Lab 2: HigherOrder Functions, Lambda Expressions
Due by 11:59pm on Wednesday, February 1.
Starter Files
Download lab02.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.
Topics
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 shortcircuit. 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 
Shortcircuiting happens when the operator reaches an operand that allows them to make a conclusion about the expression. For example, and
will shortcircuit 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 shortcircuit, 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
.
HigherOrder 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 higherorder 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)
18
>>> scale(square, 2, 5) # 5 times 2 squared
20
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)
30
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
15
>>> multiply_by(3)(10) # Chain together two call expressions
30
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. 
Example 


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
 Evaluate the expression on the right hand side of the
=
sign.  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
 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.
 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 builtin Python function like
max
or
 Evaluate the operator, whose value should be a function.
 Evaluate the operands left to right.
 Open a new frame. Label it with the sequential frame number, the intrinsic name of the function, and its parent.
 Bind the formal parameters of the function to the arguments whose values you found in step 2.
 Execute the body of the function in the new environment.
Lambdas
Note: As we saw in the
lambda
expression section above,lambda
functions have no intrinsic name. When drawinglambda
functions in environment diagrams, they are labeled with the namelambda
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.
 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.
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 berkeley.edu email.
What Would Python Display?
Important: For all WWPD questions, type
Function
if you believe the answer is<function...>
,Error
if it errors, andNothing
if nothing is displayed.
Q1: WWPD: The True Will Prevail
Use Ok to test your knowledge with the following "What Would Python Display?" questions:
python3 ok q shortcircuitnochaining u
>>> True and 13
______13
>>> False or 0
______0
>>> not 10
______False
>>> not None
______True
>>> True and 1 / 0
______Error (ZeroDivisionError)
>>> True or 1 / 0
______True
>>> 1 and 1 > 0
______True
>>> 1 or 5
______1
>>> (1 + 1) and 1
______1
>>> print(3) or ""
______3
''
>>> def f(x):
... if x == 0:
... return "I am zero!"
... elif x > 0:
... return "Positive!"
... else:
... return ""
>>> 0 or f(1)
______'Positive!'
>>> f(0) or f(1)
______'I am zero!'
>>> f(0) and f(1)
______''
Q2: WWPD: Lambda the Free
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 anything in the 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)
______5
>>> (lambda: 3)() # Using a lambda expression as an operator in a call exp.
______3
>>> b = lambda x, y: lambda: x + y # Lambdas can return other lambdas!
>>> c = b(88, 43)
>>> c
______<function <lambda> at ...
>>> c()
______131
>>> d = lambda f: f(4) # They can have functions as arguments as well.
>>> def square(x):
... return x * x
>>> d(square)
______16
>>> z = 3
>>> e = lambda x: lambda y: lambda: x + y + z
>>> e(0)(1)()
______4
>>> f = lambda z: x + z
>>> f(3)
______NameError: name 'x' is not defined
>>> x = None # remember to review the rules of WWPD given above!
>>> x
______# x evaluates to None, so nothing gets displayed
>>> 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?
______Error
>>> higher_order_lambda(g)(2)
______4
>>> call_thrice = lambda f: lambda x: f(f(f(x)))
>>> call_thrice(lambda y: y + 1)(0)
______3
>>> print_lambda = lambda z: print(z) # When is the return expression of a lambda expression executed?
>>> print_lambda
______Function
>>> one_thousand = print_lambda(1000)
______1000
>>> one_thousand # What did the call to print_lambda return?
______# print_lambda returned None, so nothing gets displayed
Q3: WWPD: Higher Order Functions
Use Ok to test your knowledge with the following "What Would Python Display?" questions:
python3 ok q hofwwpd u
>>> def cake():
... print('beets')
... def pie():
... print('sweets')
... return 'cake'
... return pie
>>> chocolate = cake()
______beets
>>> chocolate
______Function
>>> chocolate()
______sweets
'cake'
>>> more_chocolate, more_cake = chocolate(), cake
______sweets
>>> more_chocolate
______'cake'
>>> def snake(x, y):
... if cake == more_cake:
... return chocolate
... else:
... return x + y
>>> snake(10, 20)
______Function
>>> snake(10, 20)()
______30
>>> cake = 'cake'
>>> snake(10, 20)
______30
Coding Practice
Q4: Composite Identity Function
Write a function that takes in two singleargument 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))
. You can
assume the output of g(x)
is a valid input for f
and vice versa.
Try to use the composer
function defined below for more HOF practice.
def composer(f, g):
"""Return the composition function which given x, computes f(g(x)).
>>> add_one = lambda x: x + 1 # adds one to x
>>> square = lambda x: x**2
>>> a1 = composer(square, add_one) # (x + 1)^2
>>> a1(4)
25
>>> mul_three = lambda x: x * 3 # multiplies 3 to x
>>> a2 = composer(mul_three, a1) # ((x + 1)^2) * 3
>>> a2(4)
75
>>> a2(5)
108
"""
return lambda x: f(g(x))
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
>>> b1 = composite_identity(square, add_one)
>>> b1(0) # (0 + 1)^2 == 0^2 + 1
True
>>> b1(4) # (4 + 1)^2 != 4^2 + 1
False
"""
"*** YOUR CODE HERE ***"
Use Ok to test your code:
python3 ok q composite_identity
Q5: Count van Count
Consider the following implementations of count_factors
and count_primes
:
def count_factors(n):
"""Return the number of positive factors that n has.
>>> count_factors(6)
4 # 1, 2, 3, 6
>>> count_factors(4)
3 # 1, 2, 4
"""
i = 1
count = 0
while i <= n:
if n % i == 0:
count += 1
i += 1
return count
def count_primes(n):
"""Return the number of prime numbers up to and including n.
>>> count_primes(6)
3 # 2, 3, 5
>>> count_primes(13)
6 # 2, 3, 5, 7, 11, 13
"""
i = 1
count = 0
while i <= n:
if is_prime(i):
count += 1
i += 1
return count
def is_prime(n):
return count_factors(n) == 2 # only factors are 1 and n
The implementations look quite similar! Generalize this logic by writing a
function count_cond
, which takes in a twoargument predicate function
condition(n, i)
. count_cond
returns a oneargument 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 returnTrue
orFalse
based on some specified condition in its body.
def count_cond(condition):
"""Returns a function with one parameter N that counts all the numbers from
1 to N that satisfy the twoargument predicate function Condition, where
the first argument for Condition is N and the second argument is the
number from 1 to N.
>>> count_factors = count_cond(lambda n, i: n % i == 0)
>>> count_factors(2) # 1, 2
2
>>> count_factors(4) # 1, 2, 4
3
>>> count_factors(12) # 1, 2, 3, 4, 6, 12
6
>>> is_prime = lambda n, i: count_factors(i) == 2
>>> count_primes = count_cond(is_prime)
>>> count_primes(2) # 2
1
>>> count_primes(3) # 2, 3
2
>>> count_primes(4) # 2, 3
2
>>> count_primes(5) # 2, 3, 5
3
>>> count_primes(20) # 2, 3, 5, 7, 11, 13, 17, 19
8
"""
"*** 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
Make sure to submit this assignment by uploading any files you've edited to the appropriate Gradescope assignment. For a refresher on how to do this, refer to Lab 00.
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.
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, but you must complete them in order to be checked off before the end of the lab period. They are also useful practice!
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)
12
>>> multiple(14, 21)
42
"""
"*** 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
,
f3
, as arguments. cycle
will return another function that should
take in an integer argument n
and return another function. That
final function 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 should do to x
for a few
values of n
:
n = 0
, returnx
n = 1
, applyf1
tox
, or returnf1(x)
n = 2
, applyf1
tox
and thenf2
to the result of that, or returnf2(f1(x))
n = 3
, applyf1
tox
,f2
to the result of applyingf1
, and thenf3
to the result of applyingf2
, orf3(f2(f1(x)))
n = 4
, start the cycle again applyingf1
, thenf2
, thenf3
, thenf1
again, orf1(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 higherorder 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)
5
>>> add_one_then_double = my_cycle(2)
>>> add_one_then_double(1)
4
>>> do_all_functions = my_cycle(3)
>>> do_all_functions(2)
9
>>> do_more_than_a_cycle = my_cycle(4)
>>> do_more_than_a_cycle(2)
10
>>> do_two_cycles = my_cycle(6)
>>> do_two_cycles(1)
19
"""
"*** YOUR CODE HERE ***"
Use Ok to test your code:
python3 ok q cycle