Homework 3:

Due by 11:59pm on Tuesday 10/29


Download hw03.zip. Inside the archive, you will find a file called hw03.py.

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.

Required questions


Q1: Taxicab Distance

An intersection in midtown Manhattan can be identified by an avenue and a street, which are both indexed by positive integers. The Manhattan distance or taxicab distance between two intersections is the number of blocks that must be traversed to reach one from the other, ignoring one-way street restrictions and construction. For example, Times Square is on 46th Street and 7th Avenue. Ess-a-Bagel is on 51st Street and 3rd Avenue. The taxicab distance between them is 9 blocks (5 blocks from 46th to 51st street and 4 blocks from 7th avenue to 3rd avenue). Taxicabs cannot cut diagonally through buildings to reach their destination!

Implement taxicab, which computes the taxicab distance between two intersections using the following data abstraction. Hint: You don't need to know what a Cantor pairing function is; just use the abstraction.

def intersection(st, ave):
    """Represent an intersection using the Cantor pairing function."""
    return (st+ave)*(st+ave+1)//2 + ave

def street(inter):
    return w(inter) - avenue(inter)

def avenue(inter):
    return inter - (w(inter) ** 2 + w(inter)) // 2

w = lambda z: int(((8*z+1)**0.5-1)/2)

def taxicab(a, b):
    """Return the taxicab distance between two intersections.

    >>> times_square = intersection(46, 7)
    >>> ess_a_bagel = intersection(51, 3)
    >>> taxicab(times_square, ess_a_bagel)
    >>> taxicab(ess_a_bagel, times_square)
    "*** YOUR CODE HERE ***"


Q2: Flatten

Write a function flatten that takes a (possibly deep) list and "flattens" it. For example:

>>> lst = [1, [[2], 3], 4, [5, 6]]
>>> flatten(lst)
[1, 2, 3, 4, 5, 6]

Make sure your solution does not mutate the input list.

Hint: you can check if something is a list by using the built-in type function. For example,

>>> type(3) == list
>>> type([1, 2, 3]) == list
def flatten(lst):
    """Returns a flattened version of lst.

    >>> flatten([1, 2, 3])     # normal list
    [1, 2, 3]
    >>> x = [1, [2, 3], 4]      # deep list
    >>> flatten(x)
    [1, 2, 3, 4]
    >>> x # Ensure x is not mutated
    [1, [2, 3], 4]
    >>> x = [[1, [1, 1]], 1, [1, 1]] # deep list
    >>> flatten(x)
    [1, 1, 1, 1, 1, 1]
    >>> x
    [[1, [1, 1]], 1, [1, 1]]
    "*** YOUR CODE HERE ***"


Q3: Replace Leaf

Define replace_leaf, which takes a tree t, a value old, and a value new. replace_leaf returns a new tree that's the same as t except that every leaf value equal to old has been replaced with new.

def replace_leaf(t, old, new):
    """Returns a new tree where every leaf value equal to old has
    been replaced with new.

    >>> yggdrasil = tree('odin',
    ...                  [tree('balder',
    ...                        [tree('thor'),
    ...                         tree('freya')]),
    ...                   tree('frigg',
    ...                        [tree('thor')]),
    ...                   tree('thor',
    ...                        [tree('sif'),
    ...                         tree('thor')]),
    ...                   tree('thor')])
    >>> laerad = copy_tree(yggdrasil) # copy yggdrasil for testing purposes
    >>> print_tree(replace_leaf(yggdrasil, 'thor', 'freya'))
    >>> laerad == yggdrasil # Make sure original tree is unmodified
    "*** YOUR CODE HERE ***"


Acknowledgements. This mobile example is based on a classic problem from Structure and Interpretation of Computer Programs, Section 2.2.2.

Hint: for more information on this problem (with more pictures!) please refer to this document

Mobile example

A mobile is a type of hanging sculpture. A binary mobile consists of two sides. Each side is a rod of a certain length, from which hangs either a weight or another mobile.

Labeled Mobile example

We will represent a binary mobile using the data abstractions below.

  • A mobile has a left side and a right side.
  • A side has a positive length and something hanging at the end, either a mobile or weight.
  • A weight has a positive size.

Q4: Weights

Implement the weight data abstraction by completing the weight constructor and the size selector so that a weight is represented using a two-element list where the first element is the string 'weight'. The total_weight example is provided to demonstrate use of the mobile, side, and weight abstractions.

def mobile(left, right):
    """Construct a mobile from a left side and a right side."""
    assert is_side(left), "left must be a side"
    assert is_side(right), "right must be a side"
    return ['mobile', left, right]

def is_mobile(m):
    """Return whether m is a mobile."""
    return type(m) == list and len(m) == 3 and m[0] == 'mobile'

def left(m):
    """Select the left side of a mobile."""
    assert is_mobile(m), "must call left on a mobile"
    return m[1]

def right(m):
    """Select the right side of a mobile."""
    assert is_mobile(m), "must call right on a mobile"
    return m[2]
def side(length, mobile_or_weight):
    """Construct a side: a length of rod with a mobile or weight at the end."""
    assert is_mobile(mobile_or_weight) or is_weight(mobile_or_weight)
    return ['side', length, mobile_or_weight]

def is_side(s):
    """Return whether s is a side."""
    return type(s) == list and len(s) == 3 and s[0] == 'side'

def length(s):
    """Select the length of a side."""
    assert is_side(s), "must call length on a side"
    return s[1]

def end(s):
    """Select the mobile or weight hanging at the end of a side."""
    assert is_side(s), "must call end on a side"
    return s[2]
def weight(size):
    """Construct a weight of some size."""
    assert size > 0
    "*** YOUR CODE HERE ***"

def size(w):
    """Select the size of a weight."""
    assert is_weight(w), 'must call size on a weight'
    "*** YOUR CODE HERE ***"

def is_weight(w):
    """Whether w is a weight."""
    return type(w) == list and len(w) == 2 and w[0] == 'weight'

Q5: Balanced

Hint: for more information on this problem (with more pictures!) please refer to this document

Implement the balanced function, which returns whether m is a balanced mobile. A mobile is balanced if two conditions are met:

  1. The torque applied by its left side is equal to that applied by its right side. Torque of the left side is the length of the left rod multiplied by the total weight hanging from that rod. Likewise for the right.
  2. Each of the mobiles hanging at the end of its sides is balanced.

Hint: You may find it helpful to assume that weights themselves are balanced.

def balanced(m):
    """Return whether m is balanced.

    >>> t, u, v = examples()
    >>> balanced(t)
    >>> balanced(v)
    >>> w = mobile(side(3, t), side(2, u))
    >>> balanced(w)
    >>> balanced(mobile(side(1, v), side(1, w)))
    >>> balanced(mobile(side(1, w), side(1, v)))
    "*** YOUR CODE HERE ***"

Q6: Totals

Implement totals_tree, which takes a mobile (or weight) and returns a tree whose root is its total weight and whose branches are trees for the ends of the sides.

def totals_tree(m):
    """Return a tree representing the mobile with its total weight at the root.

    >>> t, u, v = examples()
    >>> print_tree(totals_tree(t))
    >>> print_tree(totals_tree(u))
    >>> print_tree(totals_tree(v))
    "*** YOUR CODE HERE ***"

Just for fun Question

This question is out of scope for 61a. Do it if you want an extra challenge or some practice with HOF and abstraction!

Q7: Church numerals

The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. The purpose was to show that functions are sufficient to describe all of number theory: if we have functions, we do not need to assume that numbers exist, but instead we can invent them.

Your goal in this problem is to rediscover this representation known as Church numerals. Here are the definitions of zero, as well as a function that returns one more than its argument:

def zero(f):
    return lambda x: x

def successor(n):
    return lambda f: lambda x: f(n(f)(x))

First, define functions one and two such that they have the same behavior as successor(zero) and successsor(successor(zero)) respectively, but do not call successor in your implementation.

Next, implement a function church_to_int that converts a church numeral argument to a regular Python integer.

Finally, implement functions add_church, mul_church, and pow_church that perform addition, multiplication, and exponentiation on church numerals.

def one(f):
    """Church numeral 1: same as successor(zero)"""
    "*** YOUR CODE HERE ***"

def two(f):
    """Church numeral 2: same as successor(successor(zero))"""
    "*** YOUR CODE HERE ***"

three = successor(two)

def church_to_int(n):
    """Convert the Church numeral n to a Python integer.

    >>> church_to_int(zero)
    >>> church_to_int(one)
    >>> church_to_int(two)
    >>> church_to_int(three)
    "*** YOUR CODE HERE ***"

def add_church(m, n):
    """Return the Church numeral for m + n, for Church numerals m and n.

    >>> church_to_int(add_church(two, three))
    "*** YOUR CODE HERE ***"

def mul_church(m, n):
    """Return the Church numeral for m * n, for Church numerals m and n.

    >>> four = successor(three)
    >>> church_to_int(mul_church(two, three))
    >>> church_to_int(mul_church(three, four))
    "*** YOUR CODE HERE ***"

def pow_church(m, n):
    """Return the Church numeral m ** n, for Church numerals m and n.

    >>> church_to_int(pow_church(two, three))
    >>> church_to_int(pow_church(three, two))
    "*** YOUR CODE HERE ***"