Add array indexing specification

spec/API_specification/indexing.md

@@ -1 +1,159 @@
+.. _indexing:
+
 # Indexing
+
+> Array API specification for indexing arrays.
+
+A conforming implementation of the array API standard must adhere to the following conventions.
+
+## Single-axis Indexing
+
+To index a single array axis, an array must support standard Python indexing rules. Let `n` be the axis (dimension) size.
+
+-   An integer index must be an object satisfying [`operator.index`](https://www.python.org/dev/peps/pep-0357/) (e.g., `int`).
+
+-   Nonnegative indices must start at `0` (i.e., zero-based indexing).
+
+-   **Valid** nonnegative indices must reside on the half-open interval `[0, n)`.
+
+    .. note::
+
+        This specification does not explicitly require bounds checking. The behavior for out-of-bounds integer indices is left unspecified.
+
+-   Negative indices must count backward from the last array index, starting from `-1` (i.e., negative-one-based indexing, where `-1` refers to the last array index).
+
+    .. note::
+
+        A negative index `j` is equivalent to `n-j`; the former is syntactic sugar for the latter, providing a shorthand for indexing elements that would otherwise need to be specified in terms of the axis (dimension) size.
+
+-   **Valid** negative indices must reside on the closed interval `[-n, -1]`.
+
+    .. note::
+
+        This specification does not explicitly require bounds checking. The behavior for out-of-bounds integer indices is left unspecified.
+
+-   A negative index `j` is related to a zero-based nonnegative index `i` via `i = n+j`.
+
+-   Colons `:` must be used for [slices](https://docs.python.org/3/library/functions.html#slice): `start:stop:step`, where `start` is inclusive and `stop` is exclusive.
+
+### Slice Syntax
+
+The basic slice syntax is `i:j:k` where `i` is the starting index, `j` is the stopping index, and `k` is the step (`k != 0`). A slice may contain either one or two colons, with either an integer value or nothing on either side of each colon. The following are valid slices.
+
+```text
+A[:]
+A[i:]
+A[:j]
+A[i:k]
+A[::]
+A[i::]
+A[:j:]
+A[::k]
+A[i:j:]
+A[i::k]
+A[:j:k]
+A[i::k]
+A[i:j:k]
+```
+
+.. note::
+
+    Slice syntax can be equivalently achieved using the Python built-in [`slice()`](https://docs.python.org/3/library/functions.html#slice) API. From the perspective from `A`, the behavior of `A[i:j:k]` and `A[slice(i, j, k)]` is indistinguishable (i.e., both retrieve the same set of items from `__getitem__`).
+
+Using a slice to index a single array axis must select `m` elements with index values
+
+```text
+i, i+k, i+2k, i+3k, ..., i+(m-1)k
+```
+
+where
+
+```text
+m = q + r
+```
+
+and `q` and `r` (`r != 0`) are the quotient and remainder obtained by dividing `j-i` by `k`
+
+```text
+j - i = qk + r
+```
+
+such that
+
+```text
+j > i + (m-1)k
+```
+
+.. note::
+
+    For `i` on the interval `[0, n)` (where `n` is the axis size), `j` on the interval `(0, n]`, `i` less than `j`, and positive step `k`, a starting index `i` is **always** included, while the stopping index `j` is **always** excluded. This preserves `x[:i]+x[i:]` always being equal to `x`.
+
+.. note::
+
+    Using a slice to index into a single array axis should select the same elements as using a slice to index a Python list of the same size.
+
+Slice syntax must have the following defaults. Let `n` be the axis (dimension) size.
+
+-   If `k` is not provided (e.g., `0:10`), `k` must equal `1`.
+-   If `k` is greater than `0` and `i` is not provided (e.g., `:10:2`), `i` must equal `0`.
+-   If `k` is greater than `0` and `j` is not provided (e.g., `0::2`), `j` must equal `n`.
+-   If `k` is less than `0` and `i` is not provided (e.g., `:10:-2`), `i` must equal `n-1`.
+-   If `k` is less than `0` and `j` is not provided (e.g., `0::-2`), `j` must equal `-n-1`.
+
+Using a slice to index a single array axis must adhere to the following rules. Let `n` be the axis (dimension) size.
+
+-   If `i` equals `j`, a slice must return an empty array, whose axis (dimension) size along the indexed axis is `0`.
+
+-   If `i` and `j` resolve to starting and stopping indices, respectively, which are out-of-bounds (i.e., a starting index less than `0` and/or a stopping index greater than `n`), then the respective index is clipped to the array axis bounds. For a starting index, the bound is `0`. For a stopping index, the bound is `n`.
+
+-   Indexing a single array axis with an out-of-bounds slice (i.e., a slice which does not select any array axis elements) must return an empty array, whose axis (dimension) size along the indexed axis is `0`.
+
+-   Indexing via `:` and `::` must be equivalent and have defaults derived from the rules above. Both `:` and `::` indicate to select all elements along a single axis (dimension).
+
+## Multi-axis Indexing
+
+Multi-dimensional arrays must extend the concept of single-axis indexing to multiple axes by applying single-axis indexing rules along each axis (dimension) and supporting the following additional rules. Let `N` be the number of dimensions ("rank") of a multi-dimensional array `A`.
+
+-   Each axis may be independently indexed via single-axis indexing by providing a comma-separated sequence ("selection tuple") of single-axis indexing expressions (e.g., `A[:, 2:10, :, 5]`).
+
+    .. note::
+
+        In Python, `x[(exp1, exp2, ..., expN)]` is equivalent to `x[exp1, exp2, ..., expN]`; the latter is syntactic sugar for the former.
+
+-   Providing a single nonnegative integer `i` as a single-axis index must index the same elements as the slice `i:i+1`.
+
+-   Providing a single negative integer `i` as a single-axis index must index the same elements as the slice `n+i:n`, where `n` is the axis (dimension) size.
+
+-   Providing a single integer as a single-axis index must reduce the number of array dimensions by `1` (i.e., the array rank should decrease by one; if `A` has rank `2`, `rank(A)-1 == rank(A[0, :])`). In particular, a selection tuple with the `m`th element an integer (and all other entries `:`) indexes a sub-array with rank `N-1`.
+
+-   Providing a slice must retain array dimensions (i.e., the array rank must remain the same; `rank(A) == rank(A[:])`).
+
+-   For each slice which attempts to select elements along a particular axis, but whose starting index is out-of-bounds, the axis (dimension) size of the result array must be `0`.
+
+-   If the number of provided single-axis indexing expressions is less than `N`, then `:` must be assumed for the remaining dimensions (e.g., if `A` has rank `2`, `A[2:10] == A[2:10, :]`).
+
+-   An `IndexError` exception must be raised if the number of provided single-axis indexing expressions is greater than `N`.
+
+-   Providing [ellipsis](https://docs.python.org/3/library/constants.html#Ellipsis) must apply `:` to each dimension necessary to index all dimensions (e.g., if `A` has rank `4`, `A[1:, ..., 2:5] == A[1:, :, :, 2:5]`). Only a single ellipsis must be allowed. An `IndexError` exception must be raised if more than one ellipsis is provided.
+
+-   The result of multi-axis indexing must be an array of the same data type as the indexed array.
+
+## Boolean Array Indexing
+
+An array must support indexing via a **single** `M`-dimensional boolean array `B` with shape `S1 = (s1, ..., sM)` according to the following rules. Let `A` be an `N`-dimensional array with shape `S2 = (s1, ..., sM, ..., sN)`.
+
+-   If `N >= M`, then `A[B]` must replace the first `M` dimensions of `A` with a single dimension having a size equal to the number of `True` elements in `B`.
+
+    .. note::
+
+        For example, if `N == M == 2`, indexing `A` via a boolean array `B` will return a one-dimensional array whose size is equal to the number of `True` elements in `B`.
+
+-   If `N < M`, then an `IndexError` exception must be raised.
+
+-   The size of each dimension in `B` must equal the size of the corresponding dimension in `A` or be `0`, beginning with the first dimension in `A`. If a dimension size does not equal the size of the corresponding dimension in `A` and is not `0`, then an `IndexError` exception must be raised.
+
+-   The elements of a boolean index array must be iterated in row-major, C-style order, with the exception of zero-dimensional boolean arrays.
+
+-   A zero-dimensional boolean index array (equivalent to `True` or `False`) must follow the same axis replacement rules stated above. Namely, a zero-dimensional boolean index array removes zero dimensions and adds a single dimension of length `1` if the index array's value is `True` and of length `0` if the index array's value is `False`. Accordingly, for a zero-dimensional boolean index array `B`, the result of `A[B]` has shape `S = (1, s1, ..., sN)` if the index array's value is `True` and has shape `S = (0, s1, ..., sN)` if the index array's value is `False`.
+
+-   The result of indexing into an array via a boolean index array must be an array of the same data type as the indexed array.

spec/API_specification/searching_functions.md

@@ -1,3 +1,5 @@
+.. _searching-functions:
+
 # Searching Functions
 
 > Array API specification for functions for searching arrays.