Lab 02: Higher Order Functions

1. Instructions

Please download lab materials from our QQ group if you don't have one.

In this lab, you have three tasks:

Submission: As instructed above, you just need to submit your answer for problems described in section 4 to our OJ website. You may submit more than once before the deadline; only the final submission will be scored. See lab00 for more instructions on submitting assignments.

Readings: You might find the following references to the textbook useful:

2. Review

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

2.1 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 square(3) # 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 square(3)

2.2 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:

2.2.1 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.

Try to paste the following code in the Python tutor and understand each execution step. You can also click this link.

x = 10 y = x x = 20 x, y = y + 1, x - 1

2.2.2 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.

Try to paste the following code in the Python tutor and understand each execution step. You can also click this link.

def f(x): return x + 1 def g(y): return y - 1 def f(z): return x * 2

2.2.3 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.

Try to paste the following code in the Python tutor and understand each execution step. You can also click this link.

def f(a, b, c): return a * (b + c) def g(x): return 3 * x f(1 + 2, g(2), 6)

2.2.4 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.

Try to paste the following code in the Python tutor and understand each execution step. You can also click this link.

lambda x: x * x #no binding created square = lambda x: x * x square(4) #calling a lambda function

3. What Would Python Display?

In this section, you need to think about what python would display if the code below were input to a python interpreter.

You don't have to submit your answers, which means the questions in this section don't count for your final score. However, they are great practice for future assignments, projects, and exams. Attempting these questions is valuable in helping cement your knowledge of course concepts.

To check the correctness of your answer, you can start a python interpreter, input the code into it, and compare the output displayed in the terminal with yours. It is ok for the interpreter to output nothing or raise an error.

Question 1: Lambda the Free

>>> lambda x: x # A lambda expression with one parameter x ______ >>> 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: lambda: x # Lambdas can return other lambdas! >>> c = b(88) >>> c ______ >>> c() ______ >>> d = lambda f: f(4) # They can have functions as arguments as well. >>> def square(x): ... return x * x >>> d(square) ______
>>> x = None # remember to review the rules of WWPD given above! >>> x >>> lambda x: x ______
>>> z = 3 >>> e = lambda x: lambda y: lambda: x + y + z >>> e(0)(1)() ______ >>> f = lambda z: x + z >>> f(3) ______
>>> 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 ______

Question 2: Higher Order Functions

>>> 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 ______ >>> stewart(61) ______ >>> stewart(-4) ______
>>> def cake(): ... print('beets') ... def pie(): ... print('sweets') ... return 'cake' ... return pie >>> chocolate = cake() ______ >>> chocolate ______ >>> chocolate() ______ >>> 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) ______

4. Required Problems

In this section, you are required to complete the problems below and submit your code to Contest lab02 in our OJ website as instructed in lab00 to get your answer scored.

Problem 1: Lambdas and Currying (100pts)

We can transform multiple-argument functions into a chain of single-argument, higher order functions by taking advantage of lambda expressions. This is useful when dealing with functions that take only single-argument functions. We will see some examples of these later on.

Write a function lambda_curry2 that will curry any two argument function using lambdas. See the doctest or refer to Section 1.6.6 in the textbook if you're not sure what this means.

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 to test your understanding of this topic.

def lambda_curry2(func): """ Returns a Curried version of a two-argument 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 """ "*** YOUR CODE HERE ***" return ______

Problem 2: Count van Count (100pts)

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 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.

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_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 ***"

Problem 3: Composite Identity Function (50pts)

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)). You can assume the output of g(x) is a valid input for f and vice versa. You may use the compose1 function defined below.

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 """ "*** YOUR CODE HERE ***"

Problem 4: I Heard You Liked Functions... (150pts)

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:

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) 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 ***"

5. Draw Environment Diagrams

You don't have to submit your answers to these questions but we still encourage you to do them on paper. Similar problems will appear on the exam.

Question 1: 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 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 2: Lambda the Environment Diagram

Try drawing an environment diagram for the following code and predict what Python will output.

>>> a = lambda x: x * 2 + 1 >>> def b(b, x): ... return b(x + a(x)) >>> x = 3 >>> b(a, x) ______