Lab 2 Solutions
Solution Files
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.
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
What Would Python Display?
Q1: WWPD: Lambda the Free
Use Ok to test your knowledge with the following "What Would Python Display?" questions:
python3 ok q lambda u
For all WWPD questions, type
Function
if you believe the answer is<function...>
,Error
if it errors, andNothing
if nothing is displayed. 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: lambda: x # Lambdas can return other lambdas!
>>> c = b(88)
>>> c
______<function <lambda> at ...
>>> c()
______88
>>> d = lambda f: f(4) # They can have functions as arguments as well.
>>> def square(x):
... return x * x
>>> d(square)
______16
>>> x = None # remember to review the rules of WWPD given above!
>>> x
>>> lambda x: x
______Function
>>> 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
>>> 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
______# print_lambda returned None, so nothing gets displayed
Q2: WWPD: Higher Order Functions
Use Ok to test your knowledge with the following "What Would Python Display?" questions:
python3 ok q hofwwpd u
For all WWPD questions, type
Function
if you believe the answer is<function...>
,Error
if it errors, andNothing
if nothing is displayed.
>>> def even(f):
... def odd(x):
... if x < 0:
... return f(x)
... return f(x)
... return odd
>>> steven = lambda x: x
>>> stewart = even(steven)
>>> stewart
______<function ...>
>>> stewart(61)
______61
>>> stewart(4)
______4
>>> 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
Q3: Lambdas and Currying
We can transform multipleargument functions into a chain of
singleargument, higher order functions by taking advantage of lambda
expressions. For example, we can write a function f(x, y)
as a different function g(x)(y)
. This is known as currying. It's useful when dealing with functions that take only singleargument functions. We will see some examples of these later on.
Write a function lambda_curry2
that will curry any two argument function using lambdas. Refer to the
textbook for more details about currying.
Your solution to this problem should fit entirely on the return line. You can try writing it first without this restriction, but rewrite it after in one line.
def lambda_curry2(func):
"""
Returns a Curried version of a twoargument function FUNC.
>>> from operator import add, mul, mod
>>> curried_add = lambda_curry2(add)
>>> add_three = curried_add(3)
>>> add_three(5)
8
>>> curried_mul = lambda_curry2(mul)
>>> mul_5 = curried_mul(5)
>>> mul_5(42)
210
>>> lambda_curry2(mod)(123)(10)
3
"""
return lambda arg1: lambda arg2: func(arg1, arg2)
Use Ok to test your code:
python3 ok q lambda_curry2
Q4: 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, count = 1, 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, count = 1, 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.
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
"""
def counter(n):
i, count = 1, 0
while i <= n:
if condition(n, i):
count += 1
i += 1
return count
return counter
Use Ok to test your code:
python3 ok q count_cond
Environment Diagram Practice
There is no submission for this component. However, we still encourage you to do these problems on paper to develop familiarity with Environment Diagrams, which might appear in an alternate form on the exam.
Q5: Make Adder
Draw the environment diagram for the following code:
n = 9
def make_adder(n):
return lambda k: k + n
add_ten = make_adder(n+1)
result = add_ten(n)
There are 3 frames total (including the Global frame). In addition, consider the following questions:
 In the Global frame, the name
add_ten
points to a function object. What is the intrinsic name of that function object, and what frame is its parent?  What name is frame
f2
labeled with (add_ten
or λ)? Which frame is the parent off2
?  What value is the variable
result
bound to in the Global frame?
You can try out the environment diagram at tutor.cs61a.org. To see the environment diagram for this question, click here.
 The intrinsic name of the function object that
add_ten
points to is λ (specifically, the lambda whose parameter isk
). The parent frame of this lambda isf1
. f2
is labeled with the name λ the parent frame off2
isf1
, since that is where λ is defined. The variable
result
is bound to 19.
Q6: Lambda the Environment Diagram
Try drawing an environment diagram for the following code and predict what Python will output.
You do not need to submit or unlock this question through Ok. Instead, you can check your work with the Online Python Tutor, but try drawing it yourself first!
>>> a = lambda x: x * 2 + 1
>>> def b(b, x):
... return b(x + a(x))
>>> x = 3
>>> b(a, x)
______21 # Interactive solution: https://goo.gl/Lu99QR
Submit
Make sure to submit this assignment by running:
python3 ok submit
Optional Questions
Q7: 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 compose1
function defined below for more HOF practice.
def compose1(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 = compose1(square, add_one) # (x + 1)^2
>>> a1(4)
25
>>> mul_three = lambda x: x * 3 # multiplies 3 to x
>>> a2 = compose1(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
"""
def identity(x):
return compose1(f, g)(x) == compose1(g, f)(x)
return identity
# Alternative solution
return lambda x: f(g(x)) == g(f(x))
Use Ok to test your code:
python3 ok q composite_identity
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
"""
def ret_fn(n):
def ret(x):
i = 0
while i < n:
if i % 3 == 0:
x = f1(x)
elif i % 3 == 1:
x = f2(x)
else:
x = f3(x)
i += 1
return x
return ret
return ret_fn
Use Ok to test your code:
python3 ok q cycle