bkietz commented on a change in pull request #10296:
URL: https://github.com/apache/arrow/pull/10296#discussion_r635520473



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File path: docs/source/cpp/authoring_compute_functions.rst
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+.. Licensed to the Apache Software Foundation (ASF) under one
+.. or more contributor license agreements.  See the NOTICE file
+.. distributed with this work for additional information
+.. regarding copyright ownership.  The ASF licenses this file
+.. to you under the Apache License, Version 2.0 (the
+.. "License"); you may not use this file except in compliance
+.. with the License.  You may obtain a copy of the License at
+
+..   http://www.apache.org/licenses/LICENSE-2.0
+
+.. Unless required by applicable law or agreed to in writing,
+.. software distributed under the License is distributed on an
+.. "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+.. KIND, either express or implied.  See the License for the
+.. specific language governing permissions and limitations
+.. under the License.
+
+.. default-domain:: cpp
+.. highlight:: cpp
+.. cpp:namespace:: arrow::compute
+
+===========================
+Authoring Compute Functions
+===========================
+
+Compute Functions
+=================
+
+An introduction to compute functions is provided in 
https://arrow.apache.org/docs/cpp/compute.html.
+
+The [compute 
submodule](https://github.com/edponce/arrow/tree/master/cpp/src/arrow/compute) 
contains analytical functions that process primarily columnar data for either 
scalar or Arrow-based array inputs. These are intended for use inside query 
engines, data frame libraries, etc.
+
+Many functions have SQL-like semantics in that they perform element-wise or 
scalar operations on whole arrays at a time. Other functions are not SQL-like 
and compute results that may be a different length or whose results depend on 
the order of the values.
+
+Terminology:
+* The term compute "function" refers to a particular general operation that 
may have many different implementations corresponding to different combinations 
of types or function behavior options.
+* A specific implementation of a function is a "kernel". Selecting a viable 
kernel for executing a function is referred to as "dispatching". When executing 
a function on inputs, we must first select a suitable kernel corresponding to 
the value types of the inputs is selected.
+* Functions along with their kernel implementations are collected in a 
"function registry". Given a function name and argument types, we can look up 
that function and dispatch to a compatible kernel.
+
+[Compute 
functions](https://github.com/apache/arrow/blob/master/cpp/src/arrow/compute/function.h)
 have the following principal attributes:
+* A unique 
["name"](https://arrow.apache.org/docs/cpp/api/compute.html#_CPPv4NK5arrow7compute8Function4nameEv)
 used for function invocation and language bindings
+* A 
["kind"](https://arrow.apache.org/docs/cpp/api/compute.html#_CPPv4N5arrow7compute8Function4KindE)
+  which indicates in what context it is valid for use
+    * Input/output 
[types](https://arrow.apache.org/docs/cpp/compute.html#type-categories) and 
[shapes](https://arrow.apache.org/docs/cpp/compute.html#input-shapes)
+    * Compute functions can also be further "categorized" based on the type of 
operation performed. For example, `Scalar Arithmetic` vs `Scalar String`.
+* Compute functions (see [FunctionImpl and 
subclasses](https://github.com/apache/arrow/blob/master/cpp/src/arrow/compute/function.h))
 contain 
["kernels"](https://github.com/edponce/arrow/tree/master/cpp/src/arrow/compute/kernels)
 which are implementations for specific argument signatures.
+* An 
["arity"](https://arrow.apache.org/docs/cpp/api/compute.html#_CPPv4N5arrow7compute5ArityE)
 which states the number of required arguments
+for its core operation. Functions are commonly nullary, unary, binary, or 
ternary, but can also be variadic.
+* 
["Documentation"](https://arrow.apache.org/docs/cpp/api/compute.html#_CPPv4N5arrow7compute11FunctionDocE)
 describing the function's functionality and behavior
+* 
["Options"](https://arrow.apache.org/docs/cpp/api/compute.html#_CPPv4N5arrow7compute15FunctionOptionsE)
 specifying configuration of the function's behavior.
+
+Compute functions are grouped in source files based on their "kind" in 
https://github.com/apache/arrow/tree/master/cpp/src/arrow/compute.
+Kernels of compute functions are grouped in source files based on their "kind" 
and category, see 
https://github.com/edponce/arrow/tree/master/cpp/src/arrow/compute/kernels.
+
+
+Kinds of compute functions
+--------------------------
+
+Arrow uses an enumerated type to identify the kind of a compute function, 
refer to
+https://github.com/edponce/arrow/tree/master/cpp/src/arrow/compute/function.h
+
+Scalar
+~~~~~~
+
+A function that performs scalar data operations on whole arrays of
+data. Can generally process Array or Scalar values. The size of the
+output will be the same as the size (or broadcasted size, in the case
+of mixing Array and Scalar inputs) of the input.
+
+https://arrow.apache.org/docs/cpp/compute.html#arithmetic-functions
+
+**Categories of Scalar functions**
+
+* Arithmetic
+* Comparisons
+* Logical
+* String
+    * predicates
+    * transforms
+    * trimming
+    * splitting
+    * extraction
+* Containment tests
+* Structural transforms
+* Conversions
+
+
+Vector
+~~~~~~
+
+A function with array input and output whose behavior depends on the
+values of the entire arrays passed, rather than the value of each scalar value.
+
+**Categories of Vector functions**
+
+* Associative transforms
+* Selections
+* Sorts and partitions
+* Structural transforms
+
+
+Scalar aggregate
+~~~~~~~~~~~~~~~~
+
+A function that computes scalar summary statistics from array input.
+
+### Hash aggregate

Review comment:
       In addition, please ensure that all your links are in the RST format. 
For example, to create a link to the doxygen doc for a specific class member, 
use:
   ```rst
    :member:`ScalarKernel::exec`
   ```
   
   To create a link to a specific source file on the `master` branch, use:
   ```rst
   `The scalar API header 
<https://github.com/apache/arrow/blob/master/cpp/src/arrow/compute/api_scalar.h>`__
   ```




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