python-peps/pep-3141.txt

PEP: 3141
Title: A Type Hierarchy for Numbers
Version: $Revision$
Last-Modified: $Date$
Author: Jeffrey Yasskin <jyasskin@gmail.com>
Status: Draft
Type: Standards Track
Content-Type: text/x-rst
Created: 23-Apr-2007
Post-History: 25-Apr-2007, 16-May-2007, 02-Aug-2007


Abstract
========

This proposal defines a hierarchy of Abstract Base Classes (ABCs) (PEP
3119) to represent number-like classes. It proposes a hierarchy of
``Number :> Complex :> Real :> Rational :> Integral`` where ``A :> B``
means "A is a supertype of B", and a pair of ``Exact``/``Inexact``
classes to capture the difference between ``floats`` and
``ints``. These types are significantly inspired by Scheme's numeric
tower [#schemetower]_.

Rationale
=========

Functions that take numbers as arguments should be able to determine
the properties of those numbers, and if and when overloading based on
types is added to the language, should be overloadable based on the
types of the arguments. For example, slicing requires its arguments to
be ``Integrals``, and the functions in the ``math`` module require
their arguments to be ``Real``.

Specification
=============

This PEP specifies a set of Abstract Base Classes, and suggests a
general strategy for implementing some of the methods. It uses
terminology from PEP 3119, but the hierarchy is intended to be
meaningful for any systematic method of defining sets of classes.

The type checks in the standard library should use these classes
instead of the concrete built-ins.


Numeric Classes
---------------

We begin with a Number class to make it easy for people to be fuzzy
about what kind of number they expect. This class only helps with
overloading; it doesn't provide any operations. ::

    class Number(metaclass=ABCMeta): pass


Most implementations of complex numbers will be hashable, but if you
need to rely on that, you'll have to check it explicitly: mutable
numbers are supported by this hierarchy. **Open issue:** Should
__pos__ coerce the argument to be an instance of the type it's defined
on? Why do the builtins do this? ::

    class Complex(Number):
        """Complex defines the operations that work on the builtin complex type.

        In short, those are: a conversion to complex, .real, .imag, +, -,
        *, /, abs(), .conjugate, ==, and !=.

        If it is given heterogenous arguments, and doesn't have special
        knowledge about them, it should fall back to the builtin complex
        type as described below.
        """

        @abstractmethod
        def __complex__(self):
            """Return a builtin complex instance."""

        def __bool__(self):
            """True if self != 0."""
            return self != 0

        @abstractproperty
        def real(self):
            """Retrieve the real component of this number.

            This should subclass Real.
            """
            raise NotImplementedError

        @abstractproperty
        def imag(self):
            """Retrieve the real component of this number.

            This should subclass Real.
            """
            raise NotImplementedError

        @abstractmethod
        def __add__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __radd__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __neg__(self):
            raise NotImplementedError

        def __pos__(self):
            return self

        def __sub__(self, other):
            return self + -other

        def __rsub__(self, other):
            return -self + other

        @abstractmethod
        def __mul__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rmul__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __div__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rdiv__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __pow__(self, exponent):
            """Like division, a**b should promote to complex when necessary."""
            raise NotImplementedError

        @abstractmethod
        def __rpow__(self, base):
            raise NotImplementedError

        @abstractmethod
        def __abs__(self):
            """Returns the Real distance from 0."""
            raise NotImplementedError

        @abstractmethod
        def conjugate(self):
            """(x+y*i).conjugate() returns (x-y*i)."""
            raise NotImplementedError

        @abstractmethod
        def __eq__(self, other):
            raise NotImplementedError

        def __ne__(self, other):
            return not (self == other)


The ``Real`` ABC indicates that the value is on the real line, and
supports the operations of the ``float`` builtin. Real numbers are
totally ordered except for NaNs (which this PEP basically ignores). ::

    class Real(Complex):
        """To Complex, Real adds the operations that work on real numbers.

        In short, those are: a conversion to float, trunc(), divmod,
        %, <, <=, >, and >=.

        Real also provides defaults for the derived operations.
        """

        @abstractmethod
        def __float__(self):
            """Any Real can be converted to a native float object."""
            raise NotImplementedError

        @abstractmethod
        def __trunc__(self):
            """Truncates self to an Integral.

            Returns an Integral i such that:
              * i>=0 iff self>0
              * abs(i) <= abs(self).
              * for any Integral j satisfying the first two conditions,
                abs(i) >= abs(j) [i.e. i has "maximal" abs among those]
            i.e. "truncate towards 0".
            """
            raise NotImplementedError

        def __divmod__(self, other):
            """The pair (self // other, self % other).

            Sometimes this can be computed faster than the pair of
            operations.
            """
            return (self // other, self % other)

        def __rdivmod__(self, other):
            """The pair (self // other, self % other).

            Sometimes this can be computed faster than the pair of
            operations.
            """
            return (other // self, other % self)

        @abstractmethod
        def __floordiv__(self, other):
            """The floor() of self/other. Integral."""
            raise NotImplementedError

        @abstractmethod
        def __rfloordiv__(self, other):
            """The floor() of other/self."""
            raise NotImplementedError

        @abstractmethod
        def __mod__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rmod__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __lt__(self, other):
            """< on Reals defines a total ordering, except perhaps for NaN."""
            raise NotImplementedError

        @abstractmethod
        def __le__(self, other): 
            raise NotImplementedError

        # Concrete implementations of Complex abstract methods.

        def __complex__(self):
            return complex(float(self))

        @property
        def real(self):
            return self

        @property
        def imag(self):
            return 0

        def conjugate(self):
            """Conjugate is a no-op for Reals."""
            return self


There is no built-in rational type, but it's straightforward to write,
so we provide an ABC for it. **Open issue**: Add Demo/classes/Rat.py
to the stdlib? ::

    class Rational(Real, Exact):
        """.numerator and .denominator should be in lowest terms."""

        @abstractproperty
        def numerator(self):
            raise NotImplementedError

        @abstractproperty
        def denominator(self):
            raise NotImplementedError

        # Concrete implementation of Real's conversion to float.

        def __float__(self):
            return self.numerator / self.denominator


And finally integers::

    class Integral(Rational):
        """Integral adds a conversion to int and the bit-string operations."""

        @abstractmethod
        def __int__(self):
            raise NotImplementedError

        def __index__(self):
            return int(self)

        @abstractmethod
        def __pow__(self, exponent, modulus):
            """self ** exponent % modulus, but maybe faster.

            Implement this if you want to support the 3-argument version
            of pow(). Otherwise, just implement the 2-argument version
            described in Complex. Raise a TypeError if exponent < 0 or any
            argument isn't Integral.
            """
            raise NotImplementedError

        @abstractmethod
        def __lshift__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rlshift__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rshift__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rrshift__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __and__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rand__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __xor__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __rxor__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __or__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __ror__(self, other):
            raise NotImplementedError

        @abstractmethod
        def __invert__(self):
            raise NotImplementedError

        # Concrete implementations of Rational and Real abstract methods.

        def __float__(self):
            return float(int(self))

        @property
        def numerator(self):
            return self

        @property
        def denominator(self):
            return 1


Exact vs. Inexact Classes
-------------------------

Floating point values may not exactly obey several of the properties
you would expect. For example, it is possible for ``(X + -X) + 3 ==
3``, but ``X + (-X + 3) == 0``. On the range of values that most
functions deal with this isn't a problem, but it is something to be
aware of.

Therefore, I define ``Exact`` and ``Inexact`` ABCs to mark whether
types have this problem. Every instance of ``Integral`` and
``Rational`` should be Exact, but ``Reals`` and ``Complexes`` may or
may not be. (Do we really only need one of these, and the other is
defined as ``not`` the first?) ::

    class Exact(Number): pass
    class Inexact(Number): pass


Changes to operations and __magic__ methods
-------------------------------------------

To support more precise narrowing from float to int (and more
generally, from Real to Integral), I'm proposing the following new
__magic__ methods, to be called from the corresponding library
functions. All of these return Integrals rather than Reals.

1. ``__trunc__(self)``, called from a new builtin ``trunc(x)``, which
   returns the Integral closest to ``x`` between 0 and ``x``.

2. ``__floor__(self)``, called from ``math.floor(x)``, which returns
   the greatest Integral ``<= x``.

3. ``__ceil__(self)``, called from ``math.ceil(x)``, which returns the
   least Integral ``>= x``.

4. ``__round__(self)``, called from ``round(x)``, with returns the
   Integral closest to ``x``, rounding half toward even. **Open
   issue:** We could support the 2-argument version, but then we'd
   only return an Integral if the second argument were ``<= 0``.

5. ``__properfraction__(self)``, called from a new function,
   ``math.properfraction(x)``, which resembles C's ``modf()``: returns
   a pair ``(n:Integral, r:Real)`` where ``x == n + r``, both ``n``
   and ``r`` have the same sign as ``x``, and ``abs(r) < 1``. **Open
   issue:** Oh, we already have ``math.modf``. What name do we want
   for this? Should we use divmod(x, 1) instead?

Because the ``int()`` conversion from ``float`` is equivalent to but
less explicit than ``trunc()``, let's remove it. (Or, if that breaks
too much, just add a deprecation warning.)

``complex.__{divmod,mod,floordiv,int,float}__`` should also go
away. These should continue to raise ``TypeError`` to help confused
porters, but should not appear in ``help(complex)`` to avoid confusing
more people. **Open issue:** This is difficult to do with the
``PyNumberMethods`` struct. What's the best way to accomplish it?


Notes for type implementors
---------------------------

Implementors should be careful to make equal numbers equal and
hash them to the same values. This may be subtle if there are two
different extensions of the real numbers. For example, a complex type
could reasonably implement hash() as follows::

        def __hash__(self):
	    return hash(complex(self))

but should be careful of any values that fall outside of the built in
complex's range or precision.

Adding More Numeric ABCs
~~~~~~~~~~~~~~~~~~~~~~~~

There are, of course, more possible ABCs for numbers, and this would
be a poor hierarchy if it precluded the possibility of adding
those. You can add ``MyFoo`` between ``Complex`` and ``Real`` with::

    class MyFoo(Complex): ...
    MyFoo.register(Real)

Implementing the arithmetic operations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We want to implement the arithmetic operations so that mixed-mode
operations either call an implementation whose author knew about the
types of both arguments, or convert both to the nearest built in type
and do the operation there. For subtypes of Integral, this means that
__add__ and __radd__ should be defined as::

    class MyIntegral(Integral):

        def __add__(self, other):
            if isinstance(other, MyIntegral):
                return do_my_adding_stuff(self, other)
            elif isinstance(other, OtherTypeIKnowAbout):
                return do_my_other_adding_stuff(self, other)
            else:
                return NotImplemented

        def __radd__(self, other):
            if isinstance(other, MyIntegral):
                return do_my_adding_stuff(other, self)
            elif isinstance(other, OtherTypeIKnowAbout):
                return do_my_other_adding_stuff(other, self)
            elif isinstance(other, Integral):
                return int(other) + int(self)
            elif isinstance(other, Real):
                return float(other) + float(self)
            elif isinstance(other, Complex):
                return complex(other) + complex(self)
            else:
                return NotImplemented


There are 5 different cases for a mixed-type operation on subclasses
of Complex. I'll refer to all of the above code that doesn't refer to
MyIntegral and OtherTypeIKnowAbout as "boilerplate". ``a`` will be an
instance of ``A``, which is a subtype of ``Complex`` (``a : A <:
Complex``), and ``b : B <: Complex``. I'll consider ``a + b``:

    1. If A defines an __add__ which accepts b, all is well.
    2. If A falls back to the boilerplate code, and it were to return
       a value from __add__, we'd miss the possibility that B defines
       a more intelligent __radd__, so the boilerplate should return
       NotImplemented from __add__. (Or A may not implement __add__ at
       all.)
    3. Then B's __radd__ gets a chance. If it accepts a, all is well.
    4. If it falls back to the boilerplate, there are no more possible
       methods to try, so this is where the default implementation
       should live.
    5. If B <: A, Python tries B.__radd__ before A.__add__. This is
       ok, because it was implemented with knowledge of A, so it can
       handle those instances before delegating to Complex.

If ``A<:Complex`` and ``B<:Real`` without sharing any other knowledge,
then the appropriate shared operation is the one involving the built
in complex, and both __radd__s land there, so ``a+b == b+a``.


Rejected Alternatives
=====================

The initial version of this PEP defined an algebraic hierarchy
inspired by a Haskell Numeric Prelude [#numericprelude]_ including
MonoidUnderPlus, AdditiveGroup, Ring, and Field, and mentioned several
other possible algebraic types before getting to the numbers. I had
expected this to be useful to people using vectors and matrices, but
the NumPy community really wasn't interested, and we ran into the
issue that even if ``x`` is an instance of ``X <: MonoidUnderPlus``
and ``y`` is an instance of ``Y <: MonoidUnderPlus``, ``x + y`` may
still not make sense.

Then I gave the numbers a much more branching structure to include
things like the Gaussian Integers and Z/nZ, which could be Complex but
wouldn't necessarily support things like division. The community
decided that this was too much complication for Python, so I've now
scaled back the proposal to resemble the Scheme numeric tower much
more closely.


References
==========

.. [#pep3119] Introducing Abstract Base Classes
   (http://www.python.org/dev/peps/pep-3119/)

.. [#classtree] Possible Python 3K Class Tree?, wiki page created by Bill Janssen
   (http://wiki.python.org/moin/AbstractBaseClasses)

.. [#numericprelude] NumericPrelude: An experimental alternative hierarchy of numeric type classes
   (http://darcs.haskell.org/numericprelude/docs/html/index.html)

.. [#schemetower] The Scheme numerical tower
   (http://www.swiss.ai.mit.edu/ftpdir/scheme-reports/r5rs-html/r5rs_8.html#SEC50)


Acknowledgements
================

Thanks to Neil Norwitz for encouraging me to write this PEP in the
first place, to Travis Oliphant for pointing out that the numpy people
didn't really care about the algebraic concepts, to Alan Isaac for
reminding me that Scheme had already done this, and to Guido van
Rossum and lots of other people on the mailing list for refining the
concept.

Copyright
=========

This document has been placed in the public domain.



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