Lab 2: Higher-Order Functions, Lambda Expressions

Due by 11:59pm on Thursday, June 27.

Starter Files

Download Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.


Consult this section if you need a refresher on the material for this lab. It's okay to skip directly to the questions and refer back here should you get stuck.

Short Circuiting

What do you think will happen if we type the following into Python?

1 / 0

Try it out in Python! You should see a ZeroDivisionError. But what about this expression?

True or 1 / 0

It evaluates to True because Python's and and or operators short-circuit. That is, they don't necessarily evaluate every operand.

Operator Checks if: Evaluates from left to right up to: Example
AND All values are true The first false value False and 1 / 0 evaluates to False
OR At least one value is true The first true value True or 1 / 0 evaluates to True

Short-circuiting happens when the operator reaches an operand that allows them to make a conclusion about the expression. For example, and will short-circuit as soon as it reaches the first false value because it then knows that not all the values are true.

If and and or do not short-circuit, they just return the last value; another way to remember this is that and and or always return the last thing they evaluate, whether they short circuit or not. Keep in mind that and and or don't always return booleans when using values other than True and False.

Higher-Order Functions

Variables are names bound to values, which can be primitives like 3 or 'Hello World', but they can also be functions. And since functions can take arguments of any value, other functions can be passed in as arguments. This is the basis for higher-order functions.

A higher order function is a function that manipulates other functions by taking in functions as arguments, returning a function, or both. We will introduce the basics of higher order functions in this lab and will be exploring many applications of higher order functions in our next lab.

Functions as arguments

In Python, function objects are values that can be passed around. We know that one way to create functions is by using a def statement:

def square(x):
    return x * x

The above statement created a function object with the intrinsic name square as well as binded it to the name square in the current environment. Now let's try passing it as an argument.

First, let's write a function that takes in another function as an argument:

def scale(f, x, k):
    """ Returns the result of f(x) scaled by k. """
    return k * f(x)

We can now call scale on square and some other arguments:

>>> scale(square, 3, 2) # Double square(3)
>>> scale(square, 2, 5) # 5 times 2 squared

Note that in the body of the call to scale, the function object with the intrinsic name square is bound to the parameter f. Then, we call square in the body of scale by calling f(x).

As we saw in the above section on lambda expressions, we can also pass lambda expressions into call expressions!

>>> scale(lambda x: x + 10, 5, 2)

In the frame for this call expression, the name f is bound to the function created by the lambda expression lambda x: x + 10.

Functions that return functions

Because functions are values, they are valid as return values! Here's an example:

def multiply_by(m):
    def multiply(n):
        return n * m
    return multiply

In this particular case, we defined the function multiply within the body of multiply_by and then returned it. Let's see it in action:

>>> multiply_by(3)
<function multiply_by.<locals>.multiply at ...>
>>> multiply(4)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
NameError: name 'multiply' is not defined

A call to multiply_by returns a function, as expected. However, calling multiply errors, even though that's the name we gave the inner function. This is because the name multiply only exists within the frame where we evaluate the body of multiply_by.

So how do we actually use the inner function? Here are two ways:

>>> times_three = multiply_by(3) # Assign the result of the call expression to a name
>>> times_three(5) # Call the inner function with its new name
>>> multiply_by(3)(10) # Chain together two call expressions

The point is, because multiply_by returns a function, you can use its return value just like you would use any other function.

Lambda Expressions

Lambda expressions are expressions that evaluate to functions by specifying two things: the parameters and a return expression.

lambda <parameters>: <return expression>

While both lambda expressions and def statements create function objects, there are some notable differences. lambda expressions work like other expressions; much like a mathematical expression just evaluates to a number and does not alter the current environment, a lambda expression evaluates to a function without changing the current environment. Let's take a closer look.

lambda def
Type Expression that evaluates to a value Statement that alters the environment
Result of execution Creates an anonymous lambda function with no intrinsic name. Creates a function with an intrinsic name and binds it to that name in the current environment.
Effect on the environment Evaluating a lambda expression does not create or modify any variables. Executing a def statement both creates a new function object and binds it to a name in the current environment.
Usage A lambda expression can be used anywhere that expects an expression, such as in an assignment statement or as the operator or operand to a call expression. After executing a def statement, the created function is bound to a name. You should use this name to refer to the function anywhere that expects an expression.
# A lambda expression by itself does not alter
# the environment
lambda x: x * x

# We can assign lambda functions to a name
# with an assignment statement
square = lambda x: x * x

# Lambda expressions can be used as an operator
# or operand
negate = lambda f, x: -f(x)
negate(lambda x: x * x, 3)
def square(x):
    return x * x

# A function created by a def statement
# can be referred to by its intrinsic name

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Environment Diagrams

Environment diagrams are one of the best learning tools for understanding lambda expressions and higher order functions because you're able to keep track of all the different names, function objects, and arguments to functions. We highly recommend drawing environment diagrams or using Python tutor if you get stuck doing the WWPD problems below. For examples of what environment diagrams should look like, try running some code in Python tutor. Here are the rules:

Assignment Statements

  1. Evaluate the expression on the right hand side of the = sign.
  2. If the name found on the left hand side of the = doesn't already exist in the current frame, write it in. If it does, erase the current binding. Bind the value obtained in step 1 to this name.

If there is more than one name/expression in the statement, evaluate all the expressions first from left to right before making any bindings.

def Statements

  1. Draw the function object with its intrinsic name, formal parameters, and parent frame. A function's parent frame is the frame in which the function was defined.
  2. If the intrinsic name of the function doesn't already exist in the current frame, write it in. If it does, erase the current binding. Bind the newly created function object to this name.

Call expressions

Note: you do not have to go through this process for a built-in Python function like max or print.

  1. Evaluate the operator, whose value should be a function.
  2. Evaluate the operands left to right.
  3. Open a new frame. Label it with the sequential frame number, the intrinsic name of the function, and its parent.
  4. Bind the formal parameters of the function to the arguments whose values you found in step 2.
  5. Execute the body of the function in the new environment.


Note: As we saw in the lambda expression section above, lambda functions have no intrinsic name. When drawing lambda functions in environment diagrams, they are labeled with the name lambda or with the lowercase Greek letter λ. This can get confusing when there are multiple lambda functions in an environment diagram, so you can distinguish them by numbering them or by writing the line number on which they were defined.

  1. Draw the lambda function object and label it with λ, its formal parameters, and its parent frame. A function's parent frame is the frame in which the function was defined.

This is the only step. We are including this section to emphasize the fact that the difference between lambda expressions and def statements is that lambda expressions do not create any new bindings in the environment.

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Required Questions

Getting Started Videos

These videos may provide some helpful direction for tackling the coding problems on this assignment.

To see these videos, you should be logged into your email.

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What Would Python Display?

Important: For all WWPD questions, type Function if you believe the answer is <function...>, Error if it errors, and Nothing if nothing is displayed.

Q1: WWPD: The Truth Will Prevail

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

python3 ok -q short-circuit -u

>>> True and 13
>>> False or 0
>>> not 10
>>> not None
>>> True and 1 / 0
Error (ZeroDivisionError)
>>> True or 1 / 0
>>> -1 and 1 > 0
>>> -1 or 5
>>> (1 + 1) and 1
>>> print(3) or ""
3 ''
>>> def f(x):
...     if x == 0:
...         return "zero"
...     elif x > 0:
...         return "positive"
...     else:
...         return ""
>>> 0 or f(1)
>>> f(0) or f(-1)
>>> f(0) and f(-1)

Q2: WWPD: Higher-Order Functions

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

python3 ok -q hof-wwpd -u

>>> def cake():
...    print('beets')
...    def pie():
...        print('sweets')
...        return 'cake'
...    return pie
>>> chocolate = cake()
>>> chocolate
>>> chocolate()
sweets 'cake'
>>> more_chocolate, more_cake = chocolate(), cake
>>> more_chocolate
>>> def snake(x, y): ... if cake == more_cake: ... return chocolate ... else: ... return x + y >>> snake(10, 20)
>>> snake(10, 20)()
>>> cake = 'cake' >>> snake(10, 20)

Q3: WWPD: Lambda

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

python3 ok -q lambda -u

As a reminder, the following two lines of code will not display any output in the interactive Python interpreter when executed:
>>> x = None
>>> x
>>> lambda x: x  # A lambda expression with one parameter x
<function <lambda> at ...>
>>> a = lambda x: x # Assigning the lambda function to the name a >>> a(5)
>>> (lambda: 3)() # Using a lambda expression as an operator in a call exp.
>>> b = lambda x, y: lambda: x + y # Lambdas can return other lambdas! >>> c = b(8, 4) >>> c
<function <lambda> at ...
>>> c()
>>> d = lambda f: f(4) # They can have functions as arguments as well. >>> def square(x): ... return x * x >>> d(square)
>>> higher_order_lambda = lambda f: lambda x: f(x)
>>> g = lambda x: x * x
>>> higher_order_lambda(2)(g)  # Which argument belongs to which function call?
>>> higher_order_lambda(g)(2)
>>> call_thrice = lambda f: lambda x: f(f(f(x))) >>> call_thrice(lambda y: y + 1)(0)
>>> print_lambda = lambda z: print(z) # When is the return expression of a lambda expression executed? >>> print_lambda
>>> one_thousand = print_lambda(1000)
>>> one_thousand # What did the call to print_lambda return?
# print_lambda returned None, so nothing gets displayed

Coding Practice

Q4: Composite Identity Function

Write a function that takes in two single-argument functions, f and g, and returns another function that has a single parameter x. The returned function should return True if f(g(x)) is equal to g(f(x)) and False otherwise. You can assume the output of g(x) is a valid input for f and vice versa.

def composite_identity(f, g):
    Return a function with one parameter x that returns True if f(g(x)) is
    equal to g(f(x)). You can assume the result of g(x) is a valid input for f
    and vice versa.

    >>> add_one = lambda x: x + 1        # adds one to x
    >>> square = lambda x: x**2          # squares x [returns x^2]
    >>> b1 = composite_identity(square, add_one)
    >>> b1(0)                            # (0 + 1) ** 2 == 0 ** 2 + 1
    >>> b1(4)                            # (4 + 1) ** 2 != 4 ** 2 + 1
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q composite_identity

Q5: Count Cond

Consider the following implementations of count_fives and count_primes which use the sum_digits and is_prime functions from earlier assignments:

def count_fives(n):
    """Return the number of values i from 1 to n (including n)
    where sum_digits(n * i) is 5.
    >>> count_fives(10)  # Among 10, 20, 30, ..., 100, only 50 (10 * 5) has digit sum 5
    >>> count_fives(50)  # 50 (50 * 1), 500 (50 * 10), 1400 (50 * 28), 2300 (50 * 46)
    i = 1
    count = 0
    while i <= n:
        if sum_digits(n * i) == 5:
            count += 1
        i += 1
    return count

def count_primes(n):
    """Return the number of prime numbers up to and including n.
    >>> count_primes(6)   # 2, 3, 5
    >>> count_primes(13)  # 2, 3, 5, 7, 11, 13
    i = 1
    count = 0
    while i <= n:
        if is_prime(i):
            count += 1
        i += 1
    return count

The implementations look quite similar! Generalize this logic by writing a function count_cond, which takes in a two-argument predicate function condition(n, i). count_cond returns a one-argument function that takes in n, which counts all the numbers from 1 to n that satisfy condition when called.

Note: When we say condition is a predicate function, we mean that it is a function that will return True or False.

def sum_digits(y):
    """Return the sum of the digits of non-negative integer y."""
    total = 0
    while y > 0:
        total, y = total + y % 10, y // 10
    return total

def is_prime(n):
    """Return whether positive integer n is prime."""
    if n == 1:
        return False
    k = 2
    while k < n:
        if n % k == 0:
            return False
        k += 1
    return True

def count_cond(condition):
    """Returns a function with one parameter N that counts all the numbers from
    1 to N that satisfy the two-argument predicate function Condition, where
    the first argument for Condition is N and the second argument is the
    number from 1 to N.

    >>> count_fives = count_cond(lambda n, i: sum_digits(n * i) == 5)
    >>> count_fives(10)   # 50 (10 * 5)
    >>> count_fives(50)   # 50 (50 * 1), 500 (50 * 10), 1400 (50 * 28), 2300 (50 * 46)

    >>> is_i_prime = lambda n, i: is_prime(i) # need to pass 2-argument function into count_cond
    >>> count_primes = count_cond(is_i_prime)
    >>> count_primes(2)    # 2
    >>> count_primes(3)    # 2, 3
    >>> count_primes(4)    # 2, 3
    >>> count_primes(5)    # 2, 3, 5
    >>> count_primes(20)   # 2, 3, 5, 7, 11, 13, 17, 19
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q count_cond

Check Your Score Locally

You can locally check your score on each question of this assignment by running

python3 ok --score

This does NOT submit the assignment! When you are satisfied with your score, submit the assignment to Gradescope to receive credit for it.

Submit Assignment

Submit this assignment by uploading any files you've edited to the appropriate Gradescope assignment. Lab 00 has detailed instructions.

In addition, all students who are not in the mega lab must submit the attendance form for credit. Ask your section TA for the link! Submit this form for each section, whether you attended lab or missed it for a good reason. The attendance form is not required for mega section students.

Environment Diagram Practice

There is no Gradescope submission for this component.

However, we still encourage you to do this problem on paper to develop familiarity with Environment Diagrams, which might appear in an alternate form on the exam. To check your work, you can try putting the code into PythonTutor.

Q6: HOF Diagram Practice

Draw the environment diagram that results from executing the code below on paper or a whiteboard. Use to check your work.

n = 7

def f(x):
    n = 8
    return x + 1

def g(x):
    n = 9
    def h():
        return x + 1
    return h

def f(f, x):
    return f(x + n)

f = f(g, n)
g = (lambda y: y())(f)

Optional Questions

These questions are optional. If you don't complete them, you will still receive credit for lab. They are great practice, so do them anyway!

Q7: Multiple

Write a function that takes in two numbers and returns the smallest number that is a multiple of both.

def multiple(a, b):
    """Return the smallest number n that is a multiple of both a and b.

    >>> multiple(3, 4)
    >>> multiple(14, 21)
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q multiple

Q8: I Heard You Liked Functions...

Define a function cycle that takes in three functions f1, f2, and f3, as arguments. cycle will return another function g that should take in an integer argument n and return another function h. That final function h should take in an argument x and cycle through applying f1, f2, and f3 to x, depending on what n was. Here's what the final function h should do to x for a few values of n:

  • n = 0, return x
  • n = 1, apply f1 to x, or return f1(x)
  • n = 2, apply f1 to x and then f2 to the result of that, or return f2(f1(x))
  • n = 3, apply f1 to x, f2 to the result of applying f1, and then f3 to the result of applying f2, or f3(f2(f1(x)))
  • n = 4, start the cycle again applying f1, then f2, then f3, then f1 again, or f1(f3(f2(f1(x))))
  • And so forth.

Hint: most of the work goes inside the most nested function.

def cycle(f1, f2, f3):
    """Returns a function that is itself a higher-order function.

    >>> def add1(x):
    ...     return x + 1
    >>> def times2(x):
    ...     return x * 2
    >>> def add3(x):
    ...     return x + 3
    >>> my_cycle = cycle(add1, times2, add3)
    >>> identity = my_cycle(0)
    >>> identity(5)
    >>> add_one_then_double = my_cycle(2)
    >>> add_one_then_double(1)
    >>> do_all_functions = my_cycle(3)
    >>> do_all_functions(2)
    >>> do_more_than_a_cycle = my_cycle(4)
    >>> do_more_than_a_cycle(2)
    >>> do_two_cycles = my_cycle(6)
    >>> do_two_cycles(1)
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q cycle