CS 61A: Homework 9

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

Submission: See Lab 1 for submission instructions. We have provided a hw9.py starter file for the questions below.

Readings: You might find the following references useful:

Table of Contents

This extra-large homework is worth 6 points! While graded on effort, you should make progress on all problems to earn credit. Start early!

The Brackulator language shares an evaluator with the Calculator language, but uses a more concise syntax. Instead of using operator names or symbols, Brackulator indicates operations using different kinds of brackets:

Operand expressions are separated by spaces within these brackets. The following Brackulator expressions are followed by their Calculator equivalents.

<1 2 3>              (* 1 2 3)
(5)                  (- 5)
[2{4 8}]             (+ 2 (/ 4 8))
<[2{12 6}](3 4 5)>   (* (+ 2 (/ 12 6)) (- 3 4 5))

By solving the following problems, you will implement a parser, brack_read, that returns an expression for the calc_eval evaluator implemented in the Calculator example from lecture.

All of your solutions should be defined in terms of the following dictionaries of bracket types, which configure the parser to supply the correct operator for each bracket:

# A dictionary from pairs of matching brackets to the operators they indicate.
brackets = {('[', ']'): '+',
            ('(', ')'): '-',
            ('<', '>'): '*',
            ('{', '}'): '/'}

# A dictionary with left-bracket keys and corresponding right-bracket values.
left_to_right = {left: right for left, right in brackets}

# The set of all left and right brackets.
all_brackets = set(left_to_right.keys()).union(set(left_to_right.values()))

Question 1

Complete tokenize, which splits a Brackulator expression into tokens, and raise a ValueError if any token is not a number or a known bracket. Hint: You can first surround each bracket with spaces using line.replace, then split on spaces. Afterward, check each token to ensure that it is legal. The provided coerce_to_number function may prove useful:

def tokenize(line):
    """Convert a string into a list of tokens.

    >>> tokenize('2.3')
    >>> tokenize('(2 3)')
    ['(', 2, 3, ')']
    >>> tokenize('<2 3)')
    ['<', 2, 3, ')']
    >>> tokenize('<[2{12.5 6.0}](3 -4 5)>')
    ['<', '[', 2, '{', 12.5, 6.0, '}', ']', '(', 3, -4, 5, ')', '>']

    >>> tokenize('2.3.4')
    Traceback (most recent call last):
    ValueError: invalid token 2.3.4

    >>> tokenize('?')
    Traceback (most recent call last):
    ValueError: invalid token ?

    >>> tokenize('hello')
    Traceback (most recent call last):
    ValueError: invalid token hello

    >>> tokenize('<(GO BEARS)>')
    Traceback (most recent call last):
    ValueError: invalid token GO
    # Surround all brackets by spaces so that they are separated by split.
    for b in all_brackets:
        line = line.replace(b, ' ' + b + ' ')

    # Convert numerals to numbers and raise ValueErrors for invalid tokens.
    tokens = []
    for t in line.split():
        "*** YOUR CODE HERE ***"
    return tokens

def coerce_to_number(token):
    """Coerce a string to a number or return None.

    >>> coerce_to_number('-2.3')
    >>> print(coerce_to_number('('))
        return int(token)
    except (TypeError, ValueError):
            return float(token)
        except (TypeError, ValueError):
            return None

Question 2

Implement brack_read, which returns an expression tree for the first valid Brackulator expression in a list of tokens. The expression tree should contain Calculator operators that correspond to the bracket types. Raise a SyntaxError for any malformed expression. The Pair class and nil object from lecture appear at the bottom of this file. This function is similar to scheme_read from Calculator's Scheme reader file.

Hint: Introduce another function read_tail that reads the elements in a combination until reaching a closing bracket. In brack_read make sure that the closing bracket of an expression matches the opening bracket. The left_to_right dictionary defined above gives you the matching right bracket for each type of left bracket. The brackets dictionary gives you the corresponding operator (e.g. '+' for '[' and ']').

Once you complete this problem, you can place your homework file in the same directory as scalc.py (and its supporting files), then run read_eval_print_loop to interact with the Brackulator language:

def brack_read(tokens):
    """Return an expression tree for the first well-formed Brackulator
    expression in tokens. Tokens in that expression are removed from tokens as
    a side effect.

    >>> brack_read(tokenize('100'))
    >>> brack_read(tokenize('([])'))
    Pair('-', Pair(Pair('+', nil), nil))
    >>> print(brack_read(tokenize('<[2{12 6}](3 4 5)>')))
    (* (+ 2 (/ 12 6)) (- 3 4 5))
    >>> brack_read(tokenize('(1)(1)')) # More than one expression is ok
    Pair('-', Pair(1, nil))
    >>> brack_read(tokenize('[])')) # Junk after a valid expression is ok
    Pair('+', nil)

    >>> brack_read(tokenize('([]')) # Missing right bracket
    Traceback (most recent call last):
    SyntaxError: unexpected end of line

    >>> brack_read(tokenize('[)]')) # Extra right bracket
    Traceback (most recent call last):
    SyntaxError: unexpected )

    >>> brack_read(tokenize('([)]')) # Improper nesting
    Traceback (most recent call last):
    SyntaxError: unexpected )

    >>> brack_read(tokenize('')) # No expression
    Traceback (most recent call last):
    SyntaxError: unexpected end of line
    if not tokens:
        raise SyntaxError('unexpected end of line')
    token = tokens.pop(0)
    n = coerce_to_number(token)
    if n != None:
        return n
    elif token in left_to_right:
        "*** YOUR CODE HERE ***"

Support code for Brackulator (from the Calculator example):

# Support classes for Brackulator #

class Pair:
    """A pair has two instance attributes: first and second.  For a Pair to be
    a well-formed list, second is either a well-formed list or nil.  Some
    methods only apply to well-formed lists.

    >>> s = Pair(1, Pair(2, nil))
    >>> s
    Pair(1, Pair(2, nil))
    >>> print(s)
    (1 2)
    >>> len(s)
    >>> s[1]
    >>> print(s.map(lambda x: x+4))
    (5 6)
    def __init__(self, first, second):
        self.first = first
        self.second = second

    def __repr__(self):
        return "Pair({0}, {1})".format(repr(self.first), repr(self.second))

    def __str__(self):
        s = "(" + str(self.first)
        second = self.second
        while isinstance(second, Pair):
            s += " " + str(second.first)
            second = second.second
        if second is not nil:
            s += " . " + str(second)
        return s + ")"

    def __len__(self):
        n, second = 1, self.second
        while isinstance(second, Pair):
            n += 1
            second = second.second
        if second is not nil:
            raise TypeError("length attempted on improper list")
        return n

    def __getitem__(self, k):
        if k < 0:
            raise IndexError("negative index into list")
        y = self
        for _ in range(k):
            if y.second is nil:
                raise IndexError("list index out of bounds")
            elif not isinstance(y.second, Pair):
                raise TypeError("ill-formed list")
            y = y.second
        return y.first

    def map(self, fn):
        """Return a Scheme list after mapping Python function FN to SELF."""
        mapped = fn(self.first)
        if self.second is nil or isinstance(self.second, Pair):
            return Pair(mapped, self.second.map(fn))
            raise TypeError("ill-formed list")

class nil:
    """The empty list"""

    def __repr__(self):
        return "nil"

    def __str__(self):
        return "()"

    def __len__(self):
        return 0

    def __getitem__(self, k):
        if k < 0:
            raise IndexError("negative index into list")
        raise IndexError("list index out of bounds")

    def map(self, fn):
        return self

nil = nil() # Assignment hides the nil class; there is only one instance

To use the following function, you will need to place your homework solution in the same directory as the files from the Calculator Example:

def read_eval_print_loop():
    """Run a read-eval-print loop for the Brackulator language."""
    global Pair, nil
    from scheme_reader import Pair, nil
    from scalc import calc_eval

    while True:
            src = tokenize(input('brack> '))
            while len(src) > 0:
              expression = brack_read(src)
        except (SyntaxError, ValueError, TypeError, ZeroDivisionError) as err:
            print(type(err).__name__ + ':', err)
        except (KeyboardInterrupt, EOFError):  # <Control>-D, etc.

Question 3

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:

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
    >>> m.is_balanced()
    >>> m.left.contents = Mobile(Branch(1, Weight(1)), Branch(2, Weight(1)))
    >>> m.weight
    >>> m.left.contents.is_balanced()
    >>> m.is_balanced() # All submobiles must be balanced for m to be balanced
    >>> m.left.contents.right.contents.weight = 0.5
    >>> m.left.contents.is_balanced()
    >>> m.is_balanced()
    >>> m.right.length = 1.5
    >>> m.is_balanced()

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

    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(2)
    >>> 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')
        self.length = length
        self.contents = contents

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

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

    def is_balanced(self):
        return True

Question 4

Your partner designed a beautiful balanced Mobile, but forgot to fill in the classes of each part, instead just writing T.


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
    >>> interpret_mobile(simple).is_balanced()
    >>> s = 'T(T(4,T(T(4,T(1)),T(1,T(4)))),T(2,T(10)))'
    >>> m = interpret_mobile(s)
    >>> m.weight
    >>> m.is_balanced()
    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
            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

Question 5

An enhancement of 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.

class Stream:
    """A lazily computed linked 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

    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.

        >>> it = iter(ints)
        >>> [next(it) for _ in range(6)]
        [1, 2, 3, 4, 5, 6]
        "*** YOUR CODE HERE ***"

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

        >>> ints[5]
        >>> increment_stream(ints)[7]
        >>> s = Stream(1, lambda: Stream(2))
        >>> [s[i] for i in range(2)]
        [1, 2]
        >>> s[2]
        Traceback (most recent call last):
        IndexError: Stream index out of range
        "*** 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))

Question 6

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

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

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

Question 7

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.

Now all we have to do is combine elements from these sources. For this we define the merge function 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 strictly increasing s0 and s1,
    removing repeats. Assume that s0 and s1 have no 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

Question 8

Implement a function unique that takes a stream and returns a stream in which duplicates of any value are filtered out. This should work for infinite streams as well as finite ones.

def unique(s):
    """Return a stream of the unique elements in s in the order that they
    first appear.

    >>> s = unique(to_stream([1, 2, 2, 1, 0, 4, 2, 3, 1, 9, 0]))
    >>> [s[i] for i in range(6)]
    [1, 2, 0, 4, 3, 9]
    "*** YOUR CODE HERE ***"

def to_stream(lst):
    if not lst:
        return Stream.empty
    return Stream(lst[0], lambda: to_stream(lst[1:]))

Question 9

Run-length encoding is a very simple data compression technique, whereby runs of data are compressed and stored as a single value. A run is defined to be a contiguous sequence of the same number. For example, in the (finite) sequence

1, 1, 1, 1, 1, 6, 6, 6, 6, 2, 5, 5, 5

there are four runs: one each of 1, 6, 2, and 5. We can represent the same sequence as a sequence of tuples:

(5, 1), (4, 6), (1, 2), (3, 5)

Notice that the first element of each tuple is the number of times a particular number appears in a run, and the second element is the number in the run.

We will extend this idea to (possibly infinite) streams. Write a function called rle that takes in a stream of data, and returns a corresponding stream of tuples, which represents the run-length encoded version of the stream. It will also take in the maximum size of any given run (default 10) to prevent having to compress infinite runs.

def rle(s, max_run_length=10):
    >>> example_stream = to_stream([1, 1, 1, 2, 3, 3])
    >>> encoded_example = rle(example_stream)
    >>> [encoded_example[i] for i in range(3)]
    [(3, 1), (1, 2), (2, 3)]
    >>> shorter_encoded_example = rle(example_stream, 2)
    >>> [shorter_encoded_example[i] for i in range(4)]
    [(2, 1), (1, 1), (1, 2), (2, 3)]
    >>> encoded_naturals = rle(ints)
    >>> [encoded_naturals[i] for i in range(3)]
    [(1, 1), (1, 2), (1, 3)]
    >>> ones = Stream(1, lambda: ones)
    >>> encoded_ones = rle(ones, max_run_length=3)
    >>> [encoded_ones[i] for i in range(3)]
    [(3, 1), (3, 1), (3, 1)]
    "*** YOUR CODE HERE ***"

Question 10: Challenge Problem (optional)

The Python Challenge is a website designed to teach people the many features of the Python Library. Each page of the site is a puzzle that can be solved simply in Python. The solution to each puzzle gives the URL of the next.

There is a function stub below to include your solution to puzzle 4 (the one with the picture of a wood carving). You will have to complete puzzles 0, 1, 2, and 3 to reach 4. You can start on puzzle 0 here.

Some hints:

You can include your solution to puzzle 4 below:

from urllib.request import urlopen

def puzzle_4():
    """Return the soluton to puzzle 4."""
    "*** YOUR CODE HERE ***"