From c747cbab2847d454615712403f495f73823db917 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Sat, 6 Nov 2021 22:50:34 -0400 Subject: Update Join description in spec --- docs/spec/primitive.html | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'docs') diff --git a/docs/spec/primitive.html b/docs/spec/primitive.html index cb03f6f8..b1c115f2 100644 --- a/docs/spec/primitive.html +++ b/docs/spec/primitive.html @@ -121,7 +121,7 @@

Restructuring

Enclose (<) forms a unit array that contains its argument. Enlist and Pair (⋈) form a 1- or 2-element list of all arguments, that is, ⟨𝕩⟩ or ⟨𝕨,𝕩⟩.

Merge (>) combines the outer axes of an array of arrays with inner axes: it requires that all elements of its argument have the same shape, and creates an array such that (i∾j)⊑>𝕩 is i⊑j⊑𝕩. It also accepts atom elements of 𝕩, converting them to unit arrays, or an atom argument, which is returned unchanged. Solo and Couple (≍) turn one or two arguments into major cells of the result and can be defined easily in terms of Merge.

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Join To (∾) combines its two arguments along an existing initial axis, unless both arguments are units, in which case it creates an axis and is identical to Couple (≍). The arguments must differ in rank by at most 1, and the result rank is equal to the maximum of 1 and the higher argument rank. Each argument with rank less than the result, and each major cell of an argument with rank equal to it, becomes a major cell of the result, with cells from the left argument placed before those from the right. Join (∾) generalizes the equal-rank subset of this behavior to an array of values instead of just two. The argument must be an array (unlike Merge), and its elements must all the same rank, which is at least the argument rank. Atom elements are treated as unit arrays. Then "outer" argument axes are matched up with leading "inner" element axes, and elements are joined along these axes. In order to allow this, the length of an element along a particular axis must depend only on the position along the corresponding axis in the argument. An empty argument to Join is return unchanged, as though the element rank is equal to the argument rank.

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Join To (∾) combines its two arguments along an existing initial axis, unless both arguments are units, in which case it creates an axis and is identical to Couple (≍). The arguments must differ in rank by at most 1, and the result rank is equal to the maximum of 1 and the higher argument rank. Each argument with rank less than the result, and each major cell of an argument with rank equal to it, becomes a major cell of the result, with cells from the left argument placed before those from the right. Join (∾) generalizes this behavior to an array of values instead of just two. The argument must be an array (unlike Merge), and atom elements are treated as unit arrays. "Outer" argument axes are matched up with leading "inner" element axes, and elements are joined along these axes. In order to allow this, the length of an element along a particular axis must depend only on the position along the corresponding axis in the argument. However, the axis may also be omitted, resulting in an uneven rank among arguments of 𝕩. At least one position must have the axis, so that whether a given element has it can be found by comparing it to others along that axis. Along an axis, ranks can differ by at most one because each element either has or omits that axis. If the argument to Join is empty, the result is an empty array with shape computed based on an array of fill elements: the fill rank must be greater than the rank of 𝕩, and the result shape is computed by multiplying leading lengths of the fill shape by ≢𝕩, with the trailing shape unchanged.

Deshape (⥊) differs from the provided function (which returns the element list of an array) only in that it accepts an atom, returning a one-element list containing it. Reshape (⥊) is extended in numerous ways. It accepts any list of natural numbers (including as a unit array or atom) for the left argument and any right argument; 𝕩 is deshaped first so that it is treated as a list of elements. These elements are repeated cyclically to fill the result array in ravel order. If 𝕩 is empty then a non-empty requested result shape causes an error. Furthermore, at most one element of 𝕨 can be a "length code": one of the primitives ∘⌊⌽↑. In this case, a target length is computed from the number of elements in 𝕩 divided by the product of the other elements of 𝕨 (which must not be zero). If the target length is an integer then it is used directly for the length code. Otherwise, an error is given if the length code is ∘, and the target length is rounded down if the code is ⌊ and up if it's ⌽ or ↑. With code ⌽, elements are repeated cyclically as usual, but with code ↑, the extra elements after each argument element is used are fill values for 𝕩.

Transpose (⍉) reorders axes of its argument to place the first axis last; if the argument has one or fewer axes then it's enclosed if it's an atom and otherwise returned unchanged. Reorder Axes (⍉) requires the left argument to be a list or unit of natural numbers, with length at most the rank of the right argument. This list is extended to match the right argument rank exactly by repeatedly appending the least unused natural number (for example, given 1‿3‿0‿0, 2 is appended). After extension, it specifies a result axis for each axis of the right argument. There must be no gaps in the list: that is, with the result rank equal to one plus the greatest value present, every result axis must appear at least once. Now each argument axis is "sent to" the specified result axis: in terms of indices, i⊑𝕨⍉𝕩 is (𝕨⊏i)⊑𝕩 if 𝕨 is complete. If multiple argument axes correspond to the same result axis, then a diagonal is taken, and it's as long as the shortest of those argument axes. Like Transpose, Reorder Axes encloses 𝕩 if it's an atom, so that its result is always an array.

Indices and selection

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