********************************
  Functional Programming HOWTO
********************************

:Author: \A. M. Kuchling
:Release: 0.30

(This is a first draft.  Please send comments/error reports/suggestions to
amk@amk.ca.  This URL is probably not going to be the final location of the
document, so be careful about linking to it -- you may want to add a
disclaimer.)

In this document, we'll take a tour of Python's features suitable for
implementing programs in a functional style.  After an introduction to the
concepts of functional programming, we'll look at language features such as
iterators and generators and relevant library modules such as :mod:`itertools`
and :mod:`functools`.


Introduction
============

This section explains the basic concept of functional programming; if you're
just interested in learning about Python language features, skip to the next
section.

Programming languages support decomposing problems in several different ways:

* Most programming languages are **procedural**: programs are lists of
  instructions that tell the computer what to do with the program's input.  C,
  Pascal, and even Unix shells are procedural languages.

* In **declarative** languages, you write a specification that describes the
  problem to be solved, and the language implementation figures out how to
  perform the computation efficiently.  SQL is the declarative language you're
  most likely to be familiar with; a SQL query describes the data set you want
  to retrieve, and the SQL engine decides whether to scan tables or use indexes,
  which subclauses should be performed first, etc.

* **Object-oriented** programs manipulate collections of objects.  Objects have
  internal state and support methods that query or modify this internal state in
  some way. Smalltalk and Java are object-oriented languages.  C++ and Python
  are languages that support object-oriented programming, but don't force the
  use of object-oriented features.

* **Functional** programming decomposes a problem into a set of functions.
  Ideally, functions only take inputs and produce outputs, and don't have any
  internal state that affects the output produced for a given input.  Well-known
  functional languages include the ML family (Standard ML, OCaml, and other
  variants) and Haskell.

The designers of some computer languages have chosen one approach to programming
that's emphasized.  This often makes it difficult to write programs that use a
different approach.  Other languages are multi-paradigm languages that support
several different approaches.  Lisp, C++, and Python are multi-paradigm; you can
write programs or libraries that are largely procedural, object-oriented, or
functional in all of these languages.  In a large program, different sections
might be written using different approaches; the GUI might be object-oriented
while the processing logic is procedural or functional, for example.

In a functional program, input flows through a set of functions. Each function
operates on its input and produces some output.  Functional style frowns upon
functions with side effects that modify internal state or make other changes
that aren't visible in the function's return value.  Functions that have no side
effects at all are called **purely functional**.  Avoiding side effects means
not using data structures that get updated as a program runs; every function's
output must only depend on its input.

Some languages are very strict about purity and don't even have assignment
statements such as ``a=3`` or ``c = a + b``, but it's difficult to avoid all
side effects.  Printing to the screen or writing to a disk file are side
effects, for example.  For example, in Python a ``print`` statement or a
``time.sleep(1)`` both return no useful value; they're only called for their
side effects of sending some text to the screen or pausing execution for a
second.

Python programs written in functional style usually won't go to the extreme of
avoiding all I/O or all assignments; instead, they'll provide a
functional-appearing interface but will use non-functional features internally.
For example, the implementation of a function will still use assignments to
local variables, but won't modify global variables or have other side effects.

Functional programming can be considered the opposite of object-oriented
programming.  Objects are little capsules containing some internal state along
with a collection of method calls that let you modify this state, and programs
consist of making the right set of state changes.  Functional programming wants
to avoid state changes as much as possible and works with data flowing between
functions.  In Python you might combine the two approaches by writing functions
that take and return instances representing objects in your application (e-mail
messages, transactions, etc.).

Functional design may seem like an odd constraint to work under.  Why should you
avoid objects and side effects?  There are theoretical and practical advantages
to the functional style:

* Formal provability.
* Modularity.
* Composability.
* Ease of debugging and testing.

Formal provability
------------------

A theoretical benefit is that it's easier to construct a mathematical proof that
a functional program is correct.

For a long time researchers have been interested in finding ways to
mathematically prove programs correct.  This is different from testing a program
on numerous inputs and concluding that its output is usually correct, or reading
a program's source code and concluding that the code looks right; the goal is
instead a rigorous proof that a program produces the right result for all
possible inputs.

The technique used to prove programs correct is to write down **invariants**,
properties of the input data and of the program's variables that are always
true.  For each line of code, you then show that if invariants X and Y are true
**before** the line is executed, the slightly different invariants X' and Y' are
true **after** the line is executed.  This continues until you reach the end of
the program, at which point the invariants should match the desired conditions
on the program's output.

Functional programming's avoidance of assignments arose because assignments are
difficult to handle with this technique; assignments can break invariants that
were true before the assignment without producing any new invariants that can be
propagated onward.

Unfortunately, proving programs correct is largely impractical and not relevant
to Python software. Even trivial programs require proofs that are several pages
long; the proof of correctness for a moderately complicated program would be
enormous, and few or none of the programs you use daily (the Python interpreter,
your XML parser, your web browser) could be proven correct.  Even if you wrote
down or generated a proof, there would then be the question of verifying the
proof; maybe there's an error in it, and you wrongly believe you've proved the
program correct.

Modularity
----------

A more practical benefit of functional programming is that it forces you to
break apart your problem into small pieces.  Programs are more modular as a
result.  It's easier to specify and write a small function that does one thing
than a large function that performs a complicated transformation.  Small
functions are also easier to read and to check for errors.


Ease of debugging and testing 
-----------------------------

Testing and debugging a functional-style program is easier.

Debugging is simplified because functions are generally small and clearly
specified.  When a program doesn't work, each function is an interface point
where you can check that the data are correct.  You can look at the intermediate
inputs and outputs to quickly isolate the function that's responsible for a bug.

Testing is easier because each function is a potential subject for a unit test.
Functions don't depend on system state that needs to be replicated before
running a test; instead you only have to synthesize the right input and then
check that the output matches expectations.



Composability
-------------

As you work on a functional-style program, you'll write a number of functions
with varying inputs and outputs.  Some of these functions will be unavoidably
specialized to a particular application, but others will be useful in a wide
variety of programs.  For example, a function that takes a directory path and
returns all the XML files in the directory, or a function that takes a filename
and returns its contents, can be applied to many different situations.

Over time you'll form a personal library of utilities.  Often you'll assemble
new programs by arranging existing functions in a new configuration and writing
a few functions specialized for the current task.



Iterators
=========

I'll start by looking at a Python language feature that's an important
foundation for writing functional-style programs: iterators.

An iterator is an object representing a stream of data; this object returns the
data one element at a time.  A Python iterator must support a method called
``next()`` that takes no arguments and always returns the next element of the
stream.  If there are no more elements in the stream, ``next()`` must raise the
``StopIteration`` exception.  Iterators don't have to be finite, though; it's
perfectly reasonable to write an iterator that produces an infinite stream of
data.

The built-in :func:`iter` function takes an arbitrary object and tries to return
an iterator that will return the object's contents or elements, raising
:exc:`TypeError` if the object doesn't support iteration.  Several of Python's
built-in data types support iteration, the most common being lists and
dictionaries.  An object is called an **iterable** object if you can get an
iterator for it.

You can experiment with the iteration interface manually::

    >>> L = [1,2,3]
    >>> it = iter(L)
    >>> it
    <iterator object at 0x8116870>
    >>> it.next()
    1
    >>> it.next()
    2
    >>> it.next()
    3
    >>> it.next()
    Traceback (most recent call last):
      File "<stdin>", line 1, in ?
    StopIteration
    >>>      

Python expects iterable objects in several different contexts, the most
important being the ``for`` statement.  In the statement ``for X in Y``, Y must
be an iterator or some object for which ``iter()`` can create an iterator.
These two statements are equivalent::

        for i in iter(obj):
            print(i)

        for i in obj:
            print(i)

Iterators can be materialized as lists or tuples by using the :func:`list` or
:func:`tuple` constructor functions::

    >>> L = [1,2,3]
    >>> iterator = iter(L)
    >>> t = tuple(iterator)
    >>> t
    (1, 2, 3)

Sequence unpacking also supports iterators: if you know an iterator will return
N elements, you can unpack them into an N-tuple::

    >>> L = [1,2,3]
    >>> iterator = iter(L)
    >>> a,b,c = iterator
    >>> a,b,c
    (1, 2, 3)

Built-in functions such as :func:`max` and :func:`min` can take a single
iterator argument and will return the largest or smallest element.  The ``"in"``
and ``"not in"`` operators also support iterators: ``X in iterator`` is true if
X is found in the stream returned by the iterator.  You'll run into obvious
problems if the iterator is infinite; ``max()``, ``min()``, and ``"not in"``
will never return, and if the element X never appears in the stream, the
``"in"`` operator won't return either.

Note that you can only go forward in an iterator; there's no way to get the
previous element, reset the iterator, or make a copy of it.  Iterator objects
can optionally provide these additional capabilities, but the iterator protocol
only specifies the ``next()`` method.  Functions may therefore consume all of
the iterator's output, and if you need to do something different with the same
stream, you'll have to create a new iterator.



Data Types That Support Iterators
---------------------------------

We've already seen how lists and tuples support iterators.  In fact, any Python
sequence type, such as strings, will automatically support creation of an
iterator.

Calling :func:`iter` on a dictionary returns an iterator that will loop over the
dictionary's keys::

    >>> m = {'Jan': 1, 'Feb': 2, 'Mar': 3, 'Apr': 4, 'May': 5, 'Jun': 6,
    ...      'Jul': 7, 'Aug': 8, 'Sep': 9, 'Oct': 10, 'Nov': 11, 'Dec': 12}
    >>> for key in m:
    ...     print(key, m[key])
    Mar 3
    Feb 2
    Aug 8
    Sep 9
    May 5
    Jun 6
    Jul 7
    Jan 1
    Apr 4
    Nov 11
    Dec 12
    Oct 10

Note that the order is essentially random, because it's based on the hash
ordering of the objects in the dictionary.

Applying ``iter()`` to a dictionary always loops over the keys, but dictionaries
have methods that return other iterators.  If you want to iterate over keys,
values, or key/value pairs, you can explicitly call the ``iterkeys()``,
``itervalues()``, or ``iteritems()`` methods to get an appropriate iterator.

The :func:`dict` constructor can accept an iterator that returns a finite stream
of ``(key, value)`` tuples::

    >>> L = [('Italy', 'Rome'), ('France', 'Paris'), ('US', 'Washington DC')]
    >>> dict(iter(L))
    {'Italy': 'Rome', 'US': 'Washington DC', 'France': 'Paris'}

Files also support iteration by calling the ``readline()`` method until there
are no more lines in the file.  This means you can read each line of a file like
this::

    for line in file:
        # do something for each line
        ...

Sets can take their contents from an iterable and let you iterate over the set's
elements::

    S = set((2, 3, 5, 7, 11, 13))
    for i in S:
        print(i)



Generator expressions and list comprehensions
=============================================

Two common operations on an iterator's output are 1) performing some operation
for every element, 2) selecting a subset of elements that meet some condition.
For example, given a list of strings, you might want to strip off trailing
whitespace from each line or extract all the strings containing a given
substring.

List comprehensions and generator expressions (short form: "listcomps" and
"genexps") are a concise notation for such operations, borrowed from the
functional programming language Haskell (http://www.haskell.org).  You can strip
all the whitespace from a stream of strings with the following code::

        line_list = ['  line 1\n', 'line 2  \n', ...]

        # Generator expression -- returns iterator
        stripped_iter = (line.strip() for line in line_list)

        # List comprehension -- returns list
        stripped_list = [line.strip() for line in line_list]

You can select only certain elements by adding an ``"if"`` condition::

        stripped_list = [line.strip() for line in line_list
                         if line != ""]

With a list comprehension, you get back a Python list; ``stripped_list`` is a
list containing the resulting lines, not an iterator.  Generator expressions
return an iterator that computes the values as necessary, not needing to
materialize all the values at once.  This means that list comprehensions aren't
useful if you're working with iterators that return an infinite stream or a very
large amount of data.  Generator expressions are preferable in these situations.

Generator expressions are surrounded by parentheses ("()") and list
comprehensions are surrounded by square brackets ("[]").  Generator expressions
have the form::

    ( expression for expr in sequence1 
                 if condition1
                 for expr2 in sequence2
                 if condition2
                 for expr3 in sequence3 ...
                 if condition3
                 for exprN in sequenceN
                 if conditionN )

Again, for a list comprehension only the outside brackets are different (square
brackets instead of parentheses).

The elements of the generated output will be the successive values of
``expression``.  The ``if`` clauses are all optional; if present, ``expression``
is only evaluated and added to the result when ``condition`` is true.

Generator expressions always have to be written inside parentheses, but the
parentheses signalling a function call also count.  If you want to create an
iterator that will be immediately passed to a function you can write::

        obj_total = sum(obj.count for obj in list_all_objects())

The ``for...in`` clauses contain the sequences to be iterated over.  The
sequences do not have to be the same length, because they are iterated over from
left to right, **not** in parallel.  For each element in ``sequence1``,
``sequence2`` is looped over from the beginning.  ``sequence3`` is then looped
over for each resulting pair of elements from ``sequence1`` and ``sequence2``.

To put it another way, a list comprehension or generator expression is
equivalent to the following Python code::

    for expr1 in sequence1:
        if not (condition1):
            continue   # Skip this element
        for expr2 in sequence2:
            if not (condition2):
                continue    # Skip this element
            ...
            for exprN in sequenceN:
                 if not (conditionN):
                     continue   # Skip this element

                 # Output the value of 
                 # the expression.

This means that when there are multiple ``for...in`` clauses but no ``if``
clauses, the length of the resulting output will be equal to the product of the
lengths of all the sequences.  If you have two lists of length 3, the output
list is 9 elements long::

    seq1 = 'abc'
    seq2 = (1,2,3)
    >>> [ (x,y) for x in seq1 for y in seq2]
    [('a', 1), ('a', 2), ('a', 3), 
     ('b', 1), ('b', 2), ('b', 3), 
     ('c', 1), ('c', 2), ('c', 3)]

To avoid introducing an ambiguity into Python's grammar, if ``expression`` is
creating a tuple, it must be surrounded with parentheses.  The first list
comprehension below is a syntax error, while the second one is correct::

    # Syntax error
    [ x,y for x in seq1 for y in seq2]
    # Correct
    [ (x,y) for x in seq1 for y in seq2]


Generators
==========

Generators are a special class of functions that simplify the task of writing
iterators.  Regular functions compute a value and return it, but generators
return an iterator that returns a stream of values.

You're doubtless familiar with how regular function calls work in Python or C.
When you call a function, it gets a private namespace where its local variables
are created.  When the function reaches a ``return`` statement, the local
variables are destroyed and the value is returned to the caller.  A later call
to the same function creates a new private namespace and a fresh set of local
variables. But, what if the local variables weren't thrown away on exiting a
function?  What if you could later resume the function where it left off?  This
is what generators provide; they can be thought of as resumable functions.

Here's the simplest example of a generator function::

    def generate_ints(N):
        for i in range(N):
            yield i

Any function containing a ``yield`` keyword is a generator function; this is
detected by Python's bytecode compiler which compiles the function specially as
a result.

When you call a generator function, it doesn't return a single value; instead it
returns a generator object that supports the iterator protocol.  On executing
the ``yield`` expression, the generator outputs the value of ``i``, similar to a
``return`` statement.  The big difference between ``yield`` and a ``return``
statement is that on reaching a ``yield`` the generator's state of execution is
suspended and local variables are preserved.  On the next call to the
generator's ``.next()`` method, the function will resume executing.

Here's a sample usage of the ``generate_ints()`` generator::

    >>> gen = generate_ints(3)
    >>> gen
    <generator object at 0x8117f90>
    >>> gen.next()
    0
    >>> gen.next()
    1
    >>> gen.next()
    2
    >>> gen.next()
    Traceback (most recent call last):
      File "stdin", line 1, in ?
      File "stdin", line 2, in generate_ints
    StopIteration

You could equally write ``for i in generate_ints(5)``, or ``a,b,c =
generate_ints(3)``.

Inside a generator function, the ``return`` statement can only be used without a
value, and signals the end of the procession of values; after executing a
``return`` the generator cannot return any further values.  ``return`` with a
value, such as ``return 5``, is a syntax error inside a generator function.  The
end of the generator's results can also be indicated by raising
``StopIteration`` manually, or by just letting the flow of execution fall off
the bottom of the function.

You could achieve the effect of generators manually by writing your own class
and storing all the local variables of the generator as instance variables.  For
example, returning a list of integers could be done by setting ``self.count`` to
0, and having the ``next()`` method increment ``self.count`` and return it.
However, for a moderately complicated generator, writing a corresponding class
can be much messier.

The test suite included with Python's library, ``test_generators.py``, contains
a number of more interesting examples.  Here's one generator that implements an
in-order traversal of a tree using generators recursively.

::

    # A recursive generator that generates Tree leaves in in-order.
    def inorder(t):
        if t:
            for x in inorder(t.left):
                yield x

            yield t.label

            for x in inorder(t.right):
                yield x

Two other examples in ``test_generators.py`` produce solutions for the N-Queens
problem (placing N queens on an NxN chess board so that no queen threatens
another) and the Knight's Tour (finding a route that takes a knight to every
square of an NxN chessboard without visiting any square twice).



Passing values into a generator
-------------------------------

In Python 2.4 and earlier, generators only produced output.  Once a generator's
code was invoked to create an iterator, there was no way to pass any new
information into the function when its execution is resumed.  You could hack
together this ability by making the generator look at a global variable or by
passing in some mutable object that callers then modify, but these approaches
are messy.

In Python 2.5 there's a simple way to pass values into a generator.
:keyword:`yield` became an expression, returning a value that can be assigned to
a variable or otherwise operated on::

    val = (yield i)

I recommend that you **always** put parentheses around a ``yield`` expression
when you're doing something with the returned value, as in the above example.
The parentheses aren't always necessary, but it's easier to always add them
instead of having to remember when they're needed.

(PEP 342 explains the exact rules, which are that a ``yield``-expression must
always be parenthesized except when it occurs at the top-level expression on the
right-hand side of an assignment.  This means you can write ``val = yield i``
but have to use parentheses when there's an operation, as in ``val = (yield i)
+ 12``.)

Values are sent into a generator by calling its ``send(value)`` method.  This
method resumes the generator's code and the ``yield`` expression returns the
specified value.  If the regular ``next()`` method is called, the ``yield``
returns ``None``.

Here's a simple counter that increments by 1 and allows changing the value of
the internal counter.

::

    def counter (maximum):
        i = 0
        while i < maximum:
            val = (yield i)
            # If value provided, change counter
            if val is not None:
                i = val
            else:
                i += 1

And here's an example of changing the counter:

    >>> it = counter(10)
    >>> it.next()
    0
    >>> it.next()
    1
    >>> it.send(8)
    8
    >>> it.next()
    9
    >>> it.next()
    Traceback (most recent call last):
      File ``t.py'', line 15, in ?
        it.next()
    StopIteration

Because ``yield`` will often be returning ``None``, you should always check for
this case.  Don't just use its value in expressions unless you're sure that the
``send()`` method will be the only method used resume your generator function.

In addition to ``send()``, there are two other new methods on generators:

* ``throw(type, value=None, traceback=None)`` is used to raise an exception
  inside the generator; the exception is raised by the ``yield`` expression
  where the generator's execution is paused.

* ``close()`` raises a :exc:`GeneratorExit` exception inside the generator to
  terminate the iteration.  On receiving this exception, the generator's code
  must either raise :exc:`GeneratorExit` or :exc:`StopIteration`; catching the
  exception and doing anything else is illegal and will trigger a
  :exc:`RuntimeError`.  ``close()`` will also be called by Python's garbage
  collector when the generator is garbage-collected.

  If you need to run cleanup code when a :exc:`GeneratorExit` occurs, I suggest
  using a ``try: ... finally:`` suite instead of catching :exc:`GeneratorExit`.

The cumulative effect of these changes is to turn generators from one-way
producers of information into both producers and consumers.

Generators also become **coroutines**, a more generalized form of subroutines.
Subroutines are entered at one point and exited at another point (the top of the
function, and a ``return`` statement), but coroutines can be entered, exited,
and resumed at many different points (the ``yield`` statements).


Built-in functions
==================

Let's look in more detail at built-in functions often used with iterators.

Two Python's built-in functions, :func:`map` and :func:`filter`, are somewhat
obsolete; they duplicate the features of list comprehensions but return actual
lists instead of iterators.

``map(f, iterA, iterB, ...)`` returns a list containing ``f(iterA[0], iterB[0]),
f(iterA[1], iterB[1]), f(iterA[2], iterB[2]), ...``.

::

    def upper(s):
        return s.upper()
    map(upper, ['sentence', 'fragment']) =>
      ['SENTENCE', 'FRAGMENT']

    [upper(s) for s in ['sentence', 'fragment']] =>
      ['SENTENCE', 'FRAGMENT']

As shown above, you can achieve the same effect with a list comprehension.  The
:func:`itertools.imap` function does the same thing but can handle infinite
iterators; it'll be discussed later, in the section on the :mod:`itertools` module.

``filter(predicate, iter)`` returns a list that contains all the sequence
elements that meet a certain condition, and is similarly duplicated by list
comprehensions.  A **predicate** is a function that returns the truth value of
some condition; for use with :func:`filter`, the predicate must take a single
value.

::

    def is_even(x):
        return (x % 2) == 0

    filter(is_even, range(10)) =>
      [0, 2, 4, 6, 8]

This can also be written as a list comprehension::

    >>> [x for x in range(10) if is_even(x)]
    [0, 2, 4, 6, 8]

:func:`filter` also has a counterpart in the :mod:`itertools` module,
:func:`itertools.ifilter`, that returns an iterator and can therefore handle
infinite sequences just as :func:`itertools.imap` can.

``reduce(func, iter, [initial_value])`` doesn't have a counterpart in the
:mod:`itertools` module because it cumulatively performs an operation on all the
iterable's elements and therefore can't be applied to infinite iterables.
``func`` must be a function that takes two elements and returns a single value.
:func:`reduce` takes the first two elements A and B returned by the iterator and
calculates ``func(A, B)``.  It then requests the third element, C, calculates
``func(func(A, B), C)``, combines this result with the fourth element returned,
and continues until the iterable is exhausted.  If the iterable returns no
values at all, a :exc:`TypeError` exception is raised.  If the initial value is
supplied, it's used as a starting point and ``func(initial_value, A)`` is the
first calculation.

::

    import operator
    reduce(operator.concat, ['A', 'BB', 'C']) =>
      'ABBC'
    reduce(operator.concat, []) =>
      TypeError: reduce() of empty sequence with no initial value
    reduce(operator.mul, [1,2,3], 1) =>
      6
    reduce(operator.mul, [], 1) =>
      1

If you use :func:`operator.add` with :func:`reduce`, you'll add up all the
elements of the iterable.  This case is so common that there's a special
built-in called :func:`sum` to compute it::

    reduce(operator.add, [1,2,3,4], 0) =>
      10
    sum([1,2,3,4]) =>
      10
    sum([]) =>
      0

For many uses of :func:`reduce`, though, it can be clearer to just write the
obvious :keyword:`for` loop::

    # Instead of:
    product = reduce(operator.mul, [1,2,3], 1)

    # You can write:
    product = 1
    for i in [1,2,3]:
        product *= i


``enumerate(iter)`` counts off the elements in the iterable, returning 2-tuples
containing the count and each element.

::

    enumerate(['subject', 'verb', 'object']) =>
      (0, 'subject'), (1, 'verb'), (2, 'object')

:func:`enumerate` is often used when looping through a list and recording the
indexes at which certain conditions are met::

    f = open('data.txt', 'r')
    for i, line in enumerate(f):
        if line.strip() == '':
            print('Blank line at line #%i' % i)

``sorted(iterable, [cmp=None], [key=None], [reverse=False)`` collects all the
elements of the iterable into a list, sorts the list, and returns the sorted
result.  The ``cmp``, ``key``, and ``reverse`` arguments are passed through to
the constructed list's ``.sort()`` method.

::

    import random
    # Generate 8 random numbers between [0, 10000)
    rand_list = random.sample(range(10000), 8)
    rand_list =>
      [769, 7953, 9828, 6431, 8442, 9878, 6213, 2207]
    sorted(rand_list) =>
      [769, 2207, 6213, 6431, 7953, 8442, 9828, 9878]
    sorted(rand_list, reverse=True) =>
      [9878, 9828, 8442, 7953, 6431, 6213, 2207, 769]

(For a more detailed discussion of sorting, see the Sorting mini-HOWTO in the
Python wiki at http://wiki.python.org/moin/HowTo/Sorting.)

The ``any(iter)`` and ``all(iter)`` built-ins look at the truth values of an
iterable's contents.  :func:`any` returns True if any element in the iterable is
a true value, and :func:`all` returns True if all of the elements are true
values::

    any([0,1,0]) =>
      True
    any([0,0,0]) =>
      False
    any([1,1,1]) =>
      True
    all([0,1,0]) =>
      False
    all([0,0,0]) => 
      False
    all([1,1,1]) =>
      True


Small functions and the lambda expression
=========================================

When writing functional-style programs, you'll often need little functions that
act as predicates or that combine elements in some way.

If there's a Python built-in or a module function that's suitable, you don't
need to define a new function at all::

        stripped_lines = [line.strip() for line in lines]
        existing_files = filter(os.path.exists, file_list)

If the function you need doesn't exist, you need to write it.  One way to write
small functions is to use the ``lambda`` statement.  ``lambda`` takes a number
of parameters and an expression combining these parameters, and creates a small
function that returns the value of the expression::

        lowercase = lambda x: x.lower()

        print_assign = lambda name, value: name + '=' + str(value)

        adder = lambda x, y: x+y

An alternative is to just use the ``def`` statement and define a function in the
usual way::

        def lowercase(x):
            return x.lower()

        def print_assign(name, value):
            return name + '=' + str(value)

        def adder(x,y):
            return x + y

Which alternative is preferable?  That's a style question; my usual course is to
avoid using ``lambda``.

One reason for my preference is that ``lambda`` is quite limited in the
functions it can define.  The result has to be computable as a single
expression, which means you can't have multiway ``if... elif... else``
comparisons or ``try... except`` statements.  If you try to do too much in a
``lambda`` statement, you'll end up with an overly complicated expression that's
hard to read.  Quick, what's the following code doing?

::

    total = reduce(lambda a, b: (0, a[1] + b[1]), items)[1]

You can figure it out, but it takes time to disentangle the expression to figure
out what's going on.  Using a short nested ``def`` statements makes things a
little bit better::

    def combine (a, b):
        return 0, a[1] + b[1]

    total = reduce(combine, items)[1]

But it would be best of all if I had simply used a ``for`` loop::

     total = 0
     for a, b in items:
         total += b

Or the :func:`sum` built-in and a generator expression::

     total = sum(b for a,b in items)

Many uses of :func:`reduce` are clearer when written as ``for`` loops.

Fredrik Lundh once suggested the following set of rules for refactoring uses of
``lambda``:

1) Write a lambda function.
2) Write a comment explaining what the heck that lambda does.
3) Study the comment for a while, and think of a name that captures the essence
   of the comment.
4) Convert the lambda to a def statement, using that name.
5) Remove the comment.

I really like these rules, but you're free to disagree that this lambda-free
style is better.


The itertools module
====================

The :mod:`itertools` module contains a number of commonly-used iterators as well
as functions for combining several iterators.  This section will introduce the
module's contents by showing small examples.

The module's functions fall into a few broad classes:

* Functions that create a new iterator based on an existing iterator.
* Functions for treating an iterator's elements as function arguments.
* Functions for selecting portions of an iterator's output.
* A function for grouping an iterator's output.

Creating new iterators
----------------------

``itertools.count(n)`` returns an infinite stream of integers, increasing by 1
each time.  You can optionally supply the starting number, which defaults to 0::

        itertools.count() =>
          0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
        itertools.count(10) =>
          10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...

``itertools.cycle(iter)`` saves a copy of the contents of a provided iterable
and returns a new iterator that returns its elements from first to last.  The
new iterator will repeat these elements infinitely.

::

        itertools.cycle([1,2,3,4,5]) =>
          1, 2, 3, 4, 5, 1, 2, 3, 4, 5, ...

``itertools.repeat(elem, [n])`` returns the provided element ``n`` times, or
returns the element endlessly if ``n`` is not provided.

::

    itertools.repeat('abc') =>
      abc, abc, abc, abc, abc, abc, abc, abc, abc, abc, ...
    itertools.repeat('abc', 5) =>
      abc, abc, abc, abc, abc

``itertools.chain(iterA, iterB, ...)`` takes an arbitrary number of iterables as
input, and returns all the elements of the first iterator, then all the elements
of the second, and so on, until all of the iterables have been exhausted.

::

    itertools.chain(['a', 'b', 'c'], (1, 2, 3)) =>
      a, b, c, 1, 2, 3

``itertools.izip(iterA, iterB, ...)`` takes one element from each iterable and
returns them in a tuple::

    itertools.izip(['a', 'b', 'c'], (1, 2, 3)) =>
      ('a', 1), ('b', 2), ('c', 3)

It's similiar to the built-in :func:`zip` function, but doesn't construct an
in-memory list and exhaust all the input iterators before returning; instead
tuples are constructed and returned only if they're requested.  (The technical
term for this behaviour is `lazy evaluation
<http://en.wikipedia.org/wiki/Lazy_evaluation>`__.)

This iterator is intended to be used with iterables that are all of the same
length.  If the iterables are of different lengths, the resulting stream will be
the same length as the shortest iterable.

::

    itertools.izip(['a', 'b'], (1, 2, 3)) =>
      ('a', 1), ('b', 2)

You should avoid doing this, though, because an element may be taken from the
longer iterators and discarded.  This means you can't go on to use the iterators
further because you risk skipping a discarded element.

``itertools.islice(iter, [start], stop, [step])`` returns a stream that's a
slice of the iterator.  With a single ``stop`` argument, it will return the
first ``stop`` elements.  If you supply a starting index, you'll get
``stop-start`` elements, and if you supply a value for ``step``, elements will
be skipped accordingly.  Unlike Python's string and list slicing, you can't use
negative values for ``start``, ``stop``, or ``step``.

::

    itertools.islice(range(10), 8) =>
      0, 1, 2, 3, 4, 5, 6, 7
    itertools.islice(range(10), 2, 8) =>
      2, 3, 4, 5, 6, 7
    itertools.islice(range(10), 2, 8, 2) =>
      2, 4, 6

``itertools.tee(iter, [n])`` replicates an iterator; it returns ``n``
independent iterators that will all return the contents of the source iterator.
If you don't supply a value for ``n``, the default is 2.  Replicating iterators
requires saving some of the contents of the source iterator, so this can consume
significant memory if the iterator is large and one of the new iterators is
consumed more than the others.

::

        itertools.tee( itertools.count() ) =>
           iterA, iterB

        where iterA ->
           0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

        and   iterB ->
           0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...


Calling functions on elements
-----------------------------

Two functions are used for calling other functions on the contents of an
iterable.

``itertools.imap(f, iterA, iterB, ...)`` returns a stream containing
``f(iterA[0], iterB[0]), f(iterA[1], iterB[1]), f(iterA[2], iterB[2]), ...``::

    itertools.imap(operator.add, [5, 6, 5], [1, 2, 3]) =>
      6, 8, 8

The ``operator`` module contains a set of functions corresponding to Python's
operators.  Some examples are ``operator.add(a, b)`` (adds two values),
``operator.ne(a, b)`` (same as ``a!=b``), and ``operator.attrgetter('id')``
(returns a callable that fetches the ``"id"`` attribute).

``itertools.starmap(func, iter)`` assumes that the iterable will return a stream
of tuples, and calls ``f()`` using these tuples as the arguments::

    itertools.starmap(os.path.join, 
                      [('/usr', 'bin', 'java'), ('/bin', 'python'),
                       ('/usr', 'bin', 'perl'),('/usr', 'bin', 'ruby')])
    =>
      /usr/bin/java, /bin/python, /usr/bin/perl, /usr/bin/ruby


Selecting elements
------------------

Another group of functions chooses a subset of an iterator's elements based on a
predicate.

``itertools.ifilter(predicate, iter)`` returns all the elements for which the
predicate returns true::

    def is_even(x):
        return (x % 2) == 0

    itertools.ifilter(is_even, itertools.count()) =>
      0, 2, 4, 6, 8, 10, 12, 14, ...

``itertools.ifilterfalse(predicate, iter)`` is the opposite, returning all
elements for which the predicate returns false::

    itertools.ifilterfalse(is_even, itertools.count()) =>
      1, 3, 5, 7, 9, 11, 13, 15, ...

``itertools.takewhile(predicate, iter)`` returns elements for as long as the
predicate returns true.  Once the predicate returns false, the iterator will
signal the end of its results.

::

    def less_than_10(x):
        return (x < 10)

    itertools.takewhile(less_than_10, itertools.count()) =>
      0, 1, 2, 3, 4, 5, 6, 7, 8, 9

    itertools.takewhile(is_even, itertools.count()) =>
      0

``itertools.dropwhile(predicate, iter)`` discards elements while the predicate
returns true, and then returns the rest of the iterable's results.

::

    itertools.dropwhile(less_than_10, itertools.count()) =>
      10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...

    itertools.dropwhile(is_even, itertools.count()) =>
      1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...


Grouping elements
-----------------

The last function I'll discuss, ``itertools.groupby(iter, key_func=None)``, is
the most complicated.  ``key_func(elem)`` is a function that can compute a key
value for each element returned by the iterable.  If you don't supply a key
function, the key is simply each element itself.

``groupby()`` collects all the consecutive elements from the underlying iterable
that have the same key value, and returns a stream of 2-tuples containing a key
value and an iterator for the elements with that key.

::

    city_list = [('Decatur', 'AL'), ('Huntsville', 'AL'), ('Selma', 'AL'), 
                 ('Anchorage', 'AK'), ('Nome', 'AK'),
                 ('Flagstaff', 'AZ'), ('Phoenix', 'AZ'), ('Tucson', 'AZ'), 
                 ...
                ]

    def get_state ((city, state)):
        return state

    itertools.groupby(city_list, get_state) =>
      ('AL', iterator-1),
      ('AK', iterator-2),
      ('AZ', iterator-3), ...

    where
    iterator-1 =>
      ('Decatur', 'AL'), ('Huntsville', 'AL'), ('Selma', 'AL')
    iterator-2 => 
      ('Anchorage', 'AK'), ('Nome', 'AK')
    iterator-3 =>
      ('Flagstaff', 'AZ'), ('Phoenix', 'AZ'), ('Tucson', 'AZ')

``groupby()`` assumes that the underlying iterable's contents will already be
sorted based on the key.  Note that the returned iterators also use the
underlying iterable, so you have to consume the results of iterator-1 before
requesting iterator-2 and its corresponding key.


The functools module
====================

The :mod:`functools` module in Python 2.5 contains some higher-order functions.
A **higher-order function** takes one or more functions as input and returns a
new function.  The most useful tool in this module is the
:func:`functools.partial` function.

For programs written in a functional style, you'll sometimes want to construct
variants of existing functions that have some of the parameters filled in.
Consider a Python function ``f(a, b, c)``; you may wish to create a new function
``g(b, c)`` that's equivalent to ``f(1, b, c)``; you're filling in a value for
one of ``f()``'s parameters.  This is called "partial function application".

The constructor for ``partial`` takes the arguments ``(function, arg1, arg2,
... kwarg1=value1, kwarg2=value2)``.  The resulting object is callable, so you
can just call it to invoke ``function`` with the filled-in arguments.

Here's a small but realistic example::

    import functools

    def log (message, subsystem):
        "Write the contents of 'message' to the specified subsystem."
        print('%s: %s' % (subsystem, message))
        ...

    server_log = functools.partial(log, subsystem='server')
    server_log('Unable to open socket')


The operator module
-------------------

The :mod:`operator` module was mentioned earlier.  It contains a set of
functions corresponding to Python's operators.  These functions are often useful
in functional-style code because they save you from writing trivial functions
that perform a single operation.

Some of the functions in this module are:

* Math operations: ``add()``, ``sub()``, ``mul()``, ``div()``, ``floordiv()``,
  ``abs()``, ...
* Logical operations: ``not_()``, ``truth()``.
* Bitwise operations: ``and_()``, ``or_()``, ``invert()``.
* Comparisons: ``eq()``, ``ne()``, ``lt()``, ``le()``, ``gt()``, and ``ge()``.
* Object identity: ``is_()``, ``is_not()``.

Consult the operator module's documentation for a complete list.



The functional module
---------------------

Collin Winter's `functional module <http://oakwinter.com/code/functional/>`__
provides a number of more advanced tools for functional programming. It also
reimplements several Python built-ins, trying to make them more intuitive to
those used to functional programming in other languages.

This section contains an introduction to some of the most important functions in
``functional``; full documentation can be found at `the project's website
<http://oakwinter.com/code/functional/documentation/>`__.

``compose(outer, inner, unpack=False)``

The ``compose()`` function implements function composition.  In other words, it
returns a wrapper around the ``outer`` and ``inner`` callables, such that the
return value from ``inner`` is fed directly to ``outer``.  That is,

::

        >>> def add(a, b):
        ...     return a + b
        ...
        >>> def double(a):
        ...     return 2 * a
        ...
        >>> compose(double, add)(5, 6)
        22

is equivalent to

::

        >>> double(add(5, 6))
        22
                    
The ``unpack`` keyword is provided to work around the fact that Python functions
are not always `fully curried <http://en.wikipedia.org/wiki/Currying>`__.  By
default, it is expected that the ``inner`` function will return a single object
and that the ``outer`` function will take a single argument. Setting the
``unpack`` argument causes ``compose`` to expect a tuple from ``inner`` which
will be expanded before being passed to ``outer``. Put simply,

::

        compose(f, g)(5, 6)
                    
is equivalent to::

        f(g(5, 6))
                    
while

::

        compose(f, g, unpack=True)(5, 6)
                    
is equivalent to::

        f(*g(5, 6))

Even though ``compose()`` only accepts two functions, it's trivial to build up a
version that will compose any number of functions. We'll use ``reduce()``,
``compose()`` and ``partial()`` (the last of which is provided by both
``functional`` and ``functools``).

::

        from functional import compose, partial
        
        multi_compose = partial(reduce, compose)
        
    
We can also use ``map()``, ``compose()`` and ``partial()`` to craft a version of
``"".join(...)`` that converts its arguments to string::

        from functional import compose, partial
        
        join = compose("".join, partial(map, str))


``flip(func)``
                    
``flip()`` wraps the callable in ``func`` and causes it to receive its
non-keyword arguments in reverse order.

::

        >>> def triple(a, b, c):
        ...     return (a, b, c)
        ...
        >>> triple(5, 6, 7)
        (5, 6, 7)
        >>>
        >>> flipped_triple = flip(triple)
        >>> flipped_triple(5, 6, 7)
        (7, 6, 5)

``foldl(func, start, iterable)``
                    
``foldl()`` takes a binary function, a starting value (usually some kind of
'zero'), and an iterable.  The function is applied to the starting value and the
first element of the list, then the result of that and the second element of the
list, then the result of that and the third element of the list, and so on.

This means that a call such as::

        foldl(f, 0, [1, 2, 3])

is equivalent to::

        f(f(f(0, 1), 2), 3)

    
``foldl()`` is roughly equivalent to the following recursive function::

        def foldl(func, start, seq):
            if len(seq) == 0:
                return start

            return foldl(func, func(start, seq[0]), seq[1:])

Speaking of equivalence, the above ``foldl`` call can be expressed in terms of
the built-in ``reduce`` like so::

        reduce(f, [1, 2, 3], 0)


We can use ``foldl()``, ``operator.concat()`` and ``partial()`` to write a
cleaner, more aesthetically-pleasing version of Python's ``"".join(...)``
idiom::

        from functional import foldl, partial
        from operator import concat
        
        join = partial(foldl, concat, "")


Revision History and Acknowledgements
=====================================

The author would like to thank the following people for offering suggestions,
corrections and assistance with various drafts of this article: Ian Bicking,
Nick Coghlan, Nick Efford, Raymond Hettinger, Jim Jewett, Mike Krell, Leandro
Lameiro, Jussi Salmela, Collin Winter, Blake Winton.

Version 0.1: posted June 30 2006.

Version 0.11: posted July 1 2006.  Typo fixes.

Version 0.2: posted July 10 2006.  Merged genexp and listcomp sections into one.
Typo fixes.

Version 0.21: Added more references suggested on the tutor mailing list.

Version 0.30: Adds a section on the ``functional`` module written by Collin
Winter; adds short section on the operator module; a few other edits.


References
==========

General
-------

**Structure and Interpretation of Computer Programs**, by Harold Abelson and
Gerald Jay Sussman with Julie Sussman.  Full text at
http://mitpress.mit.edu/sicp/.  In this classic textbook of computer science,
chapters 2 and 3 discuss the use of sequences and streams to organize the data
flow inside a program.  The book uses Scheme for its examples, but many of the
design approaches described in these chapters are applicable to functional-style
Python code.

http://www.defmacro.org/ramblings/fp.html: A general introduction to functional
programming that uses Java examples and has a lengthy historical introduction.

http://en.wikipedia.org/wiki/Functional_programming: General Wikipedia entry
describing functional programming.

http://en.wikipedia.org/wiki/Coroutine: Entry for coroutines.

http://en.wikipedia.org/wiki/Currying: Entry for the concept of currying.

Python-specific
---------------

http://gnosis.cx/TPiP/: The first chapter of David Mertz's book
:title-reference:`Text Processing in Python` discusses functional programming
for text processing, in the section titled "Utilizing Higher-Order Functions in
Text Processing".

Mertz also wrote a 3-part series of articles on functional programming
for IBM's DeveloperWorks site; see 
`part 1 <http://www-128.ibm.com/developerworks/library/l-prog.html>`__,
`part 2 <http://www-128.ibm.com/developerworks/library/l-prog2.html>`__, and
`part 3 <http://www-128.ibm.com/developerworks/linux/library/l-prog3.html>`__,


Python documentation
--------------------

Documentation for the :mod:`itertools` module.

Documentation for the :mod:`operator` module.

:pep:`289`: "Generator Expressions"

:pep:`342`: "Coroutines via Enhanced Generators" describes the new generator
features in Python 2.5.

.. comment

    Topics to place
    -----------------------------

    XXX os.walk()

    XXX Need a large example.

    But will an example add much?  I'll post a first draft and see
    what the comments say.

.. comment

    Original outline:
    Introduction
            Idea of FP
                    Programs built out of functions
                    Functions are strictly input-output, no internal state
            Opposed to OO programming, where objects have state

            Why FP?
                    Formal provability
                            Assignment is difficult to reason about
                            Not very relevant to Python
                    Modularity
                            Small functions that do one thing
                    Debuggability:
                            Easy to test due to lack of state
                            Easy to verify output from intermediate steps
                    Composability
                            You assemble a toolbox of functions that can be mixed

    Tackling a problem
            Need a significant example

    Iterators
    Generators
    The itertools module
    List comprehensions
    Small functions and the lambda statement
    Built-in functions
            map
            filter
            reduce

.. comment

    Handy little function for printing part of an iterator -- used
    while writing this document.

    import itertools
    def print_iter(it):
         slice = itertools.islice(it, 10)
         for elem in slice[:-1]:
             sys.stdout.write(str(elem))
             sys.stdout.write(', ')
        print(elem[-1])


