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:''This page is about array depth, a number associated with every array. For the operator that applies a function at a given depth, see [[Depth (operator)]].'' | |||
{{Built-in|Depth|≡}} is a [[monadic]] [[primitive function]] that returns an array's depth. In the APL [[array model]], the '''depth''' of an array is the number of levels of [[Nested array model|nesting]] or [[box]]ing it exhibits. In some languages, Depth returns a negative result to indicate that not all paths through the array have the same depth. | {{Built-in|Depth|≡}} is a [[monadic]] [[primitive function]] that returns an array's depth. In the APL [[array model]], the '''depth''' of an array is the number of levels of [[Nested array model|nesting]] or [[box]]ing it exhibits. In some languages, Depth returns a negative result to indicate that not all paths through the array have the same depth. | ||
== Nested array depth == | == Nested array depth == | ||
Nested APLs vary in their definition of depth. They may take into account the array's [[prototype]], or not, and may use the positive | Nested APLs vary in their definition of depth. They may take into account the array's [[prototype]], or not, and may use the positive or signed depth as defined below (the choice may also depend on [[migration level]]). The APL Wiki generally uses "depth" to mean the positive depth. | ||
In the [[nested array model]], the depth is defined using the base case of a [[simple scalar]], which by definition has depth 0. | In the [[nested array model]], the depth is defined using the base case of a [[simple scalar]], which by definition has depth 0. | ||
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The positive depth of a non-[[empty]] array other than a simple scalar is defined to be the largest depth among its [[element]]s, plus one. Thus a [[simple]] but non-[[scalar]] array has depth 1. | The positive depth of a non-[[empty]] array other than a simple scalar is defined to be the largest depth among its [[element]]s, plus one. Thus a [[simple]] but non-[[scalar]] array has depth 1. | ||
The positive depth of an [[empty]] array is usually defined (for example, in [[Dyalog APL]]) to be the depth of its [[prototype]] plus one. It can also be set to 1, since it contains no elements but is not a simple scalar. This is the case in [[ngn/apl]]. | The positive depth of an [[empty]] array is usually defined (for example, in [[Dyalog APL]]) to be the depth of its [[prototype]] plus one. It can also be set to 1, since it contains no elements but is not a simple scalar. This is the case in [[ngn/apl]] and [[dzaima/APL]]. | ||
An array has a ''consistent depth'' if it is a simple scalar, or if all of its elements (including the prototype, if | An array has a ''consistent depth'' if it is a simple scalar, or if all of its elements (including the prototype, if prototypes are used to determine depth) have a consistent depth and are equal in depth. For example, the following works in all nested APLs with [[stranding]]: | ||
For | |||
<source lang=apl> | <source lang=apl> | ||
≡('ab' 'cde')('fg' 'hi') | ≡('ab' 'cde')('fg' 'hi') | ||
3 | 3 | ||
</source> | </source> | ||
The signed depth of an array is an integer with [[Magnitude|absolute value]] equal to its positive depth. It is negative if and only if the array does not have a consistent depth. | |||
A [[simple]] array must have a consistent depth, because it is either a [[simple scalar]] or contains only simple scalars. In the latter case each element necessarily has depth 0 and a consistent depth. Because of this it is not possible to have an array with a signed depth of ¯1: any array with a positive depth of 1 must be simple, and hence have consistent depth. | A [[simple]] array must have a consistent depth, because it is either a [[simple scalar]] or contains only simple scalars. In the latter case each element necessarily has depth 0 and a consistent depth. Because of this it is not possible to have an array with a signed depth of ¯1: any array with a positive depth of 1 must be simple, and hence have consistent depth. | ||
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In the [[flat array model]], the depth is the number of levels of [[box]]ing in an array. More precisely, the depth of a non-boxed or [[empty]] array is 0, and a non-empty boxed array has depth equal to one plus the maximum of the depths of the arrays it contains. | In the [[flat array model]], the depth is the number of levels of [[box]]ing in an array. More precisely, the depth of a non-boxed or [[empty]] array is 0, and a non-empty boxed array has depth equal to one plus the maximum of the depths of the arrays it contains. | ||
The [[J]] language uses the token <source lang= | The [[J]] language uses the token <source lang=j inline>L.</source> and name "Level Of" for depth. | ||
== External links == | == External links == | ||
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* [https://code.jsoftware.com/wiki/Vocabulary/dollar#dyadic J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/lcapdot J NuVoc] (as <source lang=apl inline>L.</source> "Level of") | * [https://code.jsoftware.com/wiki/Vocabulary/dollar#dyadic J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/lcapdot J NuVoc] (as <source lang=apl inline>L.</source> "Level of") | ||
{{APL features}} | {{APL features}} | ||
{{APL built-ins}} | {{APL built-ins}}[[Category:Primitive functions]] | ||