[PEP draft 2] Adding new math operators

[Huaiyu Zhu]

This incorporates the comments I received.  It is to be merged (linked) with
the official PEP 211.  

        Python Extension Proposal: Adding New Math Operators 
                Huaiyu Zhu <hzhu at users.sourceforge.net>
                         2000-08-08, draft 2


Introduction
------------

This PEP describes a proposal to add new math operators to Python, and
summarises discussions in the news group comp.lang.python on this topic.
Issues discussed here include:

1. Background.
2. Description of proposed operators and implementation issues.
3. Analysis of alternatives to new operators.
4. Analysis of alternative forms.
5. Compatibility issues
6. 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.



Background
----------

Python provides five basic 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

1. function:   mul(a,b)
2. method:     a.mul(b)
3. 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

1. Readability: Even for moderately complicated formulas, infix operators
   are much cleaner than alternatives.
2. Familiarity: Users are familiar with ordinary math operators.  
3. 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
------------------

1.  New operators ~+ ~- ~* ~/ ~** ~+= ~-= ~*= ~/= ~**= are added to core
    Python.  They parallel the existing operators + - * / ** and the (soon
    to be added) += -= *= /= **= operators.

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

3.  Operator ~op retains the semantical properties of operator op on
    built-in number types.  They raise syntax error on other types.

4.  These operators are overloadable in classes with names that prepend
    "alt" to names of ordinary math operators.  For example, __altadd__ and
    __raltadd__ work for ~+ just as __add__ and __radd__ work for +.

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

The symbol ~ is already used in Python as the unary "bitwise not" operator.
Currently it is not allowed for binary operators, so using it as a prefix to
binary operators will not create incompatibility.

The proposed implementation is to patch several files relating to the parser
and compiler to duplicate the functionality of existing math operators as
necessary.  All new semantics are to be implemented in the application that
overloads them, but they are recommended to be conceptually similar to
existing math operators.

It is not specified which version of operators stands for elementwise or
objectwise operations, leaving the decision to applications.

A prototype implementation already exists.



Alternatives to adding new operators
------------------------------------

Some of the leading alternatives, using the multiplication 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. Implement a shadowing "elementwise class" and use casting to indicate the
   operators.  For example a*b for matrix multiply, and a.E*b for
   elementwise multiply.

   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
     class methods.  (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 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. 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.

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 creates more problems than it solves.



Alternative forms of infix operators
------------------------------------

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

1. Bracketed form

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

2. Meta character form

   .op
   @op
   ~op
   
   Alternatively the meta character is put after the operator.

3. 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

4. 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 at +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: Swapping the semantics of op and ~op (by casting the
    objects) is often advantageous, as the ~op operators would only appear
    in chunks of code where the other flavor dominate.


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:
---------------------

1. 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.

2. 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.

3. 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.

4. 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.

5. Bitwise operators.  Currently Python assigns six operators to bitwise
   operations: and (&), or (|), xor (^), complement (~), left shift (<<) and
   right shift (>>), with their own precedence levels.  This is related to
   the new math operators in several ways:

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

   - 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 bitwise operators assign special syntactical and semantical
     structures to operations that are not as fundamental as math.  Most of
     their usage could be replaced by a bitwise module with named functions.
     Removing ~ as a single operator could also allow unification between
     bitwise and logical operators (see below).  However, this issue is
     separate from the current proposed extension.

6. Lattice operators.  It was suggested that similar operators be combined
   with bitwise operators to represent lattice operations.  For example, ~|
   and ~& could represent "lattice or" and "lattice and".  But these can
   already be achieved by overloading existing logical or bitwise operators.
   On the other hand, these operations might be more deserving for infix
   operators than the built-in bitwise operators do (see below).

7. 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.

8. There was a suggestion to provide a copy operator :=, but this can
   already be done by a=b.copy.



Impact on possible future extensions:
-------------------------------------

More general extensions could lead from the current proposal. Although they
would be distinct proposals, they might have syntactical or semantical
implications on each other.  It is prudent to ensure that the current
extension do not restrict any future possibilities.


1. Named operators. 

The news group discussion 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.

The idea of named infix operators is essentially this: 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.


3. Object/element dichotomy for other types of objects.

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 homogeneous elements.  Several examples are listed here:
   
   - 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]
   
   - Tuple generation
   
      [1, 2, 3], [4, 5, 6]   # ([1,2, 3], [4, 5, 6])
      [1, 2, 3]~,[4, 5, 6]   # [(1,4), (2, 5), (3,6)]
   
      This has the same effect as the proposed zip function.
   
   - Bitwise operation (regarding integer as collection of bits, and
      removing the dissimilarity between bitwise and logical operators)
   
      5 and 6       # 5
      5 or 6        # 5
                    
      5 ~and 6      # 4
      5 ~or 6       # 7
   
   - Elementwise format operator (with broadcasting)
   
      a = [1,2,3,4,5]
      print ["%5d "] ~% a     # print ("%5s "*len(a)) % tuple(a)
      a = [[1,2],[3,4]]
      print ["%5d "] ~~% a
   
   - 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)
      ...
   
   There are probably many other similar situations.  This general approach
   seems well suited for most of them.  In any case, the current proposal
   will not negatively impact on future possibilities.

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.


-- 
Huaiyu Zhu                       hzhu at users.sourceforge.net
Matrix for Python Project        http://MatPy.sourceforge.net