PEP 225 -- Elementwise/Objectwise Operators
PEP: 225 Elementwise/Objectwise Operators \$Revision\$ \$Date\$ Huaiyu Zhu , Gregory Lielens Deferred Standards Track 19-Sep-2000 2.1

### Introduction

```    This PEP describes a proposal to add new operators to Python which
are useful for distinguishing elementwise and objectwise
operations, and summarizes discussions in the news group
comp.lang.python on this topic.  See Credits and Archives section
at end.  Issues discussed here include:

- Background.
- Description of proposed operators and implementation issues.
- Analysis of alternatives to new operators.
- Analysis of alternative forms.
- Compatibility issues
- Description of wider extensions and other related ideas.

A substantial portion of this PEP describes ideas that do not go
into the proposed extension.  They are presented because the
extension is essentially syntactic sugar, so its adoption must be
weighed against various possible alternatives.  While many
alternatives may be better in some aspects, the current proposal
appears to be overall advantageous.

The issues concerning elementwise-objectwise operations extends to
wider areas than numerical computation.  This document also
describes how the current proposal may be integrated with more
general future extensions.

```

### Background

```    Python provides six binary infix math operators: + - * / % **
hereafter generically represented by "op".  They can be overloaded
with new semantics for user-defined classes.  However, for objects
composed of homogeneous elements, such as arrays, vectors and
matrices in numerical computation, there are two essentially
distinct flavors of semantics.  The objectwise operations treat
these objects as points in multidimensional spaces.  The
elementwise operations treat them as collections of individual
elements.  These two flavors of operations are often intermixed in
the same formulas, thereby requiring syntactical distinction.

Many numerical computation languages provide two sets of math
operators.  For example, in MatLab, the ordinary op is used for
objectwise operation while .op is used for elementwise operation.
In R, op stands for elementwise operation while %op% stands for
objectwise operation.

In Python, there are other methods of representation, some of
which already used by available numerical packages, such as

- function:   mul(a,b)
- method:     a.mul(b)
- casting:    a.E*b

In several aspects these are not as adequate as infix operators.
More details will be shown later, but the key points are

- Readability: Even for moderately complicated formulas, infix
operators are much cleaner than alternatives.

- Familiarity: Users are familiar with ordinary math operators.

- Implementation: New infix operators will not unduly clutter
Python syntax.  They will greatly ease the implementation of
numerical packages.

While it is possible to assign current math operators to one
flavor of semantics, there is simply not enough infix operators to
overload for the other flavor.  It is also impossible to maintain
visual symmetry between these two flavors if one of them does not
contain symbols for ordinary math operators.

```

### Proposed extension

```    - Six new binary infix operators ~+ ~- ~* ~/ ~% ~** are added to
core Python.  They parallel the existing operators + - * / % **.

- Six augmented assignment operators ~+= ~-= ~*= ~/= ~%= ~**= are
added to core Python.  They parallel the operators += -= *= /=
%= **= available in Python 2.0.

- Operator ~op retains the syntactical properties of operator op,
including precedence.

- Operator ~op retains the semantical properties of operator op on
built-in number types.

- Operator ~op raise syntax error on non-number builtin types.
This is temporary until the proper behavior can be agreed upon.

- These operators are overloadable in classes with names that
prepend "t" (for tilde) to names of ordinary math operators.
For example, __tadd__ and __rtadd__ work for ~+ just as __add__
and __radd__ work for +.

- As with existing operators, the __r*__() methods are invoked when
the left operand does not provide the appropriate method.

It is intended that one set of op or ~op is used for elementwise
operations, the other for objectwise operations, but it is not
specified which version of operators stands for elementwise or
objectwise operations, leaving the decision to applications.

The proposed implementation is to patch several files relating to
the tokenizer, parser, grammar and compiler to duplicate the
functionality of corresponding existing operators as necessary.
All new semantics are to be implemented in the classes that
overload them.

The symbol ~ is already used in Python as the unary "bitwise not"
operator.  Currently it is not allowed for binary operators.  The
new operators are completely backward compatible.

```

### Prototype Implementation

```    Greg Lielens implemented the infix ~op as a patch against Python
2.0b1 source.

To allow ~ to be part of binary operators, the tokenizer would
treat ~+ as one token.  This means that currently valid expression
~+1 would be tokenized as ~+ 1 instead of ~ + 1.  The parser would
then treat ~+ as composite of ~ +.  The effect is invisible to
applications.

Notes about current patch:

- It does not include ~op= operators yet.

- The ~op behaves the same as op on lists, instead of raising
exceptions.

These should be fixed when the final version of this proposal is
ready.

- It reserves xor as an infix operator with the semantics
equivalent to:

def __xor__(a, b):
if not b: return a
elif not a: return b
else: 0

This preserves true value as much as possible, otherwise preserve
left hand side value if possible.

This is done so that bitwise operators could be regarded as
elementwise logical operators in the future (see below).

```

### Alternatives to adding new operators

```    The discussions on comp.lang.python and python-dev mailing list
explored many alternatives.  Some of the leading alternatives are
listed here, using the multiplication operator as an example.

1. Use function mul(a,b).

Advantage:
-  No need for new operators.

Disadvantage:
- Prefix forms are cumbersome for composite formulas.
- Unfamiliar to the intended users.
- Too verbose for the intended users.
- Unable to use natural precedence rules.

2. Use method call a.mul(b)

Advantage:
- No need for new operators.

Disadvantage:
- Asymmetric for both operands.
- Unfamiliar to the intended users.
- Too verbose for the intended users.
- Unable to use natural precedence rules.

3. Use "shadow classes".  For matrix class define a shadow array
class accessible through a method .E, so that for matrices a
and b, a.E*b would be a matrix object that is
elementwise_mul(a,b).

Likewise define a shadow matrix class for arrays accessible
through a method .M so that for arrays a and b, a.M*b would be
an array that is matrixwise_mul(a,b).

Advantage:
- No need for new operators.
- Benefits of infix operators with correct precedence rules.
- Clean formulas in applications.

Disadvantage:
- Hard to maintain in current Python because ordinary numbers
cannot have user defined class methods; i.e. a.E*b will fail
if a is a pure number.
- Difficult to implement, as this will interfere with existing
method calls, like .T for transpose, etc.
- Runtime overhead of object creation and method lookup.
- The shadowing class cannot replace a true class, because it
does not return its own type.  So there need to be a M class
with shadow E class, and an E class with shadow M class.
- Unnatural to mathematicians.

4. Implement matrixwise and elementwise classes with easy casting
to the other class.  So matrixwise operations for arrays would
be like a.M*b.M and elementwise operations for matrices would
be like a.E*b.E.  For error detection a.E*b.M would raise
exceptions.

Advantage:
- No need for new operators.
- Similar to infix notation with correct precedence rules.

Disadvantage:
- Similar difficulty due to lack of user-methods for pure numbers.
- Runtime overhead of object creation and method lookup.
- More cluttered formulas
- Switching of flavor of objects to facilitate operators
becomes persistent.  This introduces long range context
dependencies in application code that would be extremely hard
to maintain.

5. Using mini parser to parse formulas written in arbitrary
extension placed in quoted strings.

Advantage:
- Pure Python, without new operators

Disadvantage:
- The actual syntax is within the quoted string, which does not
resolve the problem itself.
- Introducing zones of special syntax.
- Demanding on the mini-parser.

6. Introducing a single operator, such as @, for matrix
multiplication.

Advantage:
- Introduces less operators

Disadvantage:
- The distinctions for operators like + - ** are equally
important.  Their meaning in matrix or array-oriented
packages would be reversed (see below).
- The new operator occupies a special character.
- This does not work well with more general object-element issues.

Among these alternatives, the first and second are used in current
applications to some extent, but found inadequate.  The third is
the most favorite for applications, but it will incur huge
implementation complexity.  The fourth would make applications
codes very context-sensitive and hard to maintain.  These two
alternatives also share significant implementational difficulties
due to current type/class split.  The fifth appears to create more
problems than it would solve.  The sixth does not cover the same
range of applications.

```

### Alternative forms of infix operators

```    Two major forms and several minor variants of new infix operators
were discussed:

- Bracketed form

(op)
[op]
{op}
<op>
:op:
~op~
%op%

- Meta character form

.op
@op
~op

Alternatively the meta character is put after the operator.

- Less consistent variations of these themes.  These are
considered unfavorably.  For completeness some are listed here

- Use @/ and /@ for left and right division
- Use [*] and (*) for outer and inner products
- Use a single operator @ for multiplication.

- Use __call__ to simulate multiplication.
a(b)  or (a)(b)

Criteria for choosing among the representations include:

- No syntactical ambiguities with existing operators.

- Higher readability in actual formulas.  This makes the
bracketed forms unfavorable.  See examples below.

- Visually similar to existing math operators.

- Syntactically simple, without blocking possible future
extensions.

With these criteria the overall winner in bracket form appear to
be {op}.  A clear winner in the meta character form is ~op.
Comparing these it appears that ~op is the favorite among them
all.

Some analysis are as follows:

- The .op form is ambiguous: 1.+a would be different from 1 .+a

- The bracket type operators are most favorable when standing
alone, but not in formulas, as they interfere with visual
parsing of parenthesis for precedence and function argument.
This is so for (op) and [op], and somewhat less so for {op}
and <op>.

- The <op> form has the potential to be confused with < > and =

- The @op is not favored because @ is visually heavy (dense,
more like a letter): a@+b is more readily read as a@ + b
than a @+ b.

- For choosing meta-characters: Most of existing ASCII symbols
have already been used.  The only three unused are @ \$ ?.

```

### Semantics of new operators

```    There are convincing arguments for using either set of operators
as objectwise or elementwise.  Some of them are listed here:

1. op for element, ~op for object

- Consistent with current multiarray interface of Numeric package
- Consistent with some other languages
- Perception that elementwise operations are more natural
- Perception that elementwise operations are used more frequently

2. op for object, ~op for element

- Consistent with current linear algebra interface of MatPy package
- Consistent with some other languages
- Perception that objectwise operations are more natural
- Perception that objectwise operations are used more frequently
- Consistent with the current behavior of operators on lists
- Allow ~ to be a general elementwise meta-character in future
extensions.

It is generally agreed upon that

- there is no absolute reason to favor one or the other
- it is easy to cast from one representation to another in a
sizable chunk of code, so the other flavor of operators is
always minority
- there are other semantic differences that favor existence of
array-oriented and matrix-oriented packages, even if their
operators are unified.
- whatever the decision is taken, codes using existing
interfaces should not be broken for a very long time.

Therefore, not much is lost, and much flexibility retained, if the
semantic flavors of these two sets of operators are not dictated
by the core language.  The application packages are responsible
for making the most suitable choice.  This is already the case for
NumPy and MatPy which use opposite semantics.  Adding new
operators will not break this.  See also observation after
subsection 2 in the Examples below.

The issue of numerical precision was raised, but if the semantics
is left to the applications, the actual precisions should also go
there.

```

### Examples

```    Following are examples of the actual formulas that will appear
using various operators or other representations described above.

1. The matrix inversion formula:

- Using op for object and ~op for element:

b = a.I - a.I * u / (c.I + v/a*u) * v / a

b = a.I - a.I * u * (c.I + v*a.I*u).I * v * a.I

- Using op for element and ~op for object:

b = a.I @- a.I @* u @/ (c.I @+ v@/a@*u) @* v @/ a

b = a.I ~- a.I ~* u ~/ (c.I ~+ v~/a~*u) ~* v ~/ a

b = a.I (-) a.I (*) u (/) (c.I (+) v(/)a(*)u) (*) v (/) a

b = a.I [-] a.I [*] u [/] (c.I [+] v[/]a[*]u) [*] v [/] a

b = a.I <-> a.I <*> u </> (c.I <+> v</>a<*>u) <*> v </> a

b = a.I {-} a.I {*} u {/} (c.I {+} v{/}a{*}u) {*} v {/} a

Observation: For linear algebra using op for object is preferable.

Observation: The ~op type operators look better than (op) type
in complicated formulas.

- using named operators

b = a.I @sub a.I @mul u @div (c.I @add v @div a @mul u) @mul v @div a

b = a.I ~sub a.I ~mul u ~div (c.I ~add v ~div a ~mul u) ~mul v ~div a

Observation: Named operators are not suitable for math formulas.

2. Plotting a 3d graph

- Using op for object and ~op for element:

z = sin(x~**2 ~+ y~**2);    plot(x,y,z)

- Using op for element and ~op for object:

z = sin(x**2 + y**2);   plot(x,y,z)

Observation: Elementwise operations with broadcasting allows
much more efficient implementation than MatLab.

Observation: It is useful to have two related classes with the
semantics of op and ~op swapped.  Using these the ~op
operators would only need to appear in chunks of code where
the other flavor dominates, while maintaining consistent
semantics of the code.

3. Using + and - with automatic broadcasting

a = b - c;  d = a.T*a

Observation: This would silently produce hard-to-trace bugs if
one of b or c is row vector while the other is column vector.

```

### Miscellaneous issues:

```    - Need for the ~+ ~- operators.  The objectwise + - are important
because they provide important sanity checks as per linear
algebra.  The elementwise + - are important because they allow
broadcasting that are very efficient in applications.

- Left division (solve).  For matrix, a*x is not necessarily equal
to x*a.  The solution of a*x==b, denoted x=solve(a,b), is
therefore different from the solution of x*a==b, denoted
x=div(b,a).  There are discussions about finding a new symbol
for solve.  [Background: MatLab use b/a for div(b,a) and a\b for
solve(a,b).]

It is recognized that Python provides a better solution without
requiring a new symbol: the inverse method .I can be made to be
delayed so that a.I*b and b*a.I are equivalent to Mat lab's a\b
and b/a.  The implementation is quite simple and the resulting
application code clean.

- Power operator.  Python's use of a**b as pow(a,b) has two
perceived disadvantages:

- Most mathematicians are more familiar with a^b for this purpose.
- It results in long augmented assignment operator ~**=.

However, this issue is distinct from the main issue here.

- Additional multiplication operators.  Several forms of
multiplications are used in (multi-)linear algebra.  Most can be
seen as variations of multiplication in linear algebra sense
(such as Kronecker product).  But two forms appear to be more
fundamental: outer product and inner product.  However, their
specification includes indices, which can be either

- associated with the operator, or
- associated with the objects.

The latter (the Einstein notation) is used extensively on paper,
and is also the easier one to implement.  By implementing a
tensor-with-indices class, a general form of multiplication
would cover both outer and inner products, and specialize to
linear algebra multiplication as well.  The index rule can be
defined as class methods, like,

a = b.i(1,2,-1,-2) * c.i(4,-2,3,-1)   # a_ijkl = b_ijmn c_lnkm

Therefore, one objectwise multiplication is sufficient.

- Bitwise operators.

- The proposed new math operators use the symbol ~ that is
"bitwise not" operator.  This poses no compatibility problem
but somewhat complicates implementation.

- The symbol ^ might be better used for pow than bitwise xor.
But this depends on the future of bitwise operators.  It does
not immediately impact on the proposed math operator.

- The symbol | was suggested to be used for matrix solve.  But
the new solution of using delayed .I is better in several
ways.

- The current proposal fits in a larger and more general
extension that will remove the need for special bitwise
operators.  (See elementization below.)

- Alternative to special operator names used in definition,

def "+"(a, b)      in place of       def __add__(a, b)

This appears to require greater syntactical change, and would
only be useful when arbitrary additional operators are allowed.

```

### Impact on general elementization

```    The distinction between objectwise and elementwise operations are
meaningful in other contexts as well, where an object can be
conceptually regarded as a collection of elements.  It is
important that the current proposal does not preclude possible
future extensions.

One general future extension is to use ~ as a meta operator to
"elementize" a given operator.  Several examples are listed here:

1. Bitwise operators.  Currently Python assigns six operators to
bitwise operations: and (&), or (|), xor (^), complement (~),
left shift (<<) and right shift (>>), with their own precedence
levels.

Among them, the & | ^ ~ operators can be regarded as
elementwise versions of lattice operators applied to integers
regarded as bit strings.

5 and 6                # 6
5 or 6                 # 5

5 ~and 6               # 4
5 ~or 6                # 7

These can be regarded as general elementwise lattice operators,
not restricted to bits in integers.

In order to have named operators for xor ~xor, it is necessary
to make xor a reserved word.

2. List arithmetics.

[1, 2] + [3, 4]        # [1, 2, 3, 4]
[1, 2] ~+ [3, 4]       # [4, 6]

['a', 'b'] * 2         # ['a', 'b', 'a', 'b']
'ab' * 2               # 'abab'

['a', 'b'] ~* 2        # ['aa', 'bb']
[1, 2] ~* 2            # [2, 4]

It is also consistent to Cartesian product

[1,2]*[3,4]            # [(1,3),(1,4),(2,3),(2,4)]

3. List comprehension.

a = [1, 2]; b = [3, 4]
~f(a,b)                # [f(x,y) for x, y in zip(a,b)]
~f(a*b)                # [f(x,y) for x in a for y in b]
a ~+ b                 # [x + y for x, y in zip(a,b)]

4. Tuple generation (the zip function in Python 2.0)

[1, 2, 3], [4, 5, 6]   # ([1,2, 3], [4, 5, 6])
[1, 2, 3]~,[4, 5, 6]   # [(1,4), (2, 5), (3,6)]

5. Using ~ as generic elementwise meta-character to replace map

~f(a, b)               # map(f, a, b)
~~f(a, b)              # map(lambda *x:map(f, *x), a, b)

More generally,

def ~f(*x): return map(f, *x)
def ~~f(*x): return map(~f, *x)
...

6. Elementwise format operator (with broadcasting)

a = [1,2,3,4,5]
print ["%5d "] ~% a
a = [[1,2],[3,4]]
print ["%5d "] ~~% a

7.  Rich comparison

[1, 2, 3]  ~< [3, 2, 1]  # [1, 0, 0]
[1, 2, 3] ~== [3, 2, 1]  # [0, 1, 0]

8. Rich indexing

[a, b, c, d] ~[2, 3, 1]  # [c, d, b]

9. Tuple flattening

a = (1,2);  b = (3,4)
f(~a, ~b)                # f(1,2,3,4)

10. Copy operator

a ~= b                   # a = b.copy()

There can be specific levels of deep copy

a ~~= b                  # a = b.copy(2)

Notes:

1. There are probably many other similar situations.  This general
approach seems well suited for most of them, in place of
several separated extensions for each of them (parallel and
cross iteration, list comprehension, rich comparison, etc).

2. The semantics of "elementwise" depends on applications.  For
example, an element of matrix is two levels down from the
list-of-list point of view.  This requires more fundamental
change than the current proposal.  In any case, the current
proposal will not negatively impact on future possibilities of
this nature.

Note that this section describes a type of future extensions that
is consistent with current proposal, but may present additional
compatibility or other problems.  They are not tied to the current
proposal.

```

### Impact on named operators

```    The discussions made it generally clear that infix operators is a
scarce resource in Python, not only in numerical computation, but
in other fields as well.  Several proposals and ideas were put
forward that would allow infix operators be introduced in ways
similar to named functions.  We show here that the current
extension does not negatively impact on future extensions in this
regard.

1. Named infix operators.

Choose a meta character, say @, so that for any identifier
"opname", the combination "@opname" would be a binary infix
operator, and

a @opname b == opname(a,b)

Other representations mentioned include .name ~name~ :name:
(.name) %name% and similar variations.  The pure bracket based
operators cannot be used this way.

This requires a change in the parser to recognize @opname, and
parse it into the same structure as a function call.  The
precedence of all these operators would have to be fixed at
one level, so the implementation would be different from
additional math operators which keep the precedence of
existing math operators.

The current proposed extension do not limit possible future
extensions of such form in any way.

2. More general symbolic operators.

One additional form of future extension is to use meta
character and operator symbols (symbols that cannot be used in
syntactical structures other than operators).  Suppose @ is
the meta character.  Then

a + b,    a @+ b,    a @@+ b,  a @+- b

would all be operators with a hierarchy of precedence, defined by

def "+"(a, b)
def "@+"(a, b)
def "@@+"(a, b)
def "@+-"(a, b)

One advantage compared with named operators is greater
flexibility for precedences based on either the meta character
or the ordinary operator symbols.  This also allows operator
composition.  The disadvantage is that they are more like
"line noise".  In any case the current proposal does not
impact its future possibility.

These kinds of future extensions may not be necessary when
Unicode becomes generally available.

Note that this section discusses compatibility of the proposed
extension with possible future extensions.  The desirability
or compatibility of these other extensions themselves are
specifically not considered here.

```

### Credits and archives

```    The discussions mostly happened in July to August of 2000 on news
group comp.lang.python and the mailing list python-dev.  There are
altogether several hundred postings, most can be retrieved from
these two pages (and searching word "operator"):

http://www.python.org/pipermail/python-list/2000-July/
http://www.python.org/pipermail/python-list/2000-August/

The names of contributors are too numerous to mention here,
suffice to say that a large proportion of ideas discussed here are
not our own.

Several key postings (from our point of view) that may help to
navigate the discussions include:

http://www.python.org/pipermail/python-list/2000-July/108893.html
http://www.python.org/pipermail/python-list/2000-July/108777.html
http://www.python.org/pipermail/python-list/2000-July/108848.html
http://www.python.org/pipermail/python-list/2000-July/109237.html
http://www.python.org/pipermail/python-list/2000-July/109250.html
http://www.python.org/pipermail/python-list/2000-July/109310.html
http://www.python.org/pipermail/python-list/2000-July/109448.html
http://www.python.org/pipermail/python-list/2000-July/109491.html
http://www.python.org/pipermail/python-list/2000-July/109537.html
http://www.python.org/pipermail/python-list/2000-July/109607.html
http://www.python.org/pipermail/python-list/2000-July/109709.html
http://www.python.org/pipermail/python-list/2000-July/109804.html
http://www.python.org/pipermail/python-list/2000-July/109857.html
http://www.python.org/pipermail/python-list/2000-July/110061.html
http://www.python.org/pipermail/python-list/2000-July/110208.html
http://www.python.org/pipermail/python-list/2000-August/111427.html
http://www.python.org/pipermail/python-list/2000-August/111558.html
http://www.python.org/pipermail/python-list/2000-August/112551.html
http://www.python.org/pipermail/python-list/2000-August/112606.html
http://www.python.org/pipermail/python-list/2000-August/112758.html

http://www.python.org/pipermail/python-dev/2000-July/013243.html
http://www.python.org/pipermail/python-dev/2000-July/013364.html
http://www.python.org/pipermail/python-dev/2000-August/014940.html

These are earlier drafts of this PEP:

http://www.python.org/pipermail/python-list/2000-August/111785.html
http://www.python.org/pipermail/python-list/2000-August/112529.html
http://www.python.org/pipermail/python-dev/2000-August/014906.html

There is an alternative PEP (officially, PEP 211) by Greg Wilson,
titled "Adding New Linear Algebra Operators to Python".

Its first (and current) version is at:

http://www.python.org/pipermail/python-dev/2000-August/014876.html
http://www.python.org/dev/peps/pep-0211/

```

### Additional References

```     http://MatPy.sourceforge.net/Misc/index.html

```