4.2   Implicit Sequences

A sequence can be represented without each element being stored explicitly in the memory of the computer. That is, we can construct an object that provides access to all of the elements of some sequential dataset without computing all of those elements in advance and storing them. Instead, we compute elements on demand.

A simple example of this idea arises in the range sequence type introduced in Chapter 2. A range represents a consecutive, bounded sequence of integers. However, it is not the case that each element of that sequence is represented explicitly in memory. Instead, when an element is requested from a range, it is computed. Hence, we can represent very large ranges of integers without using large blocks of memory. Only the end points of the range are stored as part of the range object.

>>> r = range(10000, 1000000000)
>>> r[45006230]

In this example, not all 999,990,000 integers in this range are stored when the range instance is constructed. Instead, the range object adds the first element 10,000 to the index 45,006,230 to produce the element 45,016,230. Computing values on demand, rather than retrieving them from an existing representation, is an example of lazy computation. In computer science, lazy computation describes any program that delays the computation of a value until that value is needed.

An iterator is an object that provides sequential access to an underlying sequential dataset. Iterators are built-in objects in many programming languages, including Python. The iterator abstraction has two components: a mechanism for retrieving the next element in some underlying series of elements and a mechanism for signaling that the end of the series has been reached and no further elements remain. In programming languages with built-in object systems, this abstraction typically corresponds to a particular interface that can be implemented by classes. The Python interface for iterators is described in the next section.

The usefulness of iterators is derived from the fact that the underlying series of data for an iterator may not be represented explicitly in memory. An iterator provides a mechanism for considering each of a series of values in turn, but all of those elements do not need to be stored simultaneously. Instead, when the next element is requested from an iterator, that element may be computed on demand instead of being retrieved from an existing memory source.

Ranges are able to compute the elements of a sequence lazily because the sequence represented is uniform, and any element is easy to compute from the starting and ending bounds of the range. Iterators allow for lazy generation of a much broader class of underlying sequential datasets, because they do not need to provide access to arbitrary elements of the underlying series. Instead, iterators are only required to compute the next element of the series, in order, each time another element is requested. While not as flexible as accessing arbitrary elements of a sequence (called random access), sequential access to sequential data series is often sufficient for data processing applications.

4.2.1   Python Iterators

The Python iterator interface is defined using a method called __next__ that returns the next element of some underlying sequential series that it represents. In response to invoking __next__, an iterator can perform arbitrary computation in order to either retrieve or compute the next element. Calls to __next__ make a mutating change to the iterator: they advance the position of the iterator. Hence, multiple calls to __next__ will return sequential elements of an underlying series. Python signals that the end of an underlying series has been reached by raising a StopIteration exception during a call to __next__.

The LetterIter class below iterates over an underlying series of letters from some start letter up to but not including some end letter. The instance attribute next_letter stores the next letter to be returned. The __next__ method returns this letter and uses it to compute a new next_letter.

>>> class LetterIter:
        """An iterator over letters of the alphabet in ASCII order."""
        def __init__(self, start='a', end='e'):
            self.next_letter = start
            self.end = end
        def __next__(self):
            if self.next_letter == self.end:
                raise StopIteration
            letter = self.next_letter
            self.next_letter = chr(ord(letter)+1)
            return letter

Using this class, we can access letters in sequence using either the __next__ method or the built-in next function, which invokes __next__ on its argument.

>>> letter_iter = LetterIter()
>>> letter_iter.__next__()
>>> letter_iter.__next__()
>>> next(letter_iter)
>>> letter_iter.__next__()
>>> letter_iter.__next__()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 12, in next

Iterators are mutable: they track the position in some underlying sequence of values as they progress. When the end is reached, the iterator is used up. A LetterIter instance can only be iterated through once. After its __next__() method raises a StopIteration exception, it continues to do so from then on. Typically, an iterator is not reset; instead a new instance is created to start a new iteration.

Iterators also allow us to represent infinite series by implementing a __next__ method that never raises a StopIteration exception. For example, the Positives class below iterates over the infinite series of positive integers. The built-in next function in Python invokes the __next__ method on its argument.

>>> class Positives:
        def __init__(self):
            self.next_positive = 1;
        def __next__(self):
            result = self.next_positive
            self.next_positive += 1
            return result
>>> p = Positives()
>>> next(p)
>>> next(p)
>>> next(p)

4.2.2   Iterables

An object is iterable if it returns an iterator when its __iter__ method is invoked. Iterable values represent data collections, and they provide a fixed representation that may produce more than one iterator.

For example, an instance of the Letters class below represents a sequence of consecutive letters. Each time its __iter__ method is invoked, a new LetterIter instance is constructed, which allows for sequential access to the contents of the sequence.

>>> class Letters:
        def __init__(self, start='a', end='e'):
            self.start = start
            self.end = end
        def __iter__(self):
            return LetterIter(self.start, self.end)

The built-in iter function invokes the __iter__ method on its argument. In the sequence of expressions below, two iterators derived from the same iterable sequence independently yield letters in sequence.

>>> b_to_k = Letters('b', 'k')
>>> first_iterator = b_to_k.__iter__()
>>> next(first_iterator)
>>> next(first_iterator)
>>> second_iterator = iter(b_to_k)
>>> second_iterator.__next__()
>>> first_iterator.__next__()
>>> first_iterator.__next__()
>>> second_iterator.__next__()
>>> second_iterator.__next__()

The iterable Letters instance b_to_k and the LetterIter iterator instances first_iterator and second_iterator are different in that the Letters instance does not change, while the iterator instances do change with each call to next (or equivalently, each invocation of __next__). The iterator tracks progress through sequential data, while an iterable represents the data itself.

Many built-in functions in Python take iterable arguments and return iterators. The map function, for example, takes a function and an iterable. It returns an iterator over the result of applying the function argument to each element in the iterable argument.

>>> caps = map(lambda x: x.upper(), b_to_k)
>>> next(caps)
>>> next(caps)

The map function is lazy: calling it does not perform the computation required to compute elements of its result. Instead, an iterator object is created that can return results if queried using next. We can observe this fact in the following example, in which the call to print is delayed until the corresponding element is requested from the doubled iterator.

>>> def double_and_print(x):
        print('***', x, '=>', 2*x, '***')
        return 2*x
>>> s = range(3, 7)
>>> doubled = map(double_and_print, s)  # double_and_print not yet called
>>> next(doubled)                       # double_and_print called once
*** 3 => 6 ***
>>> next(doubled)                       # double_and_print called again
*** 4 => 8 ***
>>> list(doubled)                       # double_and_print called twice more
*** 5 => 10 ***
*** 6 => 12 ***
[10, 12]

The filter, zip, and reversed functions also return iterators.

4.2.3   For Statements

The for statement in Python operates on iterators. Objects are iterable (an interface) if they have an __iter__ method that returns an iterator. Iterable objects can be the value of the <expression> in the header of a for statement:

for <name> in <expression>:

To execute a for statement, Python evaluates the header <expression>, which must yield an iterable value. Then, the __iter__ method is invoked on that value. Until a StopIteration exception is raised, Python repeatedly invokes the __next__ method on that iterator and binds the result to the <name> in the for statement. Then, it executes the <suite>.

>>> counts = [1, 2, 3]
>>> for item in counts:

In the above example, the counts list returns an iterator from its __iter__() method. The for statement then calls that iterator's __next__() method repeatedly, and assigns the returned value to item each time. This process continues until the iterator raises a StopIteration exception, at which point execution of the for statement concludes.

With our knowledge of iterators, we can implement the execution rule of a for statement in terms of while, assignment, and try statements.

>>> items = counts.__iter__()
>>> try:
        while True:
            item = items.__next__()
    except StopIteration:

Above, the iterator returned by invoking the __iter__ method of counts is bound to a name items so that it can be queried for each element in turn. The handling clause for the StopIteration exception does nothing, but handling the exception provides a control mechanism for exiting the while loop.

To use an iterator in a for loop, the iterator must also have an __iter__ method. The Iterator types <http://docs.python.org/3/library/stdtypes.html#iterator-types> `_ section of the Python docs suggest that an iterator have an ``__iter__` method that returns the iterator itself, so that all iterators are iterable.

4.2.4   Generators and Yield Statements

The Letters and Positives objects above require us to introduce a new field self.current into our object to keep track of progress through the sequence. With simple sequences like those shown above, this can be done easily. With complex sequences, however, it can be quite difficult for the __next__ method to save its place in the calculation. Generators allow us to define more complicated iterations by leveraging the features of the Python interpreter.

A generator is an iterator returned by a special class of function called a generator function. Generator functions are distinguished from regular functions in that rather than containing return statements in their body, they use yield statement to return elements of a series.

Generators do not use attributes of an object to track their progress through a series. Instead, they control the execution of the generator function, which runs until the next yield statement is executed each time the generator's __next__ method is invoked. The Letters iterator can be implemented much more compactly using a generator function.

>>> def letters_generator():
        current = 'a'
        while current <= 'd':
            yield current
            current = chr(ord(current)+1)
>>> for letter in letters_generator():

Even though we never explicitly defined __iter__ or __next__ methods, the yield statement indicates that we are defining a generator function. When called, a generator function doesn't return a particular yielded value, but instead a generator (which is a type of iterator) that itself can return the yielded values. A generator object has __iter__ and __next__ methods, and each call to __next__ continues execution of the generator function from wherever it left off previously until another yield statement is executed.

The first time __next__ is called, the program executes statements from the body of the letters_generator function until it encounters the yield statement. Then, it pauses and returns the value of current. yield statements do not destroy the newly created environment, they preserve it for later. When __next__ is called again, execution resumes where it left off. The values of current and of any other bound names in the scope of letters_generator are preserved across subsequent calls to __next__.

We can walk through the generator by manually calling ____next__():

>>> letters = letters_generator()
>>> type(letters)
<class 'generator'>
>>> letters.__next__()
>>> letters.__next__()
>>> letters.__next__()
>>> letters.__next__()
>>> letters.__next__()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>

The generator does not start executing any of the body statements of its generator function until the first time __next__ is invoked. The generator raises a StopIteration exception whenever its generator function returns.

4.2.5   Creating Iterables with Yield

In Python, iterators only make a single pass over the elements of an underlying series. After that pass, the iterator will continue to raise a StopIteration exception when __next__ is invoked. Many applications require iteration over elements multiple times. For example, we have to iterate over a list many times in order to enumerate all pairs of elements.

>>> def all_pairs(s):
        for item1 in s:
            for item2 in s:
                yield (item1, item2)
>>> list(all_pairs([1, 2, 3]))
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)]

Sequences are not themselves iterators, but instead iterable objects. The iterable interface in Python consists of a single message, __iter__, that returns an iterator. The built-in sequence types in Python return new instances of iterators when their __iter__ methods are invoked. If an iterable object returns a fresh instance of an iterator each time __iter__ is called, then it can be iterated over multiple times.

New iterable classes can be defined by implementing the iterable interface. For example, the iterable LettersWithYield class below returns a new iterator over letters each time __iter__ is invoked.

>>> class LettersWithYield:
        def __init__(self, start='a', end='e'):
            self.start = start
            self.end = end
        def __iter__(self):
            next_letter = self.start
            while next_letter < self.end:
                yield next_letter
                next_letter = chr(ord(next_letter)+1)

The __iter__ method is a generator function; it returns a generator object that yields the letters 'a' through 'd' and then stops. Each time we invoke this method, a new generator starts a fresh pass through the sequential data.

>>> letters = LettersWithYield()
>>> list(all_pairs(letters))[:5]
[('a', 'a'), ('a', 'b'), ('a', 'c'), ('a', 'd'), ('b', 'a')]

4.2.6   Streams

Streams offer another way to represent sequential data implicitly. A stream is a lazily computed linked list. Like the Link class from Chapter 2, a Stream instance responds to requests for its first element and the rest of the stream. Like an Link, the rest of a Stream is itself a Stream. Unlike an Link, the rest of a stream is only computed when it is looked up, rather than being stored in advance. That is, the rest of a stream is computed lazily.

To achieve this lazy evaluation, a stream stores a function that computes the rest of the stream. Whenever this function is called, its returned value is cached as part of the stream in an attribute called _rest, named with an underscore to indicate that it should not be accessed directly.

The accessible attribute rest is a property method that returns the rest of the stream, computing it if necessary. With this design, a stream stores how to compute the rest of the stream, rather than always storing the rest explicitly.

>>> class Stream:
        """A lazily computed linked list."""
        class empty:
            def __repr__(self):
                return 'Stream.empty'
        empty = empty()
        def __init__(self, first, compute_rest=lambda: empty):
            assert callable(compute_rest), 'compute_rest must be callable.'
            self.first = first
            self._compute_rest = compute_rest
        def rest(self):
            """Return the rest of the stream, computing it if necessary."""
            if self._compute_rest is not None:
                self._rest = self._compute_rest()
                self._compute_rest = None
            return self._rest
        def __repr__(self):
            return 'Stream({0}, <...>)'.format(repr(self.first))

A linked list is defined using a nested expression. For example, we can create an Link that represents the elements 1 then 5 as follows:

>>> r = Link(1, Link(2+3, Link(9)))

Likewise, we can create a Stream representing the same series. The Stream does not actually compute the second element 5 until the rest of the stream is requested. We achieve this effect by creating anonymous functions.

>>> s = Stream(1, lambda: Stream(2+3, lambda: Stream(9)))

Here, 1 is the first element of the stream, and the lambda expression that follows returns a function for computing the rest of the stream.

Accessing the elements of linked list r and stream s proceed similarly. However, while 5 is stored within r, it is computed on demand for s via addition, the first time that it is requested.

>>> r.first
>>> s.first
>>> r.rest.first
>>> s.rest.first
>>> r.rest
Link(5, Link(9))
>>> s.rest
Stream(5, <...>)

While the rest of r is a two-element linked list, the rest of s includes a function to compute the rest; the fact that it will return the empty stream may not yet have been discovered.

When a Stream instance is constructed, the field self._rest is None, signifying that the rest of the Stream has not yet been computed. When the rest attribute is requested via a dot expression, the rest property method is invoked, which triggers computation with self._rest = self._compute_rest(). Because of the caching mechanism within a Stream, the compute_rest function is only ever called once, then discarded.

The essential properties of a compute_rest function are that it takes no arguments, and it returns a Stream or Stream.empty.

Lazy evaluation gives us the ability to represent infinite sequential datasets using streams. For example, we can represent increasing integers, starting at any first value.

>>> def integer_stream(first):
        def compute_rest():
            return integer_stream(first+1)
        return Stream(first, compute_rest)
>>> positives = integer_stream(1)
>>> positives
Stream(1, <...>)
>>> positives.first

When integer_stream is called for the first time, it returns a stream whose first is the first integer in the sequence. However, integer_stream is actually recursive because this stream's compute_rest calls integer_stream again, with an incremented argument. We say that integer_stream is lazy because the recursive call to integer_stream is only made whenever the rest of an integer stream is requested.

>>> positives.first
>>> positives.rest.first
>>> positives.rest.rest
Stream(3, <...>)

The same higher-order functions that manipulate sequences -- map and filter -- also apply to streams, although their implementations must change to apply their argument functions lazily. The function map_stream maps a function over a stream, which produces a new stream. The locally defined compute_rest function ensures that the function will be mapped onto the rest of the stream whenever the rest is computed.

>>> def map_stream(fn, s):
        if s is Stream.empty:
            return s
        def compute_rest():
            return map_stream(fn, s.rest)
        return Stream(fn(s.first), compute_rest)

A stream can be filtered by defining a compute_rest function that applies the filter function to the rest of the stream. If the filter function rejects the first element of the stream, the rest is computed immediately. Because filter_stream is recursive, the rest may be computed multiple times until a valid first element is found.

>>> def filter_stream(fn, s):
        if s is Stream.empty:
            return s
        def compute_rest():
            return filter_stream(fn, s.rest)
        if fn(s.first):
            return Stream(s.first, compute_rest)
            return compute_rest()

The map_stream and filter_stream functions exhibit a common pattern in stream processing: a locally defined compute_rest function recursively applies a processing function to the rest of the stream whenever the rest is computed.

To inspect the contents of a stream, we can coerce up to the first k elements to a Python list.

>>> def first_k_as_list(s, k):
        first_k = []
        while s is not Stream.empty and k > 0:
            s, k = s.rest, k-1
        return first_k

These convenience functions allow us to verify our map_stream implementation with a simple example that squares the integers from 3 to 7.

>>> s = integer_stream(3)
>>> s
Stream(3, <...>)
>>> m = map_stream(lambda x: x*x, s)
>>> m
Stream(9, <...>)
>>> first_k_as_list(m, 5)
[9, 16, 25, 36, 49]

We can use our filter_stream function to define a stream of prime numbers using the sieve of Eratosthenes, which filters a stream of integers to remove all numbers that are multiples of its first element. By successively filtering with each prime, all composite numbers are removed from the stream.

>>> def primes(pos_stream):
        def not_divible(x):
            return x % pos_stream.first != 0
        def compute_rest():
            return primes(filter_stream(not_divible, pos_stream.rest))
        return Stream(pos_stream.first, compute_rest)

By truncating the primes stream, we can enumerate any prefix of the prime numbers.

>>> prime_numbers = primes(integer_stream(2))
>>> first_k_as_list(prime_numbers, 7)
[2, 3, 5, 7, 11, 13, 17]

Streams contrast with iterators in that they can be passed to pure functions multiple times and yield the same result each time. The primes stream is not "used up" by converting it to a list. That is, the first element of prime_numbers is still 2 after converting the prefix of the stream to a list.

>>> prime_numbers.first

Just as linked lists provide a simple implementation of the sequence abstraction, streams provide a simple, functional, recursive data structure that implements lazy evaluation through the use of higher-order functions.

Continue: 4.3 Declarative Programming