Extra Content: Functional Programming

Overview

Questions

• What is functional programming?
• Which situations/problems is functional programming well suited for?

Objectives

• Describe the core concepts that define the functional programming paradigm
• Describe the main characteristics of code that is written in functional programming style
• Learn how to generate and process data collections efficiently using MapReduce and Python’s comprehensions

Functional programming is a programming paradigm where programs are constructed by applying and composing/chaining functions. Functional programming is based on the mathematical definition of a function f(), which applies a transformation to some input data giving us some other data as a result (i.e. a mapping from input x to output f(x)). Thus, a program written in a functional style becomes a series of transformations on data which are performed to produce a desired output. Each function (transformation) taken by itself is simple and straightforward to understand; complexity is handled by composing functions in various ways.

Often when we use the term function we are referring to a construct containing a block of code which performs a particular task and can be reused. We have already seen this in procedural programming - so how are functions in functional programming different? The key difference is that functional programming is focussed on what transformations are done to the data, rather than how these transformations are performed (i.e. a detailed sequence of steps which update the state of the code to reach a desired state). Let us compare and contrast examples of these two programming paradigms.

Functional vs Procedural Programming

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The following two code examples implement the calculation of a factorial in procedural and functional styles, respectively. Recall that the factorial of a number n (denoted by n!) is calculated as the product of integer numbers from 1 to n.

The first example provides a procedural style factorial function.

PYTHON

def factorial(n):
    """Calculate the factorial of a given number.

    :param int n: The factorial to calculate
    :return: The resultant factorial
    """
    if n < 0:
        raise ValueError('Only use non-negative integers.')

    factorial = 1
    for i in range(1, n + 1): # iterate from 1 to n
        # save intermediate value to use in the next iteration
        factorial = factorial * i

    return factorial

Functions in procedural programming are procedures that describe a detailed list of instructions to tell the computer what to do step by step and how to change the state of the program and advance towards the result. They often use iteration to repeat a series of steps. Functional programming, on the other hand, typically uses recursion - an ability of a function to call/repeat itself until a particular condition is reached. Let us see how it is used in the functional programming example below to achieve a similar effect to that of iteration in procedural programming.

PYTHON

# Functional style factorial function
def factorial(n):
    """Calculate the factorial of a given number.

    :param int n: The factorial to calculate
    :return: The resultant factorial
    """
    if n < 0:
        raise ValueError('Only use non-negative integers.')

    if n == 0 or n == 1:
        return 1 # exit from recursion, prevents infinite loops
    else:
        return  n * factorial(n-1) # recursive call to the same function

Note: You may have noticed that both functions in the above code examples have the same signature (i.e. they take an integer number as input and return its factorial as output). You could easily swap these equivalent implementations without changing the way that the function is invoked. Remember, a single piece of software may well contain instances of multiple programming paradigms - including procedural, functional and object-oriented - it is up to you to decide which one to use and when to switch based on the problem at hand and your personal coding style.

Functional computations only rely on the values that are provided as inputs to a function and not on the state of the program that precedes the function call. They do not modify data that exists outside the current function, including the input data - this property is referred to as the immutability of data. This means that such functions do not create any side effects, i.e. do not perform any action that affects anything other than the value they return. For example: printing text, writing to a file, modifying the value of an input argument, or changing the value of a global variable. Functions without side affects that return the same data each time the same input arguments are provided are called pure functions.

Exercise: Pure Functions

Which of these functions are pure? If you are not sure, explain your reasoning to someone else, do they agree?

PYTHON

def add_one(x):
    return x + 1

def say_hello(name):
    print('Hello', name)

def append_item_1(a_list, item):
    a_list += [item]
    return a_list

def append_item_2(a_list, item):
    result = a_list + [item]
    return result

Show me the solution

1. add_one is pure - it has no effects other than to return a value and this value will always be the same when given the same inputs
2. say_hello is not pure - printing text counts as a side effect, even though it is the clear purpose of the function
3. append_item_1 is not pure - the argument a_list gets modified as a side effect - try this yourself to prove it
4. append_item_2 is pure - the result is a new variable, so this time a_list does not get modified - again, try this yourself

Benefits of Functional Code

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There are a few benefits we get when working with pure functions:

• Testability
• Composability
• Parallelisability

Testability indicates how easy it is to test the function - usually meaning unit tests. It is much easier to test a function if we can be certain that a particular input will always produce the same output. If a function we are testing might have different results each time it runs (e.g. a function that generates random numbers drawn from a normal distribution), we need to come up with a new way to test it. Similarly, it can be more difficult to test a function with side effects as it is not always obvious what the side effects will be, or how to measure them.

Composability refers to the ability to make a new function from a chain of other functions by piping the output of one as the input to the next. If a function does not have side effects or non-deterministic behaviour, then all of its behaviour is reflected in the value it returns. As a consequence of this, any chain of combined pure functions is itself pure, so we keep all these benefits when we are combining functions into a larger program. As an example of this, we could make a function called add_two, using the add_one function we already have.

PYTHON

def add_two(x):
    return add_one(add_one(x))

Parallelisability is the ability for operations to be performed at the same time (independently). If we know that a function is fully pure and we have got a lot of data, we can often improve performance by splitting data and distributing the computation across multiple processors. The output of a pure function depends only on its input, so we will get the right result regardless of when or where the code runs.

Everything in Moderation

Despite the benefits that pure functions can bring, we should not be trying to use them everywhere. Any software we write needs to interact with the rest of the world somehow, which requires side effects. With pure functions you cannot read any input, write any output, or interact with the rest of the world in any way, so we cannot usually write useful software using just pure functions. Python programs or libraries written in functional style will usually not be as extreme as to completely avoid reading input, writing output, updating the state of internal local variables, etc.; instead, they will provide a functional-appearing interface but may use non-functional features internally. An example of this is the Python Pandas library for data manipulation built on top of NumPy - most of its functions appear pure as they return new data objects instead of changing existing ones.

There are other advantageous properties that can be derived from the functional approach to coding. In languages which support functional programming, a function is a first-class object like any other object - not only can you compose/chain functions together, but functions can be used as inputs to, passed around or returned as results from other functions (remember, in functional programming code is data). This is why functional programming is suitable for processing data efficiently - in particular in the world of Big Data, where code is much smaller than the data, sending the code to where data is located is cheaper and faster than the other way round. Let us see how we can do data processing using functional programming.

MapReduce Data Processing Approach

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When working with data you will often find that you need to apply a transformation to each datapoint of a dataset and then perform some aggregation across the whole dataset. One instance of this data processing approach is known as MapReduce and is applied when processing (but not limited to) Big Data, e.g. using tools such as Spark or Hadoop. The name MapReduce comes from applying an operation to (mapping) each value in a dataset, then performing a reduction operation which collects/aggregates all the individual results together to produce a single result. MapReduce relies heavily on composability and parallelisability of functional programming - both map and reduce can be done in parallel and on smaller subsets of data, before aggregating all intermediate results into the final result.

Mapping

map(f, C) is a function takes another function f() and a collection C of data items as inputs. Calling map(f, C) applies the function f(x) to every data item x in a collection C and returns the resulting values as a new collection of the same size.

This is a simple mapping that takes a list of names and returns a list of the lengths of those names using the built-in function len():

PYTHON

name_lengths = map(len, ["Mary", "Isla", "Sam"])
print(list(name_lengths))

OUTPUT

[4, 4, 3]

This is a mapping that squares every number in the passed collection using anonymous, inlined lambda expression (a simple one-line mathematical expression representing a function):

PYTHON

squares = map(lambda x: x * x, [0, 1, 2, 3, 4])
print(list(squares))

OUTPUT

[0, 1, 4, 9, 16]

Lambda

Lambda expressions are used to create anonymous functions that can be used to write more compact programs by inlining function code. A lambda expression takes any number of input parameters and creates an anonymous function that returns the value of the expression. So, we can use the short, one-line lambda x, y, z, ...: expression code instead of defining and calling a named function f() as follows:

PYTHON

def f(x, y, z, ...):
  return expression

The major distinction between lambda functions and ‘normal’ functions is that lambdas do not have names. We could give a name to a lambda expression if we really wanted to - but at that point we should be using a ‘normal’ Python function instead.

PYTHON

# Do not do this
add_one = lambda x: x + 1

# Do this instead
def add_one(x):
  return x + 1

In addition to using built-in or inlining anonymous lambda functions, we can also pass a named function that we have defined ourselves to the map() function.

PYTHON

def add_one(num):
    return num + 1

result = map(add_one, [0, 1, 2])
print(list(result))

OUTPUT

[1, 2, 3]

Exercise: Check Inflammation Patient Data Against A Threshold Using Map

Write a new function called daily_above_threshold() in our inflammation models.py that determines whether or not each daily inflammation value for a given patient exceeds a given threshold.

Given a patient row number in our data, the patient dataset itself, and a given threshold, write the function to use map() to generate and return a list of booleans, with each value representing whether or not the daily inflammation value for that patient exceeded the given threshold.

Ordinarily we would use Numpy’s own map feature, but for this exercise, let us try a solution without it.

Show me the solution

PYTHON

def daily_above_threshold(patient_num, data, threshold):
    """Determine whether or not each daily inflammation value exceeds a given threshold for a given patient.

    :param patient_num: The patient row number
    :param data: A 2D data array with inflammation data
    :param threshold: An inflammation threshold to check each daily value against
    :returns: A boolean list representing whether or not each patient's daily inflammation exceeded the threshold
    """

    return list(map(lambda x: x > threshold, data[patient_num]))

Note: map() function returns a map iterator object which needs to be converted to a collection object (such as a list, dictionary, set, tuple) using the corresponding “factory” function (in our case list()).

Comprehensions for Mapping/Data Generation

Another way you can generate new collections of data from existing collections in Python is using comprehensions, which are an elegant and concise way of creating data from iterable objects using for loops. While not a pure functional concept, comprehensions provide data generation functionality and can be used to achieve the same effect as the built-in “pure functional” function map(). They are commonly used and actually recommended as a replacement of map() in modern Python. Let us have a look at some examples.

PYTHON

integers = range(5)
double_ints = [2 * i for i in integers]

print(double_ints)

OUTPUT

[0, 2, 4, 6, 8]

The above example uses a list comprehension to double each number in a sequence. Notice the similarity between the syntax for a list comprehension and a for loop - in effect, this is a for loop compressed into a single line. In this simple case, the code above is equivalent to using a map operation on a sequence, as shown below:

PYTHON

integers = range(5)
double_ints = map(lambda i: 2 * i, integers)
print(list(double_ints))

OUTPUT

[0, 2, 4, 6, 8]

We can also use list comprehensions to filter data, by adding the filter condition to the end:

PYTHON

double_even_ints = [2 * i for i in integers if i % 2 == 0]
print(double_even_ints)

OUTPUT

[0, 4, 8]

Set and Dictionary Comprehensions and Generators

We also have set comprehensions and dictionary comprehensions, which look similar to list comprehensions but use the set literal and dictionary literal syntax, respectively.

PYTHON

double_even_int_set = {2 * i for i in integers if i % 2 == 0}
print(double_even_int_set)

double_even_int_dict = {i: 2 * i for i in integers if i % 2 == 0}
print(double_even_int_dict)

OUTPUT

{0, 4, 8}
{0: 0, 2: 4, 4: 8}

Finally, there is one last ‘comprehension’ in Python - a generator expression - a type of an iterable object which we can take values from and loop over, but does not actually compute any of the values until we need them. Iterable is the generic term for anything we can loop or iterate over - lists, sets and dictionaries are all iterables.

The range function is an example of a generator - if we created a range(1000000000), but didn’t iterate over it, we’d find that it takes almost no time to do. Creating a list containing a similar number of values would take much longer, and could be at risk of running out of memory.

We can build our own generators using a generator expression. These look much like the comprehensions above, but act like a generator when we use them. Note the syntax difference for generator expressions - parenthesis are used in place of square or curly brackets.

PYTHON

doubles_generator = (2 * i for i in integers)
for x in doubles_generator:
   print(x)

OUTPUT

0
2
4
6
8

Let us now have a look at reducing the elements of a data collection into a single result.

Reducing

reduce(f, C, initialiser) function accepts a function f(), a collection C of data items and an optional initialiser, and returns a single cumulative value which aggregates (reduces) all the values from the collection into a single result. The reduction function first applies the function f() to the first two values in the collection (or to the initialiser, if present, and the first item from C). Then for each remaining value in the collection, it takes the result of the previous computation and the next value from the collection as the new arguments to f() until we have processed all of the data and reduced it to a single value. For example, if collection C has 5 elements, the call reduce(f, C) calculates:

f(f(f(f(C[0], C[1]), C[2]), C[3]), C[4])

One example of reducing would be to calculate the product of a sequence of numbers.

PYTHON

from functools import reduce

sequence = [1, 2, 3, 4]

def product(a, b):
    return a * b

print(reduce(product, sequence))

# The same reduction using a lambda function
print(reduce((lambda a, b: a * b), sequence))

OUTPUT

24
24

Note that reduce() is not a built-in function like map() - you need to import it from library functools.

Exercise: Calculate the Sum of a Sequence of Numbers Using Reduce

Using reduce calculate the sum of a sequence of numbers. Although in practice we would use the built-in sum() function for this - try doing it without it.

Show me the solution

PYTHON

from functools import reduce

sequence = [1, 2, 3, 4]

def add(a, b):
    return a + b

print(reduce(add, sequence))

# The same reduction using a lambda function
print(reduce((lambda a, b: a + b), sequence))

OUTPUT

10
10

Putting It All Together

Let us now put together what we have learned about map and reduce so far by writing a function that calculates the sum of the squares of the values in a list using the MapReduce approach.

PYTHON

from functools import reduce

def sum_of_squares(sequence):
    squares = [x * x for x in sequence]  # use list comprehension for mapping
    return reduce(lambda a, b: a + b, squares)

We should see the following behaviour when we use it:

PYTHON

print(sum_of_squares([0]))
print(sum_of_squares([1]))
print(sum_of_squares([1, 2, 3]))
print(sum_of_squares([-1]))
print(sum_of_squares([-1, -2, -3]))

OUTPUT

0
1
14
1
14

Now let’s assume we’re reading in these numbers from an input file, so they arrive as a list of strings. We will modify the function so that it passes the following tests:

PYTHON

print(sum_of_squares(['1', '2', '3']))
print(sum_of_squares(['-1', '-2', '-3']))

OUTPUT

14
14

The code may look like:

PYTHON

from functools import reduce

def sum_of_squares(sequence):
    integers = [int(x) for x in sequence]
    squares = [x * x for x in integers]
    return reduce(lambda a, b: a + b, squares)

Finally, like comments in Python, we’d like it to be possible for users to comment out numbers in the input file they give to our program. We will finally extend our function so that the following tests pass:

PYTHON

print(sum_of_squares(['1', '2', '3']))
print(sum_of_squares(['-1', '-2', '-3']))
print(sum_of_squares(['1', '2', '#100', '3']))

OUTPUT

14
14
14

To do so, we may filter out certain elements and have:

PYTHON

from functools import reduce

def sum_of_squares(sequence):
    integers = [int(x) for x in sequence if x[0] != '#']
    squares = [x * x for x in integers]
    return reduce(lambda a, b: a + b, squares)

Exercise: Extend Inflammation Threshold Function Using Reduce

Extend the daily_above_threshold() function you wrote previously to return a count of the number of days a patient’s inflammation is over the threshold. Use reduce() over the boolean array that was previously returned to generate the count, then return that value from the function.

You may choose to define a separate function to pass to reduce(), or use an inline lambda expression to do it (which is a bit trickier!).

Hints:

• Remember that you can define an initialiser value with reduce() > to help you start the counter
• If defining a lambda expression, > note that it can conditionally return different values using the syntax > <value> if <condition> else <another_value> in the expression.

Show me the solution

Using a separate function:

PYTHON

def daily_above_threshold(patient_num, data, threshold):
    """Count how many days a given patient's inflammation exceeds a given threshold.

    :param patient_num: The patient row number
    :param data: A 2D data array with inflammation data
    :param threshold: An inflammation threshold to check each daily value against
    :returns: An integer representing the number of days a patient's inflammation is over a given threshold
    """
    def count_above_threshold(a, b):
        if b:
            return a + 1
        else:
            return a

   # Use map to determine if each daily inflammation value exceeds a given threshold for a patient
   above_threshold = map(lambda x: x > threshold, data[patient_num])
   # Use reduce to count on how many days inflammation was above the threshold for a patient
   return reduce(count_above_threshold, above_threshold, 0)

Note that the count_above_threshold function used by reduce() was defined within the daily_above_threshold() function to limit its scope and clarify its purpose (i.e. it may only be useful as part of daily_above_threshold() hence being defined as an inner function).

The equivalent code using a lambda expression may look like:

PYTHON

from functools import reduce

...

def daily_above_threshold(patient_num, data, threshold):
    """Count how many days a given patient's inflammation exceeds a given threshold.

    :param patient_num: The patient row number
    :param data: A 2D data array with inflammation data
    :param threshold: An inflammation threshold to check each daily value against
    :returns: An integer representing the number of days a patient's inflammation is over a given threshold
    """

    above_threshold = map(lambda x: x > threshold, data[patient_num])
    return reduce(lambda a, b: a + 1 if b else a, above_threshold, 0)

Where could this be useful? For example, you may want to define the success criteria for a trial if, say, 80% of patients do not exhibit inflammation in any of the trial days, or some similar metrics.

Decorators

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Finally, we will look at one last aspect of Python where functional programming is coming handy. As we have seen in the episode on parametrising our unit tests, a decorator can take a function, modify/decorate it, then return the resulting function. This is possible because Python treats functions as first-class objects that can be passed around as normal data. Here, we discuss decorators in more detail and learn how to write our own. Let us look at the following code for ways on how to “decorate” functions.

PYTHON

def with_logging(func):

    """A decorator which adds logging to a function."""
    def inner(*args, **kwargs):
        print("Before function call")
        result = func(*args, **kwargs)
        print("After function call")
        return result

    return inner


def add_one(n):
    print("Adding one")
    return n + 1

# Redefine function add_one by wrapping it within with_logging function
add_one = with_logging(add_one)

# Another way to redefine a function - using a decorator
@with_logging
def add_two(n):
    print("Adding two")
    return n + 2

print(add_one(1))
print(add_two(1))

OUTPUT

Before function call
Adding one
After function call
2
Before function call
Adding two
After function call
3

In this example, we see a decorator (with_logging) and two different syntaxes for applying the decorator to a function. The decorator is implemented here as a function which encloses another function. Because the inner function (inner()) calls the function being decorated (func()) and returns its result, it still behaves like this original function. Part of this is the use of *args and **kwargs - these allow our decorated function to accept any arguments or keyword arguments and pass them directly to the function being decorated. Our decorator in this case does not need to modify any of the arguments, so we do not need to know what they are. Any additional behaviour we want to add as part of our decorated function, we can put before or after the call to the original function. Here we print some text both before and after the decorated function, to show the order in which events happen.

We also see in this example the two different ways in which a decorator can be applied. The first of these is to use a normal function call (with_logging(add_one)), where we then assign the resulting function back to a variable - often using the original name of the function, so replacing it with the decorated version. The second syntax is the one we have seen previously (@with_logging). This syntax is equivalent to the previous one - the result is that we have a decorated version of the function, here with the name add_two. Both of these syntaxes can be useful in different situations: the @ syntax is more concise if we never need to use the un-decorated version, while the function-call syntax gives us more flexibility - we can continue to use the un-decorated function if we make sure to give the decorated one a different name, and can even make multiple decorated versions using different decorators.

Exercise: Measuring Performance Using Decorators

One small task you might find a useful case for a decorator is measuring the time taken to execute a particular function. This is an important part of performance profiling.

Write a decorator which you can use to measure the execution time of the decorated function using the time.process_time_ns() function. There are several different timing functions each with slightly different use-cases, but we won’t worry about that here.

For the function to measure, you may wish to use this as an example:

PYTHON

def measure_me(n):
    total = 0
    for i in range(n):
        total += i * i

    return total

Show me the solution

PYTHON

import time

def profile(func):
    def inner(*args, **kwargs):
        start = time.process_time_ns()
        result = func(*args, **kwargs)
        stop = time.process_time_ns()

        print("Took {0} seconds".format((stop - start) / 1e9))
        return result

    return inner

@profile
def measure_me(n):
    total = 0
    for i in range(n):
        total += i * i

    return total

print(measure_me(1000000))

OUTPUT

Took 0.124199753 seconds
333332833333500000

Key Points

• Functional programming is a programming paradigm where programs are constructed by applying and composing smaller and simple functions into more complex ones (which describe the flow of data within a program as a sequence of data transformations).
• In functional programming, functions tend to be pure - they do not exhibit side-effects (by not affecting anything other than the value they return or anything outside a function). Functions can also be named, passed as arguments, and returned from other functions, just as any other data type.
• MapReduce is an instance of a data generation and processing approach, in particular suited for functional programming and handling Big Data within parallel and distributed environments.
• Python provides comprehensions for lists, dictionaries, sets and generators - a concise (if not strictly functional) way to generate new data from existing data collections while performing sophisticated mapping, filtering and conditional logic on original dataset’s members.