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:''This page describes the array datatype as defined by array languages. For the role of arrays in [[APL syntax]], see [[Array]].'' | |||
The distinguishing feature of APL and the array language family is its focus on '''arrays'''. In most array languages the array is the only first class datatype. While this sounds like a very strict model of language design, in fact it imposes no restrictions at all: any kind of data can be treated as a [[scalar]], or array with rank 0! | The distinguishing feature of APL and the array language family is its focus on '''arrays'''. In most array languages the array is the only first class datatype. While this sounds like a very strict model of language design, in fact it imposes no restrictions at all: any kind of data can be treated as a [[scalar]], or array with rank 0! | ||
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A second version of the APL array model was developed in order to more transparently handle nested data, without the need to explicitly box and unbox arrays. The Nested Array Research System ([[NARS]]) was developed to study this model. In it, arrays contain other arrays directly. In this way they resemble the [[wikipedia:inductive type|inductive type]]s used in type theory. | A second version of the APL array model was developed in order to more transparently handle nested data, without the need to explicitly box and unbox arrays. The Nested Array Research System ([[NARS]]) was developed to study this model. In it, arrays contain other arrays directly. In this way they resemble the [[wikipedia:inductive type|inductive type]]s used in type theory. | ||
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|{{quote | "An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes."|[[John Scholes]]}} | |||
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In nested APLs each individual number or character is encapsulated in a [[simple scalar]]. Such a scalar may be referred to as "a number" or "a character" but it maintains the properties of an array. Other arrays used by the language are defined inductively: an array can be formed which contains as elements any array which has already been defined. Such an array cannot, by the nature of inductive definition, contain itself, even within many levels of nesting inside it. Arrays which contain only simple scalars, or are themselves simple scalars, are called [[Simple array|simple]]. Non-simple arrays are called "nested". The simple arrays are a superset of the arrays allowed in flat array theory without boxes: they include all arrays of numbers and characters, as well as arrays which mix numbers and characters. Arrays which would not be representable in flat array theory—those which contain a mixture of simple scalar types, or contain both simple scalars and other arrays—are called [[Mixed array|mixed]]. | In nested APLs each individual number or character is encapsulated in a [[simple scalar]]. Such a scalar may be referred to as "a number" or "a character" but it maintains the properties of an array. Other arrays used by the language are defined inductively: an array can be formed which contains as elements any array which has already been defined. Such an array cannot, by the nature of inductive definition, contain itself, even within many levels of nesting inside it. Arrays which contain only simple scalars, or are themselves simple scalars, are called [[Simple array|simple]]. Non-simple arrays are called "nested". The simple arrays are a superset of the arrays allowed in flat array theory without boxes: they include all arrays of numbers and characters, as well as arrays which mix numbers and characters. Arrays which would not be representable in flat array theory—those which contain a mixture of simple scalar types, or contain both simple scalars and other arrays—are called [[Mixed array|mixed]]. | ||
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Flat array theory is often called "grounded" in contrast to "floating" nested array theory. | Flat array theory is often called "grounded" in contrast to "floating" nested array theory. | ||
== Based array theory == | |||
Based array theory discards the principle that all data should be stored in arrays, instead defining basic types such as characters and numbers independently of arrays and arrays as a collection type—possibly one of many—that can contain any data. This model does not have any widely accepted name, with the term "based system" introduced in an [[APL Quote Quad]] paper in 1981.<ref>Randall Mercer. [https://dl.acm.org/doi/abs/10.1145/586656.586663 "A based system for general arrays"]. [[APL Quote Quad]] Volume 12, Issue 2. 1981-12.</ref> However, as it is the natural model when arrays are added to an existing programming system, it is common in array libraries such as [[NumPy]], [[wikipedia:ILNumerics|ILNumerics]], and [[wikipedia:Haskell (programming language)|Haskell]]'s [https://hackage.haskell.org/package/repa Repa], as well as the language [[Julia]]. It has also been promoted with the mnemonic "APL+1" by Jacob Brickman<ref>Jacob Brickman. "Behavior of APL primitives in a system with depth-1 scalars (APL+1)". [[APL-Germany]] ''APL-Journal'' 2019 1-2 ([https://apl-germany.de/wp-content/uploads/2020/06/APL_Journal_2019_12.pdf#page=5 pdf]).</ref>, and is used by the APL-family language [[BQN]]. | |||
=== Mutable based arrays === | |||
In many languages with this array style, such as NumPy and Julia, the arrays are [[wikipedia:Immutable object|mutable]], meaning that copies of an array can be made, so that one copy reflects changes made to any copy. In contrast, APL operations that appear to modify an array, like [[indexed assignment]], will only change the particular copy of the array used, and can be said to create a new array rather than change an existing one: there is no special connection between the old and modified array. Mutable arrays make it possible for an array to contain itself, by replacing one element of an existing array with the whole array. This means that more values are possible than in an immutable based array language, and that some properties of immutable arrays, such as a finite [[depth]], do not hold. | |||
== Other features of the array model == | == Other features of the array model == | ||
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=== Numeric type coercion === | === Numeric type coercion === | ||
Most APLs, flat or nested, implicitly store simple numeric arrays as one of many [[numeric type]]s. When a numeric array is formed from numbers with different types, all numbers are converted to a common type in order to be represented as a flat array. If the hierarchy of numeric types is not strict, that is, there are some pairs of numeric types for which neither type is a subset of the other, then this coercion may affect the behavior of the numbers in the array. For example, [[J]] on a 64-bit machine uses both 64-bit integers and [[wikipedia:IEEE_754|double-precision floats]]. [[Catenate|Catenating]] the two results in an array of doubles, which will lose precision for integers whose absolute value is larger than 2<sup>53</sup>. In [[Dyalog APL]] a similar issue occurs with [[decimal float]]s and [[complex number]]s: combining the two results in an array of complex numbers, but this loses precision since Dyalog's complex numbers are stored as pairs of double-precision floats and its 128-bit decimal floats have higher precision | Most APLs, flat or nested, implicitly store simple numeric arrays as one of many [[numeric type]]s. When a numeric array is formed from numbers with different types, all numbers are converted to a common type in order to be represented as a flat array. If the hierarchy of numeric types is not strict, that is, there are some pairs of numeric types for which neither type is a subset of the other, then this coercion may affect the behavior of the numbers in the array. For example, [[J]] on a 64-bit machine uses both 64-bit integers and [[wikipedia:IEEE_754|double-precision floats]]. [[Catenate|Catenating]] the two results in an array of doubles, which will lose precision for integers whose absolute value is larger than 2<sup>53</sup>. In [[Dyalog APL]] a similar issue occurs with [[decimal float]]s and [[complex number]]s: combining the two results in an array of complex numbers, but this loses precision since Dyalog's complex numbers are stored as pairs of double-precision floats and its 128-bit decimal floats have higher precision than doubles. | ||
== Array characteristics == | == Array characteristics == | ||
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In nested APLs, a simple non-scalar array has depth 1, an array containing only depth 1 arrays has depth 2, and a simple scalar (e.g a number or character) has depth 0. | In nested APLs, a simple non-scalar array has depth 1, an array containing only depth 1 arrays has depth 2, and a simple scalar (e.g a number or character) has depth 0. | ||
Most APLs provide a Depth function < | Most APLs provide a Depth function <syntaxhighlight lang=apl inline>≡</syntaxhighlight> to find an array's depth. For example: | ||
< | <syntaxhighlight lang=apl> | ||
≡('ab' 'cde')('fg' 'hi') | ≡('ab' 'cde')('fg' 'hi') | ||
3 | 3 | ||
</ | </syntaxhighlight> | ||
APLs vary in their definition of depth: for example some may return the depth with a sign to indicate that some level of the array mixes elements of different depths. | APLs vary in their definition of depth: for example some may return the depth with a sign to indicate that some level of the array mixes elements of different depths. | ||
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== External links == | == External links == | ||
* [ | * [https://help.dyalog.com/latest/index.htm#Language/Introduction/Variables/Arrays.htm Dyalog array model] | ||
* [https://chat.stackexchange.com/rooms/52405/conversation/lesson-1-introduction-to-arrays-in-apl APL Cultivation] | * [https://chat.stackexchange.com/rooms/52405/conversation/lesson-1-introduction-to-arrays-in-apl APL Cultivation] | ||
* [https://www.sacrideo.us/tag/apl-a-day/ APL a Day] series | * [https://www.sacrideo.us/tag/apl-a-day/ APL a Day] series | ||
{{APL features}} | * [https://www.jsoftware.com/papers/array.htm What is an Array?] by [[Roger Hui]] | ||
== References == | |||
<references /> | |||
{{APL features}}[[Category:Arrays| ]] | |||