Contents

• Introduction
• Background
• Proposed extension
• Prototype Implementation
• Alternatives to adding new operators
• Alternative forms of infix operators
• Semantics of new operators
• Examples
• Miscellaneous issues
• Impact on general elementization
  • Notes
• Impact on named operators
• Credits and archives
• Additional References

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 1.

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 parentheses 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 Matlab’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)

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

1

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