Homework 2: Higher Order Functions

Due by 11:59pm on Sunday, 10/20


Download hw02.zip. Inside the archive, you will find files hw02.py and construct_check.py.

Submission: Email hw02.py to TA.

Readings: You might find the following references useful:

Grading: Homework is graded based on effort, not correctness. However, there is no partial credit; you must show substantial effort on every problem to receive any points.

The construct_check module is used in this assignment, which defines a function check. For example, a call such as

check("foo.py", "func1", ["While", "For", "Recursion"])

checks that the function func1 in file foo.py does not contain any while or for constructs, and is not an overtly recursive function (i.e., one in which a function contains a call to itself by name.)

Required questions

Several doctests refer to these functions:

from operator import add, mul, sub

square = lambda x: x * x

identity = lambda x: x

triple = lambda x: 3 * x

increment = lambda x: x + 1

Q1: Product

The summation(n, term) function from the higher-order functions lecture adds up term(1) + ... + term(n). Write a similar function called product that returns term(1) * ... * term(n). Do not use recursion.

def product(n, term):
    """Return the product of the first n terms in a sequence.
    n    -- a positive integer
    term -- a function that takes one argument

    >>> product(3, identity)  # 1 * 2 * 3
    >>> product(5, identity)  # 1 * 2 * 3 * 4 * 5
    >>> product(3, square)    # 1^2 * 2^2 * 3^2
    >>> product(5, square)    # 1^2 * 2^2 * 3^2 * 4^2 * 5^2
    >>> product(3, increment) # (1+1) * (2+1) * (3+1)
    >>> product(3, triple)    # 1*3 * 2*3 * 3*3
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'product', ['Recursion'])
    "*** YOUR CODE HERE ***"

Now, define the factorial function in terms of product in one line.

def factorial(n):
    """Return n factorial for n >= 0 by calling product.

    >>> factorial(4)  # 4 * 3 * 2 * 1
    >>> factorial(6)  # 6 * 5 * 4 * 3 * 2 * 1
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'factorial', ['Recursion', 'For', 'While'])
    "*** YOUR CODE HERE ***"

Q2: Accumulate

Let's take a look at how summation and product are instances of a more general function called accumulate:

def accumulate(combiner, base, n, term):
    """Return the result of combining the first n terms in a sequence and base.
    The terms to be combined are term(1), term(2), ..., term(n).  combiner is a
    two-argument commutative, associative function.

    >>> accumulate(add, 0, 5, identity)  # 0 + 1 + 2 + 3 + 4 + 5
    >>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5
    >>> accumulate(add, 11, 0, identity) # 11
    >>> accumulate(add, 11, 3, square)   # 11 + 1^2 + 2^2 + 3^2
    >>> accumulate(mul, 2, 3, square)    # 2 * 1^2 * 2^2 * 3^2
    >>> accumulate(lambda x, y: x + y + 1, 2, 3, square)
    "*** YOUR CODE HERE ***"

accumulate has the following parameters:

  • term and n: the same parameters as in summation and product
  • combiner: a two-argument function that specifies how the current term is combined with the previously accumulated terms.
  • base: value at which to start the accumulation.

For example, the result of accumulate(add, 11, 3, square) is

11 + square(1) + square(2) + square(3) = 25

Note: You may assume that combiner is associative and commutative. That is, combiner(a, combiner(b, c)) == combiner(combiner(a, b), c) and combiner(a, b) == combiner(b, a) for all a, b, and c. However, you may not assume combiner is chosen from a fixed function set and hard-code the solution.

After implementing accumulate, show how summation and product can both be defined as simple calls to accumulate:

def summation_using_accumulate(n, term):
    """Returns the sum of term(1) + ... + term(n). The implementation
    uses accumulate.

    >>> summation_using_accumulate(5, square)
    >>> summation_using_accumulate(5, triple)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'summation_using_accumulate',
    ...       ['Recursion', 'For', 'While'])
    "*** YOUR CODE HERE ***"

def product_using_accumulate(n, term):
    """An implementation of product using accumulate.

    >>> product_using_accumulate(4, square)
    >>> product_using_accumulate(6, triple)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'product_using_accumulate',
    ...       ['Recursion', 'For', 'While'])
    "*** YOUR CODE HERE ***"

Q3: Make Repeater

Implement a function make_repeater so that make_repeater(f, n)(x) returns f(f(...f(x)...)), where f is applied n times. That is, make_repeater(f, n) returns another function that can then be applied to another argument. For example, make_repeater(square, 3)(42) evaluates to square(square(square(42))). See if you can figure out a reasonable function to return for that case. You may use either loops or recursion in your implementation.

def make_repeater(f, n):
    """Return the function that computes the nth application of f.

    >>> add_three = make_repeater(increment, 3)
    >>> add_three(5)
    >>> make_repeater(triple, 5)(1) # 3 * 3 * 3 * 3 * 3 * 1
    >>> make_repeater(square, 2)(5) # square(square(5))
    >>> make_repeater(square, 4)(5) # square(square(square(square(5))))
    >>> make_repeater(square, 0)(5) # Yes, it makes sense to apply the function zero times!
    "*** YOUR CODE HERE ***"

For an extra challenge, try defining make_repeater using compose1 and your accumulate function in a single one-line return statement.

def compose1(f, g):
    """Return a function h, such that h(x) = f(g(x))."""
    def h(x):
        return f(g(x))
    return h

Q4: Num Sevens

Write a recursive function num_sevens that takes a positive integer n and returns the number of times the digit 7 appears in n.
Use recursion - the tests will fail if you use any assignment statements.

def num_sevens(n):
    """Returns the number of times 7 appears as a digit of n.

    >>> num_sevens(3)
    >>> num_sevens(7)
    >>> num_sevens(7777777)
    >>> num_sevens(2637)
    >>> num_sevens(76370)
    >>> num_sevens(12345)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'num_sevens',
    ...       ['Assign', 'AugAssign'])
    "*** YOUR CODE HERE ***"

Q5: Ping-pong

The ping-pong sequence counts up starting from 1 and is always either counting up or counting down. At the k-th element the direction switches if k is a multiple of 7 or contains the digit 7. The first 30 elements of the ping-pong sequence are listed below, with direction swaps marked using brackets at the 7th, 14th, 17th, 21st, 27th, and 28th elements:

1 2 3 4 5 6 [7] 6 5 4 3 2 1 [0] 1 2 [3] 2 1 0 [-1] 0 1 2 3 4 [5] [4] 5 6

Implement a function pingpong that returns the nth element of the ping-pong sequence without using any assignment statements.

You may use the function num_sevens, which you defined in the previous question.

Hint: If you're stuck, first try implementing pingpong using assignment statements and a while statement. Then, to convert this into a recursive solution, write a helper function that has a parameter for each variable that changes values in the body of the while loop.

def pingpong(n):
    """Return the nth element of the ping-pong sequence.

    >>> pingpong(7)
    >>> pingpong(8)
    >>> pingpong(15)
    >>> pingpong(21)
    >>> pingpong(22)
    >>> pingpong(30)
    >>> pingpong(68)
    >>> pingpong(69)
    >>> pingpong(70)
    >>> pingpong(71)
    >>> pingpong(72)
    >>> pingpong(100)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'pingpong', ['Assign', 'AugAssign'])
    "*** YOUR CODE HERE ***"

Q6: Count change

Once the machines take over, the denomination of every coin will be a power of two: 1-cent, 2-cent, 4-cent, 8-cent, 16-cent, etc. There will be no limit to how much a coin can be worth.

Given a positive integer amount, a set of coins makes change for amount if the sum of the values of the coins is amount. For example, the following sets make change for 7:

  • 7 1-cent coins
  • 5 1-cent, 1 2-cent coins
  • 3 1-cent, 2 2-cent coins
  • 3 1-cent, 1 4-cent coins
  • 1 1-cent, 3 2-cent coins
  • 1 1-cent, 1 2-cent, 1 4-cent coins

Thus, there are 6 ways to make change for 7. Write a recursive function count_change that takes a positive integer amount and returns the number of ways to make change for amount using these coins of the future.

Hint: Refer the implementation of count_partitions for an example of how to count the ways to sum up to an amount with smaller parts. If you need to keep track of more than one value across recursive calls, consider writing a helper function.

def count_change(amount):
    """Return the number of ways to make change for amount.

    >>> count_change(7)
    >>> count_change(10)
    >>> count_change(20)
    >>> count_change(100)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'count_change', ['While', 'For'])
    "*** YOUR CODE HERE ***"

Extra questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively! Feel free to skip them, but you may attempt them if you would like extra practice.

Q7: Towers of Hanoi

A classic puzzle called the Towers of Hanoi is a game that consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with n disks in a neat stack in ascending order of size on a start rod, the smallest at the top, forming a conical shape.

Towers of Hanoi

The objective of the puzzle is to move the entire stack to an end rod, obeying the following rules:

  • Only one disk may be moved at a time.
  • Each move consists of taking the top (smallest) disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
  • No disk may be placed on top of a smaller disk.

Complete the definition of move_stack, which prints out the steps required to move n disks from the start rod to the end rod without violating the rules. The provided print_move function will print out the step to move a single disk from the given origin to the given destination.

Hint: Draw out a few games with various n on a piece of paper and try to find a pattern of disk movements that applies to any n. In your solution, take the recursive leap of faith whenever you need to move any amount of disks less than n from one rod to another. If you need more help, see the following hints.

See the following animation of the Towers of Hanoi, found on Wikimedia by user Trixx

The strategy used in Towers of Hanoi is to move all but the bottom disc to the second peg, then moving the bottom disc to the third peg, then moving all but the second disc from the second to the third peg.

One thing you don't need to worry about is collecting all the steps. print effectively "collects" all the results in the terminal as long as you make sure that the moves are printed in order.
def print_move(origin, destination):
    """Print instructions to move a disk."""
    print("Move the top disk from rod", origin, "to rod", destination)

def move_stack(n, start, end):
    """Print the moves required to move n disks on the start pole to the end
    pole without violating the rules of Towers of Hanoi.

    n -- number of disks
    start -- a pole position, either 1, 2, or 3
    end -- a pole position, either 1, 2, or 3

    There are exactly three poles, and start and end must be different. Assume
    that the start pole has at least n disks of increasing size, and the end
    pole is either empty or has a top disk larger than the top n start disks.

    >>> move_stack(1, 1, 3)
    Move the top disk from rod 1 to rod 3
    >>> move_stack(2, 1, 3)
    Move the top disk from rod 1 to rod 2
    Move the top disk from rod 1 to rod 3
    Move the top disk from rod 2 to rod 3
    >>> move_stack(3, 1, 3)
    Move the top disk from rod 1 to rod 3
    Move the top disk from rod 1 to rod 2
    Move the top disk from rod 3 to rod 2
    Move the top disk from rod 1 to rod 3
    Move the top disk from rod 2 to rod 1
    Move the top disk from rod 2 to rod 3
    Move the top disk from rod 1 to rod 3
    assert 1 <= start <= 3 and 1 <= end <= 3 and start != end, "Bad start/end"
    "*** YOUR CODE HERE ***"

Q8: Anonymous factorial

The recursive factorial function can be written as a single expression by using a conditional expression.

>>> fact = lambda n: 1 if n == 1 else mul(n, fact(sub(n, 1)))
>>> fact(5)

However, this implementation relies on the fact (no pun intended) that fact has a name, to which we refer in the body of fact. To write a recursive function, we have always given it a name using a def or assignment statement so that we can refer to the function within its own body. In this question, your job is to define fact recursively without giving it a name!

Write an expression that computes n factorial using only call expressions, conditional expressions, and lambda expressions (no assignment or def statements). Note in particular that you are not allowed to use make_anonymous_factorial in your return expression. The sub and mul functions from the operator module are the only built-in functions required to solve this problem:

from operator import sub, mul

def make_anonymous_factorial():
    """Return the value of an expression that computes factorial.

    >>> make_anonymous_factorial()(5)
    >>> from construct_check import check
    >>> check(HW_SOURCE_FILE, 'make_anonymous_factorial', ['Assign', 'AugAssign', 'FunctionDef', 'Recursion'])