# Homework 5 Solutions

## Solution Files

You can find the solutions in hw05.py.

# Required questions

## Trees

### Q1: 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('loki')]),
... 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'))
odin
balder
freya
loki
frigg
freya
thor
sif
freya
freya
>>> laerad == yggdrasil # Make sure original tree is unmodified
True
"""
if is_leaf(t) and label(t) == old:
return tree(new)
else:
bs = [replace_leaf(b, old, new) for b in branches(t)]
return tree(label(t), bs)
```

Use Ok to test your code:

`python3 ok -q replace_leaf`

## Mobiles

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

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.

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.

### Q2: 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
return ['weight', size]
def size(w):
"""Select the size of a weight."""
assert is_weight(w), 'must call size on a weight'
return w[1]
def is_weight(w):
"""Whether w is a weight."""
return type(w) == list and len(w) == 2 and w[0] == 'weight'
```

Use Ok to test your code:

`python3 ok -q total_weight`

### Q3: Balanced

Implement the `balanced`

function, which returns whether `m`

is a balanced
mobile. A mobile is balanced if two conditions are met:

- 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.
- 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)
True
>>> balanced(v)
True
>>> w = mobile(side(3, t), side(2, u))
>>> balanced(w)
False
>>> balanced(mobile(side(1, v), side(1, w)))
False
>>> balanced(mobile(side(1, w), side(1, v)))
False
"""
if is_weight(m):
return True
else:
left_end, right_end = end(left(m)), end(right(m))
torque_left = length(left(m)) * total_weight(left_end)
torque_right = length(right(m)) * total_weight(right_end)
return balanced(left_end) and balanced(right_end) and torque_left == torque_right
```

Use Ok to test your code:

`python3 ok -q balanced`

The balanced weights assumption is important, since we will be solving this recursively like many other tree problems (even though this is not explicitly a tree).

**Base case:**if we are checking a weight, then we know that this is balanced. Why is this an appropriate base case? There are two possible approaches to this:- Because we know that our data structures so far are trees, weights are the simplest possible tree since we have chosen to implement them as leaves.
- We also know that from an ADT standpoint, weights are the terminal item in a mobile. There can be no further mobile structures under this weight, so it makes sense to stop check here.

**Otherwise:**note that it is important to do a recursive call to check if both sides are balanced. However, we also need to do the basic comparison of looking at the total weight of both sides as well as their length. For example if both sides are a weight, trivially, they will both be balanced. However, the torque must be equal in order for the entire mobile to balanced (i.e. it's insufficient to just check if the sides are balanced).

### Q4: 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))
3
2
1
>>> print_tree(totals_tree(u))
6
1
5
3
2
>>> print_tree(totals_tree(v))
9
3
2
1
6
1
5
3
2
"""
if is_weight(m):
return tree(size(m))
else:
branches = [totals_tree(end(f(m))) for f in [left, right]]
return tree(sum([label(b) for b in branches]), branches)
```

Use Ok to test your code:

`python3 ok -q totals_tree`

## Mutable functions

### Q5: Counter

Define a function `make_counter`

that returns a `counter`

function, which takes
a string and returns the number of times that the function has been called on
that string.

```
def make_counter():
"""Return a counter function.
>>> c = make_counter()
>>> c('a')
1
>>> c('a')
2
>>> c('b')
1
>>> c('a')
3
>>> c2 = make_counter()
>>> c2('b')
1
>>> c2('b')
2
>>> c('b') + c2('b')
5
"""
totals = {}
def counter(key):
totals[key] = totals.get(key, 0) + 1
return totals[key]
return counter
```

Use Ok to test your code:

`python3 ok -q make_counter`

Note that we do not need to use the `nonlocal`

keyword here; we are just
modifying the contents of dictionary, not changing the thing that `totals`

is storing.

`totals.get(key, 0)`

will attempt to retrieve the `key`

from the
dictionary, and returns a default value of `0`

if it is not found. We
could also implement this by always seeding the dictionary with a default
value if the item doesn't already exist:

```
def make_counter():
totals = {}
def counter(key):
if key not in totals:
totals[key] = 0
totals[key] += 1
return totals[key]
return counter
```

Video walkthrough: https://youtu.be/bSUr0OUISuE

### Q6: Next Fibonacci

Write a function `make_fib`

that returns a function that returns the
next Fibonacci number each time it is called. (The Fibonacci sequence begins with 0
and then 1, after which each element is the sum of the preceding two.)
Use a `nonlocal`

statement!

```
def make_fib():
"""Returns a function that returns the next Fibonacci number
every time it is called.
>>> fib = make_fib()
>>> fib()
0
>>> fib()
1
>>> fib()
1
>>> fib()
2
>>> fib()
3
>>> fib2 = make_fib()
>>> fib() + sum([fib2() for _ in range(5)])
12
"""
cur, next = 0, 1
def fib():
nonlocal cur, next
result = cur
cur, next = next, cur + next
return result
return fib
```

Use Ok to test your code:

`python3 ok -q make_fib`

In between calls to `fib`

, we need to remember how far we have gotten in
the sequence somehow. We do this by track the `cur`

and the `next`

Fibonacci number, but this also be done by just tracking the current
position in the sequence, and returning something like `fib_iter(n)`

.
While this solution works (and is included below), it is quite wasteful in
terms of computation and so we prefer the official solution.

```
def make_fib():
def fib_iter(n):
cur, next = 0, 1
while n > 0:
cur, next = next, cur + next
n -= 1
return cur
n = 0
def fib():
nonlocal n
result = fib_iter(n)
n += 1
return result
return fib
```

Consider that to calculate `fib_iter(n+1)`

, it takes just one more step
after `fib_iter(n)`

.

Video walkthrough: https://youtu.be/bSUr0OUISuE

### Q7: Password Protected Account

In lecture, we saw how to use functions to create mutable objects.
Here, for example, is the function `make_withdraw`

which produces a
function that can withdraw money from an account:

```
def make_withdraw(balance):
"""Return a withdraw function with BALANCE as its starting balance.
>>> withdraw = make_withdraw(1000)
>>> withdraw(100)
900
>>> withdraw(100)
800
>>> withdraw(900)
'Insufficient funds'
"""
def withdraw(amount):
nonlocal balance
if amount > balance:
return 'Insufficient funds'
balance = balance - amount
return balance
return withdraw
```

Write a version of the `make_withdraw`

function that returns
password-protected withdraw functions. That is, `make_withdraw`

should
take a password argument (a string) in addition to an initial balance.
The returned function should take two arguments: an amount to withdraw
and a password.

A password-protected `withdraw`

function should only process
withdrawals that include a password that matches the original. Upon
receiving an incorrect password, the function should:

- Store that incorrect password in a list, and
- Return the string 'Incorrect password'.

If a withdraw function has been called three times with incorrect
passwords `p1`

, `p2`

, and `p3`

, then it is locked. All subsequent
calls to the function should return:

`"Your account is locked. Attempts: [<p1>, <p2>, <p3>]"`

The incorrect passwords may be the same or different:

```
def make_withdraw(balance, password):
"""Return a password-protected withdraw function.
>>> w = make_withdraw(100, 'hax0r')
>>> w(25, 'hax0r')
75
>>> error = w(90, 'hax0r')
>>> error
'Insufficient funds'
>>> error = w(25, 'hwat')
>>> error
'Incorrect password'
>>> new_bal = w(25, 'hax0r')
>>> new_bal
50
>>> w(75, 'a')
'Incorrect password'
>>> w(10, 'hax0r')
40
>>> w(20, 'n00b')
'Incorrect password'
>>> w(10, 'hax0r')
"Your account is locked. Attempts: ['hwat', 'a', 'n00b']"
>>> w(10, 'l33t')
"Your account is locked. Attempts: ['hwat', 'a', 'n00b']"
>>> type(w(10, 'l33t')) == str
True
"""
attempts = []
def withdraw(amount, password_attempt):
nonlocal balance
if len(attempts) == 3:
return 'Your account is locked. Attempts: ' + str(attempts)
if password_attempt != password:
attempts.append(password_attempt)
return 'Incorrect password'
if amount > balance:
return 'Insufficient funds'
balance = balance - amount
return balance
return withdraw
```

Use Ok to test your code:

`python3 ok -q make_withdraw`

A couple of things to note:

- The
`attempts`

list does not need to be nonlocal. We're just mutating the list here, not reassigning it. - The last few lines of our
`withdraw`

function are the same as the`withdraw`

from lecture. There isn't much to add on top of that -- just the list operations and the password checks.

Video walkthrough: https://youtu.be/YyjQoug0Mtg

### Q8: Joint Account

Suppose that our banking system requires the ability to make joint
accounts. Define a function `make_joint`

that takes three arguments.

- A password-protected
`withdraw`

function, - The password with which that
`withdraw`

function was defined, and - A new password that can also access the original account.

The `make_joint`

function returns a `withdraw`

function that provides
additional access to the original account using *either* the new or old
password. Both functions draw from the same balance. Incorrect
passwords provided to either function will be stored and cause the
functions to be locked after three wrong attempts.

*Hint*: The solution is short (less than 10 lines) and contains no string
literals! The key is to call `withdraw`

with the right password and amount,
then interpret the result. You may assume that all failed attempts to withdraw
will return some string (for incorrect passwords, locked accounts, or
insufficient funds), while successful withdrawals will return a number.

Use `type(value) == str`

to test if some `value`

is a string:

```
def make_joint(withdraw, old_password, new_password):
"""Return a password-protected withdraw function that has joint access to
the balance of withdraw.
>>> w = make_withdraw(100, 'hax0r')
>>> w(25, 'hax0r')
75
>>> make_joint(w, 'my', 'secret')
'Incorrect password'
>>> j = make_joint(w, 'hax0r', 'secret')
>>> w(25, 'secret')
'Incorrect password'
>>> j(25, 'secret')
50
>>> j(25, 'hax0r')
25
>>> j(100, 'secret')
'Insufficient funds'
>>> j2 = make_joint(j, 'secret', 'code')
>>> j2(5, 'code')
20
>>> j2(5, 'secret')
15
>>> j2(5, 'hax0r')
10
>>> j2(25, 'password')
'Incorrect password'
>>> j2(5, 'secret')
"Your account is locked. Attempts: ['my', 'secret', 'password']"
>>> j(5, 'secret')
"Your account is locked. Attempts: ['my', 'secret', 'password']"
>>> w(5, 'hax0r')
"Your account is locked. Attempts: ['my', 'secret', 'password']"
>>> make_joint(w, 'hax0r', 'hello')
"Your account is locked. Attempts: ['my', 'secret', 'password']"
"""
error = withdraw(0, old_password)
if type(error) == str:
return error
def joint(amount, password_attempt):
if password_attempt == new_password:
return withdraw(amount, old_password)
return withdraw(amount, password_attempt)
return joint
```

Use Ok to test your code:

`python3 ok -q make_joint`

To make sure that we correctly created the joint account, we attempt to
withdraw `0`

from it using the supplied password. If this failed, we
should exit immediately without creating the account, following the
guidance in the doctests.

Otherwise, we have successfully created the joint account! We now know the old password is valid, but remember that the original password-protected account does not know about any new passwords that a joint account might accept. Therefore, when we see something matching the new password, we still have to access the account using the old password.

Video walkthrough: https://youtu.be/h5MvIM1k1II

## Generators

### Q9: Generate Paths

Define a generator function `generate_paths`

which takes in a tree `t`

, a value
`x`

, and yields each path from the root of `t`

to a node that has label `x`

.
Each path should be represented as a list of the labels along that path in the
tree. You may yield the paths in any order.

```
def generate_paths(t, x):
"""Yields all possible paths from the root of t to a node with the label x
as a list.
>>> t1 = tree(1, [tree(2, [tree(3), tree(4, [tree(6)]), tree(5)]), tree(5)])
>>> print_tree(t1)
1
2
3
4
6
5
5
>>> next(generate_paths(t1, 6))
[1, 2, 4, 6]
>>> path_to_5 = generate_paths(t1, 5)
>>> sorted(list(path_to_5))
[[1, 2, 5], [1, 5]]
>>> t2 = tree(0, [tree(2, [t1])])
>>> print_tree(t2)
0
2
1
2
3
4
6
5
5
>>> path_to_2 = generate_paths(t2, 2)
>>> sorted(list(path_to_2))
[[0, 2], [0, 2, 1, 2]]
"""
if label(t) == x:
yield [x]
for b in branches(t):
for path in generate_paths(b, x):
yield [label(t)] + path
```

Use Ok to test your code:

`python3 ok -q generate_paths`

If our current label is equal to `x`

, we've found a path from the root to a node
containing `x`

containing only our current label, so we should yield that. From there,
we'll see if there are any paths starting from one of our branches that ends at a
node containing `x`

. If we find these "partial paths" we can simply add our current
label to the beinning of a path to obtain a path starting from the root.

In order to do this, we'll create a generator for each of the branches which yields
these "partial paths". By calling `generate_paths`

on each of the branches, we'll create
exactly this generator! Then, since a generator is also an iterable, we can iterate over
the paths in this generator and yield the result of concatenating it with our current label.

# Extra Questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively!

## Interval

**Acknowledgements.** This interval arithmetic example is based on
a classic problem from Structure and Interpretation of Computer Programs,
Section 2.1.4.

**Introduction.** Alyssa P. Hacker is designing a system to help people
solve engineering problems. One feature she wants to provide in her
system is the ability to manipulate inexact quantities (such as
measured parameters of physical devices) with known precision, so that
when computations are done with such approximate quantities the results
will be numbers of known precision.

Alyssa's idea is to implement interval arithmetic as a set of arithmetic operations for combining "intervals" (objects that represent the range of possible values of an inexact quantity). The result of adding, subracting, multiplying, or dividing two intervals is itself an interval, representing the range of the result.

Alyssa postulates the existence of an abstract object called an "interval" that has two endpoints: a lower bound and an upper bound. She also presumes that, given the endpoints of an interval, she can construct the interval using the data constructor interval. Using the constructor and selectors, she defines the following operations:

```
def str_interval(x):
"""Return a string representation of interval x."""
return '{0} to {1}'.format(lower_bound(x), upper_bound(x))
def add_interval(x, y):
"""Return an interval that contains the sum of any value in interval x and
any value in interval y."""
lower = lower_bound(x) + lower_bound(y)
upper = upper_bound(x) + upper_bound(y)
return interval(lower, upper)
```

### Q10: Interval Abstraction

Alyssa's program is incomplete because she has not specified the implementation of the interval abstraction. She has implemented the constructor for you; fill in the implementation of the selectors.

```
def interval(a, b):
"""Construct an interval from a to b."""
assert a <= b, 'Lower bound cannot be greater than upper bound' return [a, b]
def lower_bound(x):
"""Return the lower bound of interval x."""
return x[0]
def upper_bound(x):
"""Return the upper bound of interval x."""
return x[1]
```

Use Ok to unlock and test your code:

```
python3 ok -q interval -u
python3 ok -q interval
```

Louis Reasoner has also provided an implementation of interval multiplication. Beware: there are some data abstraction violations, so help him fix his code before someone sets it on fire.

```
def mul_interval(x, y):
"""Return the interval that contains the product of any value in x and any
value in y."""
p1 = x[0] * y[0]
p2 = x[0] * y[1]
p3 = x[1] * y[0]
p4 = x[1] * y[1]
return [min(p1, p2, p3, p4), max(p1, p2, p3, p4)]
```

Use Ok to unlock and test your code:

```
python3 ok -q mul_interval -u
python3 ok -q mul_interval
```

Video walkthrough: https://youtu.be/RiLrqgBm4Dk

### Q11: Sub Interval

Using reasoning analogous to Alyssa's, define a subtraction function for intervals. Try to reuse functions that have already been implemented if you find yourself repeating code.

```
def sub_interval(x, y):
"""Return the interval that contains the difference between any value in x
and any value in y."""
negative_y = interval(-upper_bound(y), -lower_bound(y))
return add_interval(x, negative_y)
```

Use Ok to unlock and test your code:

```
python3 ok -q sub_interval -u
python3 ok -q sub_interval
```

Video walkthrough: https://youtu.be/s37VvawB0vE

### Q12: Div Interval

Alyssa implements division below by multiplying by the reciprocal of
`y`

. Ben Bitdiddle, an expert systems programmer, looks over Alyssa's
shoulder and comments that it is not clear what it means to divide by
an interval that spans zero. Add an `assert`

statement to Alyssa's code
to ensure that no such interval is used as a divisor:

```
def div_interval(x, y):
"""Return the interval that contains the quotient of any value in x divided by
any value in y. Division is implemented as the multiplication of x by the
reciprocal of y."""
assert not (lower_bound(y) <= 0 <= upper_bound(y)), 'Divide by zero' reciprocal_y = interval(1/upper_bound(y), 1/lower_bound(y))
return mul_interval(x, reciprocal_y)
```

Use Ok to unlock and test your code:

```
python3 ok -q div_interval -u
python3 ok -q div_interval
```

Video walkthrough: https://youtu.be/jNR7tmV1lMk

### Q13: Par Diff

After considerable work, Alyssa P. Hacker delivers her finished system. Several years later, after she has forgotten all about it, she gets a frenzied call from an irate user, Lem E. Tweakit. It seems that Lem has noticed that the formula for parallel resistors can be written in two algebraically equivalent ways:

`par1(r1, r2) = (r1 * r2) / (r1 + r2)`

or

`par2(r1, r2) = 1 / (1/r1 + 1/r2)`

He has written the following two programs, each of which computes the
`parallel_resistors`

formula differently::

```
def par1(r1, r2):
return div_interval(mul_interval(r1, r2), add_interval(r1, r2))
def par2(r1, r2):
one = interval(1, 1)
rep_r1 = div_interval(one, r1)
rep_r2 = div_interval(one, r2)
return div_interval(one, add_interval(rep_r1, rep_r2))
```

Lem complains that Alyssa's program gives different answers for the two ways of computing. This is a serious complaint.

Demonstrate that Lem is right. Investigate the behavior of the system
on a variety of arithmetic expressions. Make some intervals `r1`

and
`r2`

, and show that `par1`

and `par2`

can give different results.

```
def check_par():
"""Return two intervals that give different results for parallel resistors.
>>> r1, r2 = check_par()
>>> x = par1(r1, r2)
>>> y = par2(r1, r2)
>>> lower_bound(x) != lower_bound(y) or upper_bound(x) != upper_bound(y)
True
"""
r1 = interval(1, 2) r2 = interval(3, 4) return r1, r2
```

Use Ok to test your code:

`python3 ok -q check_par`

Video walkthrough: https://youtu.be/H8slb5KCbU4

### Q14: Multiple References

Eva Lu Ator, another user, has also noticed the different intervals computed by different but algebraically equivalent expressions. She says that the problem is multiple references to the same interval.

The Multiple References Problem: a formula to compute with intervals using Alyssa's system will produce tighter error bounds if it can be written in such a form that no variable that represents an uncertain number is repeated.

Thus, she says, `par2`

is a better program for parallel resistances
than `par1`

. Is she right? Why? Write a function that returns a string
containing a written explanation of your answer:

```
def multiple_references_explanation():
return """The multiple reference problem exists. The true value
within a particular interval is fixed (though unknown). Nested
combinations that refer to the same interval twice may assume two different
true values for the same interval, which is an error that results in
intervals that are larger than they should be.
Consider the case of i * i, where i is an interval from -1 to 1. No value
within this interval, when squared, will give a negative result. However,
our mul_interval function will allow us to choose 1 from the first
reference to i and -1 from the second, giving an erroneous lower bound of
-1.
Hence, a program like par2 is better than par1 because it never combines
the same interval more than once.
"""
```

Video walkthrough: https://youtu.be/H8slb5KCbU4

### Q15: Quadratic

Write a function `quadratic`

that returns the interval of all values
`f(t)`

such that `t`

is in the argument interval `x`

and `f(t)`

is a
quadratic function:

`f(t) = a*t*t + b*t + c`

Make sure that your implementation returns the smallest such interval, one that does not suffer from the multiple references problem.

*Hint*: the derivative `f'(t) = 2*a*t + b`

, and so the extreme
point of the quadratic is `-b/(2*a)`

:

```
def quadratic(x, a, b, c):
"""Return the interval that is the range of the quadratic defined by
coefficients a, b, and c, for domain interval x.
>>> str_interval(quadratic(interval(0, 2), -2, 3, -1))
'-3 to 0.125'
>>> str_interval(quadratic(interval(1, 3), 2, -3, 1))
'0 to 10'
"""
extremum = -b / (2*a)
f = lambda x: a * x * x + b * x + c
l, u, e = map(f, (lower_bound(x), upper_bound(x), extremum))
if extremum >= lower_bound(x) and extremum <= upper_bound(x):
return interval(min(l, u, e), max(l, u, e))
else:
return interval(min(l, u), max(l, u))
```

Use Ok to test your code:

`python3 ok -q quadratic`

Video walkthrough: https://youtu.be/qgSn_RNBs4A