PEP: 211 Title: Adding A New Outer Product Operator Author: Greg Wilson
<gvwilson@ddj.com> Status: Rejected Type: Standards Track Created:
15-Jul-2000 Python-Version: 2.1 Post-History:

The approach in the later PEP 465 was eventually accepted in lieu of
this PEP. The Rejected Ideas
<465#rejected-alternatives-to-adding-a-new-operator> of that PEP
explains the rationale in more detail.

Introduction

This PEP describes a proposal to define @ (pronounced "across") as a new
outer product operator in Python 2.2. When applied to sequences (or
other iterable objects), this operator will combine their iterators, so
that:

    for (i, j) in S @ T:
        pass

will be equivalent to:

    for i in S:
        for j in T:
            pass

Classes will be able to overload this operator using the special methods
__across__, __racross__, and __iacross__. In particular, the new Numeric
module (PEP 209) will overload this operator for multi-dimensional
arrays to implement matrix multiplication.

Background

Number-crunching is now just a small part of computing, but many
programmers --- including many Python users --- still need to express
complex mathematical operations in code. Most numerical languages, such
as APL, Fortran-90, MATLAB, IDL, and Mathematica, therefore provide two
forms of the common arithmetic operators. One form works
element-by-element, e.g. multiplies corresponding elements of its matrix
arguments. The other implements the "mathematical" definition of that
operation, e.g. performs row-column matrix multiplication.

Zhu and Lielens have proposed <225> doubling up Python's operators in
this way. Their proposal would create six new binary infix operators,
and six new in-place operators.

The original version of this proposal was much more conservative. The
author consulted the developers of GNU Octave[1], an open source clone
of MATLAB. Its developers agreed that providing an infix operator for
matrix multiplication was important: numerical programmers really do
care whether they have to write mmul(A,B) instead of A op B.

On the other hand, when asked how important it was to have infix
operators for matrix solution and other operations, Prof. James Rawlings
replied[2]:

  I DON'T think it's a must have, and I do a lot of matrix inversion. I
  cannot remember if its A\b or b\A so I always write inv(A)*b instead.
  I recommend dropping \.

Based on this discussion, and feedback from students at the US national
laboratories and elsewhere, we recommended adding only one new operator,
for matrix multiplication, to Python.

Iterators

The planned addition of iterators to Python 2.2 opens up a broader scope
for this proposal. As part of the discussion of PEP 201, Lockstep
Iteration, the author of this proposal conducted an informal usability
experiment[3]. The results showed that users are psychologically
receptive to "cross-product" loop syntax. For example, most users
expected:

    S = [10, 20, 30]
    T = [1, 2, 3]
    for x in S; y in T:
        print x+y,

to print 11 12 13 21 22 23 31 32 33. We believe that users will have the
same reaction to:

    for (x, y) in S @ T:
        print x+y

i.e. that they will naturally interpret this as a tidy way to write loop
nests.

This is where iterators come in. Actually constructing the cross-product
of two (or more) sequences before executing the loop would be very
expensive. On the other hand, @ could be defined to get its arguments'
iterators, and then create an outer iterator which returns tuples of the
values returned by the inner iterators.

Discussion

1.  Adding a named function "across" would have less impact on Python
    than a new infix operator. However, this would not make Python more
    appealing to numerical programmers, who really do care whether they
    can write matrix multiplication using an operator, or whether they
    have to write it as a function call.

2.  @ would have be chainable in the same way as comparison operators,
    i.e.:

        (1, 2) @ (3, 4) @ (5, 6)

    would have to return (1, 3, 5) ... (2, 4, 6), and not
    ((1, 3), 5) ... ((2, 4), 6). This should not require special support
    from the parser, as the outer iterator created by the first @ could
    easily be taught how to combine itself with ordinary iterators.

3.  There would have to be some way to distinguish restartable iterators
    from ones that couldn't be restarted. For example, if S is an input
    stream (e.g. a file), and L is a list, then S @ L is
    straightforward, but L @ S is not, since iteration through the
    stream cannot be repeated. This could be treated as an error, or by
    having the outer iterator detect non-restartable inner iterators and
    cache their values.

4.  Whiteboard testing of this proposal in front of three novice Python
    users (all of them experienced programmers) indicates that users
    will expect:

        "ab" @ "cd"

    to return four strings, not four tuples of pairs of characters.
    Opinion was divided on what:

        ("a", "b") @ "cd"

    ought to return...

Alternatives

1.  Do nothing --- keep Python simple.

    This is always the default choice.

2.  Add a named function instead of an operator.

    Python is not primarily a numerical language; it may not be worth
    complexifying it for this special case. However, support for real
    matrix multiplication is frequently requested, and the proposed
    semantics for @ for built-in sequence types would simplify
    expression of a very common idiom (nested loops).

3.  Introduce prefixed forms of all existing operators, such as ~* and
    ~+, as proposed in PEP 225.

    Our objections to this are that there isn't enough demand to justify
    the additional complexity (see Rawlings' comments[4]), and that the
    proposed syntax fails the "low toner" readability test.

Acknowledgments

I am grateful to Huaiyu Zhu for initiating this discussion, and to James
Rawlings and students in various Python courses for their discussions of
what numerical programmers really care about.

References

[1] http://bevo.che.wisc.edu/octave/

[2] http://www.egroups.com/message/python-numeric/4

[3] https://mail.python.org/pipermail/python-dev/2000-July/006427.html

[4] http://www.egroups.com/message/python-numeric/4