Lab 4: Lambdas and Environments
Due at 11:59:59 pm on 9/28/2020.
Starter Files
Download lab04.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the OK autograder.
Submission
By the end of this lab, you should have submitted the lab with python3 ok submit
. You may submit more than once before the deadline; only the final submission will be graded. Check that you have successfully submitted your code on okpy.org. See this article for more instructions on okpy and submitting assignments.
 To receive full credit for this lab, all questions must be attempted.
When you are ready to submit, run ok
with the submit
option:
python3 ok submit
After submitting, ok
will display a submission URL, with which you can view your submission on okpy.org.
Lambda
Lambda
expressions are oneline functions that specify two things:
the parameters and the return value.
lambda <parameters>: <return value>
While both lambda
and def
statements are related to functions, there are some differences.
lambda  def  

Type  lambda is an expression 
def is a statement 
Description  Evaluating a lambda expression does not create or modify any variables.
Lambda expressions just create new function objects. 
Executing a def statement will create a new function object and bind it to a variable in the current environment. 
Example 


A lambda
expression by itself is not very interesting. As with any objects such as numbers, booleans, strings, we usually:
 assign lambda to variables (
foo = lambda x: x
)  pass them in to other functions (
bar(lambda x: x)
)  return them as the results of other functions (
return lambda x: x
)  return them as the results of other lambdas (
lambda x: lambda y: x + y
)
In the final example above, the outer lambda (lambda x
) takes in a value x
, and it
returns another lambda (lambda y
) that takes an argument y
and returns x+y
.
Environment Diagrams
Environment diagrams are one of the best learning tools for understanding
lambda
expressions 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:
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.
WWPD
Question 1: 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
>>> 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
HOF Review
Question 2: Polynomial
A polynomial function is a function with coefficients, variables and constants. A polynomial function is said to be the nth degree polynomial if there is a term in the function with the variable to the nth degree. For example, a 4th degree polynomial must contain the term x^4 with some coefficient multiplied to it.
Complete the function polynomial
, which takes in a degree and a list of coefficients. The function should output the corresponding polynomial function.
Hint: the staff solutions is one line and uses lambda + a list comprehension.
def polynomial(degree, coeffs):
"""
>>> fourth = polynomial(4, [3,6,2,1, 100])
>>> fourth(3) # 3*(3**4) + 6*(3**3) + 2*(3**2) + 1*(3**1) + 100
526
>>> third = polynomial(3, [2, 0, 0, 0])
>>> third(4) # 2*(4**3) + 0*(4**2) + 0*(4**1) + 0
128
"""
"*** YOUR CODE HERE ***"
# Option 1
return lambda x: sum([coeffs[i]*(x ** (degree  i)) for i in range(degree + 1)])
# Option 2
def poly_func(x):
return sum([coeffs[i]*(x ** (degree  i)) for i in range(degree + 1)])
return poly_func
Use OK to test your code:
python3 ok q polynomial
Lambda
Question 3: Compose
Write a function that takes in 2 singleargument functions, f and g, and returns another lambda function that takes in a single argument x. The returned function should return the output of applying f(g(x)).
Hint: The staff solution is only 1 line!
def compose(f, g):
"""Write a function that takes in 2 singleargument functions, f and g, and returns another lambda function
that takes in a single argument x. The returned function should return the output of applying f(g(x)).
Hint: The staff solution is only 1 line!
Return the composition function which given x, computes f(g(x)).
>>> add_two = lambda x: x + 2 # adds 2 to x
>>> square = lambda x: x ** 2 # squares x
>>> a = compose(square, add_two) # (x + 2 ) ^ 2
>>> a(5)
49
>>> mul_ten = lambda x: x * 10 # multiplies 10 with x
>>> b = compose(mul_ten, a) # ((x + 2 ) ^ 2) * 10
>>> b(5)
490
>>> b(2)
160
"""
"*** YOUR CODE HERE ***"
return lambda x: f(g(x))
Use OK to test your code:
python3 ok q compose
Question 4: Mul_by_num
Using a lambda
expression, complete the mul_by_num
function. This
function should take an argument and return a one argument function
that multiplies any value passed to it by the original number. Its
body must be one line long:
def mul_by_num(num):
"""
Returns a function that takes one argument and returns num
times that argument.
>>> x = mul_by_num(5)
>>> y = mul_by_num(2)
>>> x(3)
15
>>> y(4)
8
"""
"*** YOUR CODE HERE ***"
return lambda num2: num * num2
Use OK to test your code:
python3 ok q mul_by_num
Question 5: Higher Order Lambdas
Return a lambda function that takes in a multiplier and returns a lambda function that given an input will return the input multiplied by the multiplier.
def higher_order_lambdas():
"""
Return a lambda function that takes in a multiplier and returns a lambda function that given an input will
return the input multiplied by the multiplier
>>> hol = higher_order_lambdas()
>>> doubles = hol(2)
>>> doubles(3)
6
>>> hol = higher_order_lambdas()
>>> triples = hol(3)
>>> triples(4)
12
"""
"*** YOUR CODE HERE ***"
return lambda m : lambda n : m * n
Use OK to test your code:
python3 ok q higher_order_lambdas
Submit
Make sure to submit this assignment by running:
python3 ok submit
Optional: 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 will appear on the exam.
Question 6: 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?  In frame
f2
, what name is the frame 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.
Question 7: 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