Lambda

Lambda expressions are one-line 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
lambda x: x * x
           
def square(x):
    return x * x

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 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.

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, and Nothing 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

Question 2: WWPD: Higher Order Functions

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

python3 ok -q hof -u

For all WWPD questions, type Function if you believe the answer is <function...>, Error if it errors, and Nothing 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 lambda: x + y ... else: ... return x + y >>> snake(10, 20)
______
Function
>>> snake(10, 20)()
______
30
>>> cake = 'cake' >>> snake(10, 20)
______
30

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 3: 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:

  1. 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?
  2. In frame f2, what name is the frame labeled with (add_ten or λ)? Which frame is the parent of f2?
  3. 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.

  1. The intrinsic name of the function object that add_ten points to is λ (specifically, the lambda whose parameter is k). The parent frame of this lambda is f1.
  2. f2 is labeled with the name λ the parent frame of f2 is f1, since that is where λ is defined.
  3. The variable result is bound to 19.

Question 4: 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

Lambda

Question 5: Compose

Write a function that takes in 2 single-argument 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 single-argument 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 6: 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

Higher Order Functions

Question 7: Funception

Write a function (funception) that takes in another function func_a and a number start and returns a function (func_b) that will have one parameter to take in the stop value. func_b should take the following into consideration the following in order:

  1. Takes in the stop value.
  2. If the value of start is less than 0, it should exit the function.
  3. If the value of start is greater than stop, apply func_a on start and return the result.
  4. If not, apply func_a on all the numbers from start (inclusive) up to stop (exclusive) and return the product.
def funception(func_a, start):
    """ Takes in a function (function A) and a start value.
    Returns a function (function B) that will find the product of 
    function A applied to the range of numbers from 
    start (inclusive) to stop (exclusive)

    >>> def func_a(num):
    ...     return num + 1
    >>> func_b1 = funception(func_a, 3)
    >>> func_b1(2)
    4
    >>> func_b2 = funception(func_a, -2)
    >>> func_b2(-3)
    >>> func_b3 = funception(func_a, -1)
    >>> func_b3(4)
    >>> func_b4 = funception(func_a, 0)
    >>> func_b4(3)
    6
    >>> func_b5 = funception(func_a, 1)
    >>> func_b5(4)
    24
    """
"*** YOUR CODE HERE ***"
def func_b(stop): i = start product = 1 if start < 0: return None if start > stop: return func_a(start) while i < stop: product *= func_a(i) i += 1 return product return func_b

Use OK to test your code:

python3 ok -q funception