# Depth

**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 nesting or boxing it exhibits. In some languages, Depth returns a negative result to indicate that not all paths through the array have the same depth.

## Contents

## 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 depth, signed depth, or minimum 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.

The positive depth of a non-empty array other than a simple scalar is defined to be the largest depth among its elements, 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.

An array has a *consistent depth* if it is a simple scalar, or if all of its elements (including the prototype, if prototype is used to determine depth) have a consistent depth and are equal in depth. The signed depth of an array is an integer with absolute value equal to its positive depth. It is negative if and only if it does not have a consistent depth.

dzaima/APL uses the minimum depth, which is 0 for a simple scalar and otherwise is one plus the *minimum* (rather than maximum) of the elements of a non-empty array. It defines the depth of an empty array to be one plus the depth of its prototype.

For arrays with a consistent depth the positive, signed, and minimum depth coincide. Thus the following example works in all nested APLs with stranding.

```
≡('ab' 'cde')('fg' 'hi')
3
```

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.

## Flat array depth

In the flat array model, the depth is the number of levels of boxing 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 `L.`

and name "Level Of" for depth.

## External links

### Lessons

### Documentation

- Dyalog
- APLX
- J Dictionary, J NuVoc (as
`L.`

"Level of")

APL features [edit]
| |
---|---|

Built-ins | Primitive function ∙ Primitive operator ∙ Quad name |

Array model | Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Box ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype |

Concepts and paradigms | Leading axis theory ∙ Scalar extension ∙ Conformability ∙ Scalar function ∙ Glyph ∙ Identity element |

Errors | LIMIT ERROR ∙ RANK ERROR |

APL built-ins [edit]
| |||
---|---|---|---|

Primitive functions | |||

Scalar | |||

Monadic | Conjugate ∙ Not ∙ Roll ∙ Type | ||

Dyadic | Add ∙ Subtract ∙ Equal to (Xnor) ∙ Not Equal to (Xor) | ||

Non-Scalar | |||

Structural | Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Reverse ∙ Raze ∙ Mix ∙ Cut (K) | ||

Selection | Take ∙ Drop ∙ Unique ∙ Identity | ||

Selector | Interval Index | ||

Computational | Match ∙ Not Match ∙ Nub Sieve | ||

Primitive operators | Each | ||

Quad names | |||

Arrays | Index origin ∙ Migration level | ||

Functions | |||

Operators | |||

Other | Zilde ∙ High minus ∙ Function axis |