APL syntax and symbols
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The APL programming language 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. 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. 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 only in Heaviside’s sense of a moderator of a function. 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
is equivalent to
In this case, the reduce operator moderates the multiply function. The expression ×/ evaluates to a monadic function that reduces an array by multiplication. (From a vector, it returns the product of its elements.)
There is no precedence hierarchy for functions or operators.
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,
1 ÷ 2 ⌊ 3 × 4 - 5 ¯0.3333333333 1 ÷ 2 ⌊ 3 × ¯1 ¯0.3333333333 1 ÷ 2 ⌊ ¯3 ¯0.3333333333 1 ÷ ¯3 ¯0.3333333333
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:
∘.=/⍳¨3 3 1 0 0 0 1 0 0 0 1
The left operand of the each operator ¨ is the index function. The derived function ⍳¨ ("iota") is used monadically and takes as its right 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 syntax and 2-glyph symbol of the outer-product operator are both unhappily anomalous.) 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
(⍳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
Some interpreters support the compose operator ∘ and the commute operator ⍨. The former "glues" functions together so that foo∘bar is a function that applies function foo to the result of function bar. Where a dyadic function is moderated by commute and then used monadically, its right argument is taken as its left argument as well. Thus a derived function to return an identity matrix:
im ← ∘.=⍨∘⍳ im 3 1 0 0 0 1 0 0 0 1
|Roll||?B||One integer selected randomly from the first B integers||U+003F|
|Ceiling||⌈B||Least integer greater than or equal to B||U+2308|
|Floor||⌊B||Greatest integer less than or equal to B||U+230A|
|Shape||⍴B||Number of components in each dimension of B||U+2374|
|Not||∼B||Logical: ∼1 is 0, ∼0 is 1||U+223C|
|Absolute value||∣B||Magnitude of B||U+2223|
|Index generator||⍳B||Vector of the first B integers||U+2373|
|Exponential||⋆B||e to the B power||U+22C6|
|Negation||−B||Changes sign of B||U+2212|
|Identity||+B||No change to B||U+002B|
|Signum||×B||¯1 if B<0; 0 if B=0; 1 if B>0||U+00D7|
|Reciprocal||÷B||1 divided by B||U+00F7|
|Ravel||,B||Reshapes B into a vector||U+002C|
|Matrix inverse||⌹B||Inverse of matrix B||U+2339|
|Pi times||○B||Multiply by π||U+25CB|
|Logarithm||⍟B||Natural logarithm of B||U+235F|
|Reversal||⌽B||Reverse elements of B along last axis||U+233D|
|Reversal||⊖B||Reverse elements of B along first axis||U+2296|
|Grade up||⍋B||Indices of B which will arrange B in ascending order||U+234B|
|Grade down||⍒B||Indices of B which will arrange B in descending order||U+2352|
|Execute||⍎B||Execute an APL expression||U+234E|
|Monadic format||⍕B||A character representation of B||U+2355|
|Monadic transpose||⍉B||Reverse the axes of B||U+2349|
|Factorial||!B||Product of integers 1 to B||U+0021|
|Add||A+B||Sum of A and B||U+002B|
|Subtract||A−B||A minus B||U+2212|
|Multiply||A×B||A multiplied by B||U+00D7|
|Divide||A÷B||A divided by B||U+00F7|
|Exponentiation||A⋆B||A raised to the B power||U+22C6|
|Circle||A○B||Trigonometric functions of B selected by A
A=1: sin(B) A=2: cos(B) A=3: tan(B)
|Deal||A?B||A distinct integers selected randomly from the first B integers||U+003F|
|Membership||A∈B||1 for elements of A present in B; 0 where not.||U+2208|
|Maximum||A⌈B||The greater value of A or B||U+2308|
|Minimum||A⌊B||The smaller value of A or B||U+230A|
|Reshape||A⍴B||Array of shape A with data B||U+2374|
|Take||A↑B||Select the first (or last) A elements of B according to ×A||U+2191|
|Drop||A↓B||Remove the first (or last) A elements of B according to ×A||U+2193|
|Decode||A⊥B||Value of a polynomial whose coefficients are B at A||U+22A5|
|Encode||A⊤B||Base-A representation of the value of B||U+22A4|
|Residue||A∣B||B modulo A||U+2223|
|Catenation||A,B||Elements of B appended to the elements of A||U+002C|
|Expansion||A\B||Insert zeros (or blanks) in B corresponding to zeros in A||U+005C|
|Compression||A/B||Select elements in B corresponding to ones in A||U+002F|
|Index of||A⍳B||The location (index) of B in A; 1+⌈/⍳⍴A if not found||U+2373|
|Matrix divide||A⌹B||Solution to system of linear equations Ax = B||U+2339|
|Rotation||A⌽B||The elements of B are rotated A positions||U+233D|
|Rotation||A⊖B||The elements of B are rotated A positions along the first axis||U+2296|
|Logarithm||A⍟B||Logarithm of B to base A||U+235F|
|Dyadic format||A⍕B||Format B into a character matrix according to A||U+2355|
|General transpose||A⍉B||The axes of B are ordered by A||U+2349|
|Combinations||A!B||Number of combinations of B taken A at a time||U+0021|
|Less than||A<B||Comparison: 1 if true, 0 if false||U+003C|
|Less than or equal||A≤B||Comparison: 1 if true, 0 if false||U+2264|
|Equal||A=B||Comparison: 1 if true, 0 if false||U+003D|
|Greater than or equal||A≥B||Comparison: 1 if true, 0 if false||U+2265|
|Greater than||A>B||Comparison: 1 if true, 0 if false||U+003E|
|Not equal||A≠B||Comparison: 1 if true, 0 if false||U+2260|
|Or||A∨B||Logic: 0 if A and B are 0; 1 otherwise||U+2228|
|And||A∧B||Logic: 1 if A and B are 1; 0 otherwise||U+2227|
|Nor||A⍱B||Logic: 1 if both A and B are 0; otherwise 0||U+2371|
|Nand||A⍲B||Logic: 0 if both A and B are 1; otherwise 1||U+2372|
Operators and axis indicator
|Name||Symbol||Example||Meaning (of example)||Unicode codepoint sequence|
|Reduce (last axis)||/||+/B||Sum across B||U+002F|
|Reduce (first axis)||⌿||+⌿B||Sum down B||U+233F|
|Scan (last axis)||\||+\B||Running sum across B||U+005C|
|Scan (first axis)||⍀||+⍀B||Running sum down B||U+2340|
|Inner product||.||A+.×B||Matrix product of A and B||U+002E|
|Outer product||∘.||A∘.×B||Outer product of A and B||U+2218, U+002E|
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:
A←4 3⍴⍳12 A 1 2 3 4 5 6 7 8 9 10 11 12 ⌽A 3 2 1 6 5 4 9 8 7 12 11 10 ⌽A 10 11 12 7 8 9 4 5 6 1 2 3
As a particular case, if the dyadic catenate function is followed by an axis indicater, it can be used to laminate (interpose) two arrays depending on whether the axis indicator is less than or greater than the index origin:
B←1 2 3 4 C←5 6 7 8 B,C 1 2 3 4 5 6 7 8 B,[0.5]C 1 2 3 4 5 6 7 8 B,[1.5]C 1 5 2 6 3 7 4 8
The 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, monadic or dyadic, the local names of the result and parameters, and whether it has any local variables.
Niladic function PI:
∇ RESULT←PI RESULT←○1 ∇
Monadic function CIRCLEAREA:
∇ AREA←CIRCLEAREA RADIUS AREA←PI×RADIUS⋆2 ∇
Dyadic function SEGMENTAREA, with local variables:
∇ 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 the function SEGMENTAREA may be modified to abort returning zero if the parameters are of different sign:
∇ AREA←DEGREES SEGMENTAREA RADIUS ; FRACTION ; CA ; SIGN FRACTION←DEGREES÷360 CA←CIRCLEAREA RADIUS SIGN←(×DEGREES)≠×RADIUS AREA←0 →SIGN/0 AREA←FRACTION×CA ∇
This 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: in general, operation is unpredictable since APL defines that computers with vector-processing capabilities should parallelise and may reorder array operations as far as possible. 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:
90 180 270 ¯90 ∘.SEGMENTAREA 1 ¯2 4 0 0 0 0 0 0 0 0 0 0 0 0
The correct way to formulate a function is to avoid explicit transfers of control and instead use expressions which evaluate correctly in all conditions:
∇ AREA←DEGREES SEGMENTAREA RADIUS ; FRACTION ; CA ; SIGN FRACTION←DEGREES÷360 CA←CIRCLEAREA RADIUS SIGN←(×DEGREES)≠×RADIUS AREA←FRACTION×CA×~SIGN ∇ 90 180 270 ¯90 ∘.SEGMENTAREA 1 ¯2 4 0.785398163 0 12.5663706 1.57079633 0 25.1327412 2.35619449 0 37.6991118 0 ¯3.14159265 0
|Name||Symbol||Example||Meaning (of example)||Unicode codepoint|
|High minus||¯||¯3||Denotes a negative number||U+00AF|
APL's "high minus" applies to the single number that follows, while the monadic minus function changes the sign of the entire array to its right.
Most APL implementations support a number of system variables and functions, usually preceded by the ⎕ (quad) character. 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:
X←⍳12 X 1 2 3 4 5 6 7 8 9 10 11 12 ⎕IO 1 X 1 ⎕IO←0 X 2
There are user functions for e.g. saving the current workspace and terminating the APL environment, there is very little standardisation of these.
The Unicode Basic Multilingual Plane includes the APL symbols in the Miscellaneous Technical block, which are therefore 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 codepoints. These mappings (and their national variations) were peculiar to interpreters, which made the display of APL programs on the Web problematic.
- Polivka, Raymond P.; & Pakin, Sandra (1975). APL: The Language and Its Usage. Prentice-Hall. ISBN 0-13-038885-8.