# 61A Homework 10

Due by 11:59pm on Tuesday, 11/26

Submission. See the online submission instructions. We have provided a hw10 starter file for the questions below.

Readings. Section 4.2 of Composing Programs.

Q1. (Objects and recursion review) A mobile is a type of hanging sculpture. A simple binary mobile consists of two branches, left and right. Each branch is a rod of a certain length, from which hangs either a weight or another mobile.

Improve the classes for Branch, Weight, and Mobile below in the following ways:

• The left and right attributes of a Mobile should both be Branch instances. Check that the types of constructor arguments for Mobile are Branch instances, and raise an appropriate TypeError for incorrect argument types. See the doctest for Mobile for exception details.
• The length of a Branch and the weight of a Weight should be positive numbers. Implement check_positive to check if an object x is a positive number.
• Add a property weight that gives the total weight of the mobile.
• A mobile is said to be balanced if the torque applied by its left branch is equal to that applied by its right branch (that is, if the length of the left rod multiplied by the weight hanging from that rod is equal to the corresponding product for the right side) and if each of the submobiles hanging off its branches is balanced. Add a property method is_balanced that returns True if and only if the Mobile is balanced.

When you are finished, all doctests below should pass:

```class Mobile:
"""A simple binary mobile that has branches of weights or other mobiles.

>>> Mobile(1, 2)
Traceback (most recent call last):
...
TypeError: 1 is not a Branch
>>> m = Mobile(Branch(1, Weight(2)), Branch(2, Weight(1)))
>>> m.weight
3
>>> m.is_balanced()
True
>>> m.left.contents = Mobile(Branch(1, Weight(1)), Branch(2, Weight(1)))
>>> m.weight
3
>>> m.left.contents.is_balanced()
False
>>> m.is_balanced() # All submobiles must be balanced for m to be balanced
False
>>> m.left.contents.right.contents.weight = 0.5
>>> m.left.contents.is_balanced()
True
>>> m.is_balanced()
False
>>> m.right.length = 1.5
>>> m.is_balanced()
True
"""

def __init__(self, left, right):
"*** YOUR CODE HERE ***"
self.left = left
self.right = right

@property
def weight(self):
"""The total weight of the mobile."""
"*** YOUR CODE HERE ***"

def is_balanced(self):
"""True if and only if the mobile is balanced."""
"*** YOUR CODE HERE ***"

def check_positive(x):
"""Check that x is a positive number, and raise an exception otherwise.

>>> check_positive('hello')
Traceback (most recent call last):
...
TypeError: hello is not a number
>>> check_positive('1')
Traceback (most recent call last):
...
TypeError: 1 is not a number
>>> check_positive(-2)
Traceback (most recent call last):
...
ValueError: -2 <= 0
"""
"*** YOUR CODE HERE ***"

class Branch:
"""A branch of a simple binary mobile."""

def __init__(self, length, contents):
if type(contents) not in (Weight, Mobile):
raise TypeError(str(contents) + ' is not a Weight or Mobile')
check_positive(length)
self.length = length
self.contents = contents

@property
def torque(self):
"""The torque on the branch"""
return self.length * self.contents.weight

class Weight:
"""A weight."""
def __init__(self, weight):
check_positive(weight)
self.weight = weight

def is_balanced(self):
return True
```

Q2. (Python evaluation and recursion review) Your partner designed a beautiful balanced Mobile, but forgot to fill in the classes of each part, instead just writing T.

T(T(4,T(T(4,T(1)),T(1,T(4)))),T(2,T(10)))

The built-in Python funciton eval takes a string argument, evaluates it as a Python expression, and returns its value.

Complete the definition of interpret_mobile so that it returns a well-formed mobile by guessing the class for each T. The function should exhaustively test all possible combinations of types, then attempt to eval the resulting string when no T remains, handling TypeErrors until a correct series of types is found.

Warning: Interpreting a large mobile is quite slow (can you say why?). You will want to remove the doctest for the large mobile during development:

```def interpret_mobile(s):
"""Return a Mobile described by string s by substituting one of the classes
Branch, Weight, or Mobile for each occurrenct of the letter T.

>>> simple = 'Mobile(T(2,T(1)), T(1,T(2)))'
>>> interpret_mobile(simple).weight
3
>>> interpret_mobile(simple).is_balanced()
True
>>> s = 'T(T(4,T(T(4,T(1)),T(1,T(4)))),T(2,T(10)))'
>>> m = interpret_mobile(s)
>>> m.weight
15
>>> m.is_balanced()
True
"""
next_T = s.find('T')        # The index of the first 'T' in s.
if next_T == -1:            # The string 'T' was not found in s
try:
return eval(s)      # Interpret s
except TypeError as e:
return None         # Return None if s is not a valid mobile
for t in ('Branch', 'Weight', 'Mobile'):
"*** YOUR CODE HERE ***"
return None
```

Q3. The Stream class from lecture appears below, along with a function that returns an infinite stream of integers. To extend it, implement an __iter__ method using a yield statement that returns a generator over the elements of the stream. Also add a __getitem__ method to support item selection.

The zip function in the doctest for __iter__ combines the corresponding elements of two iterables into pairs until one iterator raises StopIteration:

```class Stream:
"""A lazily computed recursive list."""

class empty:
def __repr__(self):
return 'Stream.empty'
empty = empty()

def __init__(self, first, compute_rest=lambda: Stream.empty):
assert callable(compute_rest), 'compute_rest must be callable.'
self.first = first
self._compute_rest = compute_rest

@property
def rest(self):
"""Return the rest of the stream, computing it if necessary."""
if self._compute_rest is not None:
self._rest = self._compute_rest()
self._compute_rest = None
return self._rest

def __repr__(self):
return 'Stream({0}, <...>)'.format(repr(self.first))

def __iter__(self):
"""Return an iterator over the elements in the stream.

>>> list(zip(range(6), ints))
[(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6)]
"""
"*** YOUR CODE HERE ***"

def __getitem__(self, k):
"""Return the k-th element of the stream.

>>> ints[5]
6
>>> increment_stream(ints)[7]
9
"""
"*** YOUR CODE HERE ***"

def increment_stream(s):
"""Increment all elements of a stream."""
return Stream(s.first+1, lambda: increment_stream(s.rest))

# The stream of consecutive integers starting at 1.
ints = Stream(1, lambda: increment_stream(ints))
```

Q4. Implement the function scale_stream that returns a stream over each element of an input stream, scaled by k:

```def scale_stream(s, k):
"""Return a stream over the elements of s scaled by a number k.

>>> s = scale_stream(ints, 5)
>>> s.first
5
>>> s.rest
Stream(10, <...>)
>>> scale_stream(s.rest, 10)[2]
200
"""
"*** YOUR CODE HERE ***"
```

Q5. A famous problem, first raised by Richard Hamming, is to enumerate, in ascending order with no repetitions, all positive integers with no prime factors other than 2, 3, or 5. These are called regular numbers. One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2, 3, and 5. But this is very inefficient, since, as the integers get larger, fewer and fewer of them fit the requirement. As an alternative, we can build a stream of such numbers. Let us call the required stream of numbers s and notice the following facts about it.

• s begins with 1 .
• The elements of scale_stream(s, 2) are also elements of s.
• The same is true for scale_stream(s, 3) and scale-stream(s, 5).
• These are all of the elements of s.

Now all we have to do is combine elements from these sources. For this we define a procedure merge that combines two ordered streams into one ordered result stream, eliminating repetitions.

Fill in the definition of merge, then fill in the definition of make_s below:

```def merge(s0, s1):
"""Return a stream over the elements of increasing s0 and s1, removing
repeats.

>>> twos = scale_stream(ints, 2)
>>> threes = scale_stream(ints, 3)
>>> m = merge(twos, threes)
>>> [m[i] for i in range(10)]
[2, 3, 4, 6, 8, 9, 10, 12, 14, 15]
"""
if s0 is Stream.empty:
return s1
elif s1 is Stream.empty:
return s0

e0, e1 = s0.first, s1.first
"*** YOUR CODE HERE ***"

def make_s():
"""Return a stream over all positive integers with only factors 2, 3, & 5.

>>> s = make_s()
>>> [s[i] for i in range(20)]
[1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36]
"""
def rest():
"*** YOUR CODE HERE ***"
s = Stream(1, rest)
return s
```