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1. Markov Models [95 points]

In this section, you will build a simple language model that can be used to generate random text resembling a

source document. Your use of external code should be limited to built-in Python modules, which excludes, for

example, NumPy and NLTK.

1. [10 points] Write a simple tokenization function tokenize(text) which takes as input a string of text

and returns a list of tokens derived from that text. Here, we define a token to be a contiguous sequence of

non-whitespace characters, with the exception that any punctuation mark should be treated as an

individual token. Hint: Use the built-in constant string.punctuation, found in the string module.

>>>tokenize(” This is an example. “)

[‘This’, ‘is’, ‘an’, ‘example’, ‘.’]

>>>tokenize(“‘Medium-rare,’ she said.”)

[“‘”, ‘Medium’, ‘-‘, ‘rare’, ‘,’, “‘”,

‘she’, ‘said’, ‘.’]

2. [10 points] Write a function ngrams(n, tokens) that produces a list of all n-grams of the specified size

from the input token list. Each n-gram should consist of a 2-element tuple (context, token), where the

context is itself an (n-1)-element tuple comprised of the n-1 words preceding the current token. The

sentence should be padded with n-1 “

end. If n = 1, all contexts should be empty tuples. You may assume that n ≥ 1.

>>>ngrams(1, [“a”, “b”, “c”])

[((), ‘a’), ((), ‘b’), ((), ‘c’),

((), ‘

>>>ngrams(2, [“a”, “b”, “c”])

[((‘

((‘b’,), ‘c’), ((‘c’,), ‘

>>>ngrams(3, [“a”, “b”, “c”])

[((‘

((‘

((‘a’, ‘b’), ‘c’),

((‘b’, ‘c’), ‘

3. [15 points] In the NgramModel class, write an initialization method __init__(self, n) which stores the

order n of the model and initializes any necessary internal variables. Then write a method

update(self, sentence) which computes the n-grams for the input sentence and updates the internal

counts. Lastly, write a method prob(self, context, token) which accepts an (n-1)-tuple representing a

context and a token, and returns the probability of that token occuring, given the preceding context.

>>>m = NgramModel(1)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>m.prob((), “a”)

0.3

>>>m.prob((), “c”)

0.1

>>>m.prob((), “

>>>m = NgramModel(2)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>m.prob((“

1.0

>>>m.prob((“b”,), “c”)

0.3333333333333333

>>>m.prob((“a”,), “x”)

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0.2 0.0

4. [20 points] In the NgramModel class, write a method random_token(self, context) which returns a

random token according to the probability distribution determined by the given context. Specifically, let T

= < t1, t2, . . . , tn > be the set of tokens which can occur in the given context, sorted according to Python’s

natural lexicographic ordering, and let 0 ≤ r < 1 be a random number between 0 and 1. Your method
should return the token ti
such that
You should use a single call to the random.random() function to generate r.
>>>m = NgramModel(1)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>random.seed(1)

>>>[m.random_token(())

for i in range(25)]

[‘

‘b’, ‘

‘

‘

>>>m = NgramModel(2)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>random.seed(2)

>>>[m.random_token((“

for i in range(6)]

[‘a’, ‘a’, ‘a’, ‘a’, ‘a’, ‘a’]

>>>[m.random_token((“b”,))

for i in range(6)]

[‘c’, ‘

5. [20 points] In the NgramModel class, write a method random_text(self, token_count) which returns

a string of space-separated tokens chosen at random using the random_token(self, context) method.

Your starting context should always be the (n-1)-tuple (“

should be updated as tokens are generated. If n = 1, your context should always be the empty tuple.

Whenever the special token “

>>>m = NgramModel(1)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>random.seed(1)

>>>m.random_text(13)

‘

>>>m = NgramModel(2)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>random.seed(2)

>>>m.random_text(15)

‘a b

6. [10 points] Write a function create_ngram_model(n, path) which loads the text at the given path and

creates an n-gram model from the resulting data. Each line in the file should be treated as a separate

sentence.

# No random seeds, so your results may vary

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>>>m = create_ngram_model(1, “frankenstein.txt”); m.random_text(15)

‘beat astonishment brought his for how , door

>>>m = create_ngram_model(2, “frankenstein.txt”); m.random_text(15)

‘As the great was extreme during the end of being .

>>>m = create_ngram_model(3, “frankenstein.txt”); m.random_text(15)

‘I had so long inhabited .

>>>m = create_ngram_model(4, “frankenstein.txt”); m.random_text(15)

‘We were soon joined by Elizabeth .

7. [10 points] Suppose we define the perplexity of a sequence of m tokens < w1, w2, . . . , wm > to be

For example, in the case of a bigram model under the framework used in the rest of the assignment, we

would generate the bigrams < (w0 = < START >, w1), (w1, w2), . . . , (wm-1, wm), (wm, wm+1 =

and would then compute the perplexity as

Intuitively, the lower the perplexity, the better the input sequence is explained by the model. Higher values

indicate the input was “perplexing” from the model’s point of view, hence the term perplexity.

In the NgramModel class, write a method perplexity(self, sentence) which computes the n-grams for

the input sentence and returns their perplexity under the current model. Hint: Consider performing an

intermediate computation in log-space and re-exponentiating at the end, so as to avoid numerical

overflow.

>>>m = NgramModel(1)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>m.perplexity(“a b”)

3.815714141844439

>>>m = NgramModel(2)

>>>m.update(“a b c d”)

>>>m.update(“a b a b”)

>>>m.perplexity(“a b”)

1.4422495703074083

2. Feedback [5 points]

1. [1 point] Approximately how long did you spend on this assignment?

2. [2 points] Which aspects of this assignment did you find most challenging? Were there any significant

stumbling blocks?

3. [2 points] Which aspects of this assignment did you like? Is there anything you would have changed?

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