ GREATER-THAN OR EQUAL TO |
| Greater than | A>B | Comparison: 1 if true, 0 if false | U+003E > GREATER-THAN SIGN |
| Not equal | A≠B | Comparison: 1 if true, 0 if false | U+2260 ≠ NOT EQUAL TO |
| Or | A∨B | Boolean Logic: 0 (False) if both A and B = 0, 1 otherwise. Alt: 1 (True) if A or B = 1 (True) | U+2228 ∨ LOGICAL OR |
| And | A∧B | Boolean Logic: 1 (True) if both A and B = 1, 0 (False) otherwise | U+2227 ∧ LOGICAL AND |
| Nor | A⍱B | Boolean Logic: 1 if both A and B are 0, otherwise 0. Alt: ~∨ = not Or | U+2371 ⍱ APL FUNCTIONAL SYMBOL DOWN CARET TILDE |
| Nand | A⍲B | Boolean Logic: 0 if both A and B are 1, otherwise 1. Alt: ~∧ = not And | U+2372 ⍲ APL FUNCTIONAL SYMBOL UP CARET TILDE |
| Left | A⊣B | A | U+22A3 ⊣ LEFT TACK |
| Right | A⊢B | B | U+22A2 ⊢ RIGHT TACK |
| Match | A≡B | 1 if A matches B exactly with respect to value, shape, and nesting; otherwise 0. | U+2261 ≡ IDENTICAL TO |
| Laminate | A⍪B | Concatenate along first axis | U+236A ⍪ APL FUNCTIONAL SYMBOL COMMA BAR |

## Operators and axis indicator

| Name(s) | Symbol | Example | Meaning (of example) | Unicode code point sequence |
| Reduce (last axis), Slash | / | +/B | Sum across B | U+002F / SOLIDUS |
| Reduce (first axis) | ⌿ | +⌿B | Sum down B | U+233F ⌿ APL FUNCTIONAL SYMBOL SLASH BAR |
| Scan (last axis), Backslash | \ | +\B | Running sum across B | U+005C \ REVERSE SOLIDUS |
| Scan (first axis) | ⍀ | +⍀B | Running sum down B | U+2340 ⍀ APL FUNCTIONAL SYMBOL BACKSLASH BAR |
| Inner product | . | A+.×B | Matrix product of A and B | U+002E . FULL STOP |
| Outer product | ∘. | A∘.×B | Outer product of A and B | U+2218 ∘ RING OPERATOR, U+002E . FULL STOP |
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.
The product operator "." expects a dyadic function on both its left and right, forming a dyadic composite function applied to the vectors on its left and right. If 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. An inner product intended for conventional matrix multiplication uses the + and × functions, replacing these with other dyadic functions can result in useful alternative operations.
Some 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. For 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:
As 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 (index origin = 1 in illustration below):

## Nested arrays

Arrays are structures which have elements grouped linearly as vectors or in table form as matrices—and higher dimensions (3D or cubed, 4D or cubed over time, etc.). Arrays containing both characters and numbers are termed mixed arrays. Array structures containing elements which are also arrays are called nested arrays.
Creating a nested array :
| User session with APL interpreter | Explanation |
| X←4 5⍴⍳20 X 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 X[2;2] 7 ⎕IO 1 X[1;1] 1 | X set = to matrix with 4 rows by 5 columns, consisting of 20 consecutive integers. Element X[2;2] in row 2 - column 2 currently is an integer = 7. Initial index origin ⎕IO value = 1. Thus, the first element in matrix X or X[1;1] = 1. |
| X[2;2]←⊂"Text" X[3;4]←⊂(2 2⍴⍳4) X 1 2 3 4 5 6 Text 8 9 10 11 12 13 1 2 15 3 4 16 17 18 19 20 | Element in X[row 2; col 2] is changed (from 7) to a nested vector "Text" using the enclose ⊂ function. Element in X[row 3; col 4], formerly integer 14, now becomes a mini enclosed or ⊂ nested 2x2 matrix of 4 consecutive integers. Since X contains numbers, text and nested elements, it is both a mixed and a nested array. Visual representation of the nested array |

## Flow control

A user may define custom functions which, like variables, are identified by name rather than by a non-textual symbol. The 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 ';').
User functions :
| Niladic function PI or π(pi) | Monadic function CIRCLEAREA | Dyadic function SEGMENTAREA, with local variables |
| ∇ RESULT←PI RESULT←○1 ∇ | ∇ AREA←CIRCLEAREA RADIUS AREA←PI×RADIUS⋆2 ∇ | ∇ AREA←DEGREES SEGMENTAREA RADIUS ; FRACTION ; CA FRACTION←DEGREES÷360 CA←CIRCLEAREA RADIUS AREA←FRACTION×CA ∇ |
Whether functions with the same identifier but different adicity are distinct is implementation-defined. If 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.
Custom 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. There 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.
Inside 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. The 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).
Hence function SEGMENTAREA may be modified to abort (just below), returning zero if the parameters (DEGREES and RADIUS below) are of different sign:
The 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. Operation 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. This 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:
A 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. Sometimes 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. The following is more concise than the above SEGMENTAREA function. The below importantly correctly handles vectorized arguments:
Avoiding 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. Conversely 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). Here is an APL idioms list, an IBM APL2 idioms list here and Finnish APL idiom library here.

## Miscellaneous

Miscellaneous symbols :
| Name(s) | Symbol | Example | Meaning (of example) | Unicode code point |
| High minus | ¯ | ¯3 | Denotes a negative number | U+00AF ¯ MACRON |
| Lamp, Comment | ⍝ | ⍝This is a comment | Everything to the right of ⍝ denotes a comment | U+235D ⍝ APL FUNCTIONAL SYMBOL UP SHOE JOT |
| RightArrow, Branch, GoTo | → | →This_Label | →This_Label sends APL execution to This_Label: | U+2192 → RIGHTWARDS ARROW |
| Assign, LeftArrow, Set to | ← | B←A | B←A sets values and shape of B to match A | U+2190 ← LEFTWARDS ARROW |
Most 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. Particularly important and widely implemented is the ⎕IO (Index Origin) variable, since while the original IBM APL based its arrays on 1 some newer variants base them on zero:
| User session with APL interpreter | Description |
| X←⍳12 X 1 2 3 4 5 6 7 8 9 10 11 12 ⎕IO 1 X[1] 1 | X set = to vector of 12 consecutive integers. Initial index origin ⎕IO value = 1. Thus, the first position in vector X or X[1] = 1 per vector of iota values {1 2 3 4 5 ...}. |
| ⎕IO←0 X[1] 2 X[0] 1 | Index Origin ⎕IO now changed to 0. Thus, the 'first index position' in vector X changes from 1 to 0. Consequently, X[1] then references or points to 2 from {1 2 3 4 5 ...} and X[0] now references 1. |
| ⎕WA 41226371072 | Quad WA or ⎕WA, another dynamic system variable, shows how much Work Area remains unused or 41,226 megabytes or about 41 gigabytes of unused additional total free work area available for the APL workspace and program to process using. If this number gets low or approaches zero - the computer may need more random-access memory (RAM), hard disk drive space or some combination of the two to increase virtual memory. |
| )VARS X | )VARS a system function in APL, )VARS shows user variable names existing in the current workspace. |
There 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. There is some standardization of these quad and hook functions.

## Fonts

The Unicode Basic Multilingual Plane includes the APL symbols in the Miscellaneous Technical block, which are thus usually rendered accurately from the larger Unicode fonts installed with most modern operating systems. These fonts are rarely designed by typographers familiar with APL glyphs. So, while accurate, the glyphs may look unfamiliar to APL programmers or be difficult to distinguish from one another.
Some Unicode fonts have been designed to display APL well: APLX Upright, APL385 Unicode, and SimPL.
Before 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. These 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.

## APL2 keyboard function to symbol mapping

Note the APL On/Off Key - topmost-rightmost key, just below. Also 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 character max (256 chars times 2 codes for each keys-combination). Again, in theory the keyboard pictured here would have allowed for about 472 different APL symbols/functions to be keyboard-input, actively used. In practice, early versions were only using something roughly equivalent to 55 APL special symbols (excluding letters, numbers, punctuation, etc. keys). Thus, 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 only 128.

## Solving puzzles

APL has proved to be extremely useful in solving mathematical puzzles, several of which are described below.

### Pascal's triangle

Take 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). The following is an APL one-liner function to visually depict Pascal's triangle:

### Prime numbers, contra proof via factors

Determine the number 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 is credited with the following one-liner APL solution to the problem:
Examining the converse or opposite of a mathematical solution is frequently needed (integer factors of a number): Prove for the subset of integers from 1 through 15 that they are non-prime by listing their decomposition factors. What are their non-one factors (#'s divisible by, except 1)?

### Fibonacci sequence

Generate a Fibonacci number sequence, where each subsequent number in the sequence is the sum of the prior two: