To: J3                                                     J3/25-186r2
From: Brandon Cook & Dan Bonachea
Subject:  Specifications and Syntax for Local Prefix Operation Intrinsics
Date: 2025-October-15
References: J3/25-145r1, WG5/N-2249

1. Background
=============

The Fortran 202Y work list (WG5/N-2249) includes work item US20:
"Add Intrinsic and collective subroutines for prefix operations"

Paper J3/25-145r1 "Requirements for US20: Local Prefix Operation
Intrinsics" presents illustrative use cases and requirements for local
prefix operation intrinsics. That paper was passed at J3 meeting #236
in June 2025.

This document focuses on specifications and syntax exclusively for the
local prefix reduction variant, refining previous concepts based on
community and WG5 feedback. By focusing exclusively on the local
variant our aim is to allow consideration independent of the closely
related but distinct collective subroutines.

2. SUM_PREFIX
=============

2.1 Syntax
-----------

For reference, the syntax for the existing SUM intrinsic is:

SUM(ARRAY, DIM [, MASK]) or
SUM(ARRAY [, MASK])

The new intrinsics are:

SUM_PREFIX_INCLUSIVE(ARRAY [, MASK]) or
SUM_PREFIX_INCLUSIVE(ARRAY, DIM [, MASK])
SUM_PREFIX_EXCLUSIVE(ARRAY, [, MASK]) or
SUM_PREFIX_EXCLUSIVE(ARRAY, DIM [, MASK])

2.2 Specifications
------------------

Arguments
---------

S01. ARRAY shall be an array of numeric type.

S03. DIM shall be an integer scalar with a value in the range
     1 <= DIM <= n, where n is the rank of ARRAY.

S05. MASK (optional) shall be of type logical and shall be conformable
     with ARRAY.

Result Characteristics
----------------------

S07. The result shall have the same type, rank, shape and kind
     parameter as ARRAY.

Result Value: SUM_PREFIX_INCLUSIVE
----------------------------------

S09. The inclusive prefix sum of a sequence S with n elements
     s_1, s_2, ..., s_n, is the sequence R with n elements
     r_1, r_2, ..., r_n, where r_i = \sum_{j=1}^i s_j.

S11. The result of SUM_PREFIX_INCLUSIVE(ARRAY) is the inclusive prefix
     sum of the sequence of elements of ARRAY in array element order,
     with the result sequence provided in array element order.

     Example:

     If A is | 1 2 3 |, SUM_PREFIX_INCLUSIVE(A) is | 1 3 6 |.

S13. The MASK argument affects the prefix sum as if ARRAY elements
     corresponding to false elements of MASK had been replaced by 0.
     Specifically, the result of SUM_PREFIX_INCLUSIVE(ARRAY, MASK) is
     equal to SUM_PREFIX_INCLUSIVE(MERGE(ARRAY, 0, MASK)).

     Example:

     If A is | 1 2 3| and M is | T F T |,
     SUM_PREFIX_INCLUSIVE(A, MASK = M) is
     SUM_PREFIX_INCLUSIVE( | 1 0 3 | ) == | 1 1 4 |.

S15. If ARRAY has rank one, the result of
     SUM_PREFIX_INCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK])
     has a value equal to
     SUM_PREFIX_INCLUSIVE(ARRAY [, MASK = MASK]).

     Otherwise, the result of
     SUM_PREFIX_INCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK]) is defined
     by the semantics of S13 applied independently along each sequence of
     ARRAY elements in dimension DIM. More specifically, the value of
     result elements (i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n) of
     SUM_PREFIX_INCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK]) is equal to
     SUM_PREFIX_INCLUSIVE(
       ARRAY(i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n),
       DIM = 1
    [, MASK(i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n)])

     Example:

     If A is | 1 2 3 |
             | 4 5 6 |,
     SUM_PREFIX_INCLUSIVE(A, DIM=2) is the array
         | SUM_PREFIX_INCLUSIVE(A(1,:)) |
         | SUM_PREFIX_INCLUSIVE(A(2,:)) |
     which is | 1 3 6  |
              | 4 9 15 |.

Result Value: SUM_PREFIX_EXCLUSIVE
----------------------------------

S19. The exclusive prefix sum of a sequence S with n elements
     s_1, s_2, ..., s_n, is the sequence R with n elements
     r_1, r_2, ..., r_n, where r_i = \sum_{j=0}^{i-1} s_j with s_0 = 0.

S21. The result of SUM_PREFIX_EXCLUSIVE(ARRAY) is the exclusive prefix
     sum of the sequence of elements of ARRAY in array element order,
     with the result sequence provided in array element order.

     Example:

     If A is | 1 2 3 |, SUM_PREFIX_EXCLUSIVE(A) is | 0 1 3 |.

S23. The MASK argument affects the prefix sum as if ARRAY elements
     corresponding to false elements of MASK had been replaced by 0.
     Specifically, the result of SUM_PREFIX_EXCLUSIVE(ARRAY, MASK) is
     equal to SUM_PREFIX_EXCLUSIVE(MERGE(ARRAY, 0, MASK)).

     Example:

     If A is | 1 2 3| and M is | T F T |,
     SUM_PREFIX_EXCLUSIVE(A, MASK = M) is
     SUM_PREFIX_EXCLUSIVE( | 1 0 3 | ) == | 0 1 1 |.

S25. If ARRAY has rank one, the result of
     SUM_PREFIX_EXCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK])
     has a value equal to
     SUM_PREFIX_EXCLUSIVE(ARRAY [, MASK = MASK]).

     Otherwise, the result of
     SUM_PREFIX_EXCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK]) is defined
     by the semantics of S23 applied independently along each sequence of
     ARRAY elements in dimension DIM. More specifically, the value of
     result elements (i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n) of
     SUM_PREFIX_EXCLUSIVE(ARRAY, DIM = DIM [, MASK = MASK]) is equal to
     SUM_PREFIX_EXCLUSIVE(
       ARRAY(i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n),
       DIM = 1
       [, MASK(i_1, i_2, ..., i_{DIM-1}, :, i_{DIM+1}, ..., i_n)])

     Example:

     If A is | 1 2 3 |
             | 4 5 6 |,
     SUM_PREFIX_EXCLUSIVE(A, DIM=2) is the array
        | SUM_PREFIX_EXCLUSIVE(A(1,:)) |
        | SUM_PREFIX_EXCLUSIVE(A(2,:)) |
     which is
        | 0 1 3 |
        | 0 4 9 |