From 229e2cd2f5c78b13c483a8559dead2c8f31d8e42 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Sat, 18 Jul 2020 18:26:52 -0400 Subject: Terminology changes: subject, 1/2-modifier, Box/Unbox to Enclose/Merge, blocks --- docs/doc/depth.html | 32 ++++++++++++++++---------------- 1 file changed, 16 insertions(+), 16 deletions(-) (limited to 'docs/doc/depth.html') diff --git a/docs/doc/depth.html b/docs/doc/depth.html index 9becdb67..c99a75e4 100644 --- a/docs/doc/depth.html +++ b/docs/doc/depth.html @@ -1,6 +1,6 @@

Depth

-

The depth of an array is the greatest level of array nesting it attains, or, put another way, the greatest number of times you can pick an element starting from the original array before reaching a non-array. The monadic function Depth (≡) returns the depth of its argument, while the composition Depth (⚇) can control the way its left operand is applied based on the depth of its arguments. Several primitive functions also use the depth of the left argument to decide whether it applies to a single axis of the right argument or to several axes.

+

The depth of an array is the greatest level of array nesting it attains, or, put another way, the greatest number of times you can pick an element starting from the original array before reaching a non-array. The monadic function Depth (≡) returns the depth of its argument, while the 2-modifier Depth (⚇) can control the way its left operand is applied based on the depth of its arguments. Several primitive functions also use the depth of the left argument to decide whether it applies to a single axis of the right argument or to several axes.

The Depth function

To find the depth of an array, use Depth (≡). For example, the depth of a list of numbers or characters is 1:

    ≡ 2‿3‿4
@@ -8,7 +8,7 @@
     ≡ "a string is a list of characters"
 1
-

Depth is somewhat analogous to an array's rank ≠≢𝕩, and in fact rank can be "converted" to depth by splitting rows with <⎉1, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:

+

Depth is somewhat analogous to an array's rank ≠≢𝕩, and in fact rank can be "converted" to depth by splitting rows with <⎉1, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:

    ≡ 3‿4⥊"characters"
 1
     ≡ (1+↕10)⥊"characters"
@@ -22,7 +22,7 @@
     ≡ ⟨2,<3,4,<<<5⟩
 4
-

As the above expressions suggest, the depth of an array is the maximum of its elements, plus one. The base case, a non-array (including a function, modifier, or combinator), has depth 0.

+

As the above expressions suggest, the depth of an array is the maximum of its elements' depths, plus one. The base case, a non-array (including a function or modifier), has depth 0.

    ≡'c'
 0
     F←+⋄≡f
@@ -32,8 +32,8 @@
     ≡⟨5,⟨'c',f,2⟩⟩
 2
-

If the function IsArray indicates whether its argument is an array, then we can write a recursive definition of Depth using the Choose composition.

-
Depth←IsArray◶0‿{1+0⌈´Depth¨⥊𝕩}
+
If the function IsArray indicates whether its argument is an array, then we can write a recursive definition of Depth using the Choose modifier.

+
Depth←IsArray◶0‿{1+0⌈´Depth¨⥊𝕩}
 

 
The minimum element depth of 0 implies that an empty array's depth is 1.

 
    ≡⟨⟩
@@ -74,22 +74,22 @@
   [ 2 1 ] [ 2 4 ] [ 2 1 ]
                           ┘
 

-
This means the left argument is homogeneous of depth 2. What should an argument of depth 1, or an argument that contains non-arrays, do? One option is to continue to require the left argument to be a vector, and convert any non-array argument into an array by boxing it:

-
    ⟨3‿2,1⟩ <⍟(0=≡)¨⊸⊏ ↕6‿7
+
This means the left argument is homogeneous of depth 2. What should an argument of depth 1, or an argument that contains non-arrays, do? One option is to continue to require the left argument to be a list, and convert any non-array argument into an array by enclosing it:

+
    ⟨3‿2,1⟩ <⍟(0=≡)¨⊸⊏ ↕6‿7
 [ [ 3 1 ] [ 2 1 ] ]
 

 
While very consistent, this extension represents a small convenience and makes it difficult to act on a single axis, which for Replicate and Group is probably the most common way the primitive is used:

 
    3‿2‿1‿2‿3 / "abcde"
 [ aaabbcddeee ]
 

-
With the extension above, every case like this would have to use <⊸/ instead of just /. BQN avoids this difficulty by testing the left argument's depth. A depth-1 argument applies to the first axis only, giving the behavior above.

+
With the extension above, every case like this would have to use <⊸/ instead of just /. BQN avoids this difficulty by testing the left argument's depth. A depth-1 argument applies to the first axis only, giving the behavior above.

 
For Select, the depth-1 case is still quite useful, but it may also be desirable to choose a single cell using a list of numbers. In this case the left argument depth can be increased from the bottom using <¨.

-
    2‿1‿4 <¨⊸⊏ ↕3‿4‿5‿2
+
    2‿1‿4 <¨⊸⊏ ↕3‿4‿5‿2
 [ [ 2 1 4 0 ] [ 2 1 4 1 ] ]
 

-
The Depth composition

-
The Depth composition (⚇) is a generalization of Each that allows diving deeper into an array. To illustrate it we'll use a shape 4‿3 array of lists of lists.

-
    ⊢ n ← <⎉1⍟2 4‿3‿2‿2⥊↕48
+
The Depth modifier

+
The Depth 2-modifier (⚇) is a generalization of Each that allows diving deeper into an array. To illustrate it we'll use a shape 4‿3 array of lists of lists.

+
    ⊢ n ← <⎉1⍟2 4‿3‿2‿2⥊↕48
 ┌
   [ [ 0 1 ] [ 2 3 ] ]     [ [ 4 5 ] [ 6 7 ] ]     [ [ 8 9 ] [ 10 11 ] ]
   [ [ 12 13 ] [ 14 15 ] ] [ [ 16 17 ] [ 18 19 ] ] [ [ 20 21 ] [ 22 23 ] ]
@@ -109,14 +109,14 @@
                                                                           ┘
 

 
Depth ¯1 is equivalent to Each, and reverses the larger vectors, while depth ¯2 applies Each twice to reverse the smaller vectors:

-
    ⌽⚇¯1 n
+
    ⌽⚇¯1 n
 ┌
   [ [ 2 3 ] [ 0 1 ] ]     [ [ 6 7 ] [ 4 5 ] ]     [ [ 10 11 ] [ 8 9 ] ]
   [ [ 14 15 ] [ 12 13 ] ] [ [ 18 19 ] [ 16 17 ] ] [ [ 22 23 ] [ 20 21 ] ]
   [ [ 26 27 ] [ 24 25 ] ] [ [ 30 31 ] [ 28 29 ] ] [ [ 34 35 ] [ 32 33 ] ]
   [ [ 38 39 ] [ 36 37 ] ] [ [ 42 43 ] [ 40 41 ] ] [ [ 46 47 ] [ 44 45 ] ]
                                                                           ┘
-    ⌽⚇¯2 n
+    ⌽⚇¯2 n
 ┌
   [ [ 1 0 ] [ 3 2 ] ]     [ [ 5 4 ] [ 7 6 ] ]     [ [ 9 8 ] [ 11 10 ] ]
   [ [ 13 12 ] [ 15 14 ] ] [ [ 17 16 ] [ 19 18 ] ] [ [ 21 20 ] [ 23 22 ] ]
@@ -125,11 +125,11 @@
                                                                           ┘
 

 
While a negative depth tells how many levels to go down, a non-negative depth gives the maximum depth of the argument before applying the left operand. On a depth-3 array like above, depth 2 is equivalent to ¯1 and depth 1 is equivalent to ¯2. A depth of 0 means to loop until non-arrays are reached, that is, apply pervasively, like a scalar function.

-
    ⟨'a',"bc"⟩ ≍⚇0 ⟨2‿3,4⟩
+
    ⟨'a',"bc"⟩ ≍⚇0 ⟨2‿3,4⟩
 [ [ [ a 2 ] [ a 3 ] ] [ [ b 4 ] [ c 4 ] ] ]
 

 
With a positive operand, Depth doesn't have to use the same depth everywhere. Here, Length is applied as soon as the depth for a particular element is 1 or less, including if the argument has depth 0. For example, it maps over ⟨2,⟨3,4⟩⟩, but not over ⟨11,12⟩, even though these are elements of the same array.

-
    ≠⚇1 ⟨1,⟨2,⟨3,4⟩⟩,⟨5,⟨6,7⟩,⟨8,9,10⟩⟩,⟨11,12⟩⟩
+
    ≠⚇1 ⟨1,⟨2,⟨3,4⟩⟩,⟨5,⟨6,7⟩,⟨8,9,10⟩⟩,⟨11,12⟩⟩
 [ 1 [ 1 2 ] [ 1 2 3 ] 2 ]
 

 
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