Array notation

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Array notation is a way to write most arrays literally, with no or minimal use of primitive functions, possibly over multiple code lines. It differs from the strand notation existing since APL\360 in that it can be used to write arrays of rank greater than one. Array notation is supported in dzaima/APL and BQN, and by some tools for Dyalog APL, where it is planned as an eventual language feature.

Array notation generally consists of a vector notation written with parentheses (), roughly equivalent to stranding, and a high-rank notation using square brackets [], indicating the Mix of a vector. It may also support namespaces, with dzaima/APL and Dyalog using name:value syntax in parentheses for this. In each case, elements are separated by statement separators such as or line breaks.


Medium-sized array constants are often needed in code. Due to the lack of a native multi-line notation, programmers have resorted to various ad-hoc methods of approximating such, usually at the cost of reduced readability. A very common technique is repeated concatenation:

poss1 2'fns'  ((0 1)(0.7 0)(0.7 0)×size)
poss   'fnd'  ((0 1)(0   0)(0   0)×size)
poss   'lines'((0 0)(0.7 0)(0.7 0)×size)
poss   'lnd'  ((0 0)(0   0)(0   0)×size)

Array notation allows this multi-line construction to be made with a single assignment, and also provides an alternate syntax for the inner vectors of vectors:

poss['fns'  ((0 1  0.7 0  0.7 0)×size)
      'fnd'  ((0 1  0   0  0   0)×size)
      'lines'((0 0  0.7 0  0.7 0)×size)
      'lnd'  ((0 0  0   0  0   0)×size)]


The notation is added to the language by giving meaning to previously invalid statements. The added syntax consists of three constructs that are currently SYNTAX ERRORs:

  • broken round parentheses
  • broken square brackets
  • empty round parentheses: ()

where broken means interrupted by one or more diamonds () or line breaks (outside of dfns).

  • A broken round parenthesis creates a namespace if every diamond/line break-separated statement is a name-value pair.
  • A broken round parenthesis creates a vector if every diamond/line break-separated statement is a value expression. In that case, every such statement forms an element in the resulting vector.
  • A broken square bracket creates a an array where every diamond/line break-separated statement forms a major cell in the resulting array.
  • () is equivalent to (⎕NS 0⍴⊂'')
  • A name-value pair consist of a valid APL identifier, followed by a : and a value expression.


See also Array notation in Dyalog APL

One-dimensional list syntax with surrounding brackets and delimiters, matching sequence notation in mathematics, is common in programming. It appears as early as ALGOL 68 with parentheses, and square-bracket lists feature in languages from the 1970s such as ML and Icon. MATLAB uses matrix syntax with square brackets, semicolons to separate rows, and commas to separate elements within a row. FP uses angle brackets for lists, and square brackets for function "construction", with behavior like function arrays.

List notation appears in Nial using brackets and commas like [a,b,c], and allowing function arrays called "atlases". A+ and K have a list notation using parentheses and semicolons like (a;b;c). In A+ this is related to bracket indexing and an "expression group" notation written with curly braces and semicolons. It allows line breaks, but in addition to rather than in place of semicolons. The later K version corresponds more closely to APL: the semicolon is a statement separator and is interchangeable with a line break, and because K represents arrays with nested lists, it corresponds to both vector and high-rank array notation.

The first published proposals that influenced Dyalog APL's array notation were made by Phil Last at Dyalog '15 and later in Vector Journal.[1][2] Last also used the design in Acre Desktop, a project manager for Dyalog APL, to support storing constant arrays and namespaces in text files. Following the conference presentation, Adám Brudzewsky began work on array notation and presented on it in a series of conferences, initially using parentheses for the high-rank notation[3] and later returning to square brackets.[4][5] Because Last's use of to separate namespace keys from values prevented lists from including arbitrary expressions (which might contain assignment), he proposed a change to : as in JSON. Dyalog APL 18.0, released in 2020, included support for array notation in source files loaded by Link, but not in the language itself.[6]

dzaima/APL added support for vector notation with parentheses in 2018, namespaces and function arrays in 2019, and high-rank arrays with square brackets in 2020. BQN supported lists with angle brackets ⟨⟩ in its initial implementation in 2020; square brackets were reserved for high-rank array notation, which was implemented in 2022.

Language support

The following languages support list or vector notation in some form. In K, all APLs, and BQN, the separators ; and indicate any separator, including a line break.

Language Vectors High-rank Namespaces Function arrays
Nial [,] Yes
A+ [;] First class
K [;] First class
dzaima/APL (⋄) [⋄] (key:val⋄) Yes
BQN ⟨⋄⟩ [⋄] {key⇐val⋄} First class
Dyalog (proposed) (⋄) [⋄] (key:val⋄) No

"First class" indicates that functions are first class, so that an array of functions can be formed directly; in Nial and dzaima/APL function arrays are a special form that can be applied to arguments to return a list of results. BQN's namespaces don't use a dedicated construction; instead, any block (like a dfn) with statements returns a namespace reference.

External links



APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming (Train, Hook, Split composition)
Array Numeric literalStringStrand notationObject literalArray notation (design considerations)
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionDefined function (traditional)DfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBracket indexingBranchStatement separatorQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloadingControl structure