Use a Markov chain to create a statistical model of a piece of English text. Simulate the Markov chain to generate stylized pseudo-random text.

Perspective. In the 1948 landmark paper A Mathematical Theory of Communication, Claude Shannon founded the field of information theory and revolutionized the telecommunications industry, laying the groundwork for today's Information Age. In this paper, Shannon proposed using a Markov chain to create a statistical model of the sequences of letters in a piece of English text. Markov chains are now widely used in speech recognition, handwriting recognition, information retrieval, data compression, and spam filtering. They also have many scientific computing applications including the genemark algorithm for gene prediction, the Metropolis algorithm for measuring thermodynamical properties, and Google's PageRank algorithm for web search. For this assignment, we consider a whimsical variant—generating stylized pseudo-random text.

Markov model of natural language. Shannon approximated the statistical structure of a piece of text using a mathematical model known as a Markov model. A Markov model of order 0 predicts that each letter in the alphabet occurs with a fixed probability. We can fit a Markov model of order 0 to a specific piece of text by counting the number of occurrences of each letter in that text, and using those frequencies as probabilities. For example, suppose that the input text is "gagggagaggcgagaaa". Then, the Markov model of order 0 predicts that each letter is 'a' with probability 7/17, 'c' with probability 1/17, and 'g' with probability 9/17 because these are the fraction of times each letter occurs. The following sequence of characters is a typical example generated from this model:

g a g g c g a g a a g a g a a g a a a g a g a g a g a a a g a g a a g ...

We obtain a more refined model by allowing the probability of choosing each successive letter to depend on the preceding letter or letters. A Markov model of order k predicts that each letter occurs with a fixed probability, but that probability can depend on the previous k characters, which we refer to as a k-gram. For example, suppose that the input text has 100 occurrences of "th", with 60 occurrences of "the", 25 occurrences of "thi", 10 occurrences of "tha", and 5 occurrences of "tho". Then, the Markov model of order 2 predicts that the letter immediately following any occurrence of "th" is 'e' with probability 3/5, 'i' with probability 1/4, 'a' with probability 1/10, and 'o' with probability 1/20.

A brute-force solution. Claude Shannon proposed the following brute-force scheme to generate text according to a Markov model of order 1:

“ To construct [a Markov model of order 1], for example, one opens a book at random and selects a letter at random on the page. This letter is recorded. The book is then opened to another page and one reads until this letter is encountered. The succeeding letter is then recorded. Turning to another page this second letter is searched for and the succeeding letter recorded, etc. It would be interesting if further approximations could be constructed, but the labor involved becomes enormous at the next stage. ”

Markov model data type. Create an immutable data type MarkovModel to represent a Markov model of order k from a given text string. The data type must implement the following API:

public class MarkovModel {

    // creates a Markov model of order k for the specified text
    public MarkovModel(String text, int k)

    // returns the order k of this Markov model
    public int order()

    // returns a string representation of the Markov model (as described below)
    public String toString()

    // returns the number of times the specified kgram appears in the text
    public int freq(String kgram)

    // returns the number of times the character c follows the specified
    // kgram in the text
    public int freq(String kgram, char c)

    // returns a random character that follows the specified kgram in the text,
    // chosen with weight proportional to the number of times that character
    // follows the specified kgram in the text
    public char random(String kgram)

    // unit tests all methods in this class
    public static void main(String[] args)
}

• Constructor. You may assume that the input text is a sequence of characters over the ASCII alphabet, so that all char values are between 0 and 127.

• String representation. Build a string representation of the Markov model, as illustrated below.

  aa: a 1 g 1 
  ag: a 3 g 2 
  cg: a 1 
  ga: a 1 g 4 
  gc: g 1 
  gg: a 1 c 1 g 1 
  

  Include one line for each k-gram that appears in the text. Each line contains the k-gram, followed by a colon; followed by each character that appears in the text immediately after that k-gram and the number of times it appears, with a space between each component. The k-grams must appear in lexicographic order; the characters associated with each k-gram must appear in ASCII order.

• Randomly generate a character. The random() method must return a character that immediately follows the specified k-gram and do so with probability proportional to the number of times that character follows the specified k-gram. For example if the k-gram "ga" appears in the text five times, once followed by the character 'a' and four times followed by the character 'g', then random("ga") should return 'a' with probability 1/5 and 'g' with probability 4/5, independently for each call.

• Circular string. To avoid dead ends, treat the input text as a circular string: the last character is considered to precede the first character. For example, if k = 2 and the text is the 17-character string "gagggagaggcgagaaa", then the salient features of the Markov model are captured in the table below:

                 frequency of   probability that
                  next char       next char is 
  kgram   freq    a   c   g        a    c    g
  ----------------------------------------------
   aa      2      1   0   1       1/2   0   1/2  
   ag      5      3   0   2       3/5   0   2/5  
   cg      1      1   0   0        1    0    0
   ga      5      1   0   4       1/5   0   4/5  
   gc      1      0   0   1        0    0    1  
   gg      3      1   1   1       1/3  1/3  1/3  
  ----------------------------------------------
          17      7   1   9
  

  Note that the frequency of "ag" is 5 (and not 4) because we treat the string as circular.

• Corner cases. Throw an IllegalArgumentException in freq() and random() if the argument kgram is not of length k. Throw an IllegalArgumentException in random() if kgram does not appear in the text.

• Performance requirements. If k is a fixed constant, then the constructor should take n log n time; the order() method should take constant time; random() and two freq() methods should take log n time, where n is the number of characters in the input text. To achieve these performance requirements, use one (or more) symbol tables whose keys are String k-grams and whose values enable efficient implementation of the two freq() methods.

Text generation client. A Markov chain is a stochastic process where the state change depends on only the current state. For text generation, the current state is a k-gram. The next character is selected at random, using the probabilities from the Markov model. For example, if the current state is "ga" in the Markov model of order 2 discussed above, then the next character will be 'a' with probability 1/5 and 'g' with probability 4/5. The next state in the Markov chain is obtained by appending the new character to the end of the k-gram and discarding the first character. A trajectory through the Markov chain is a sequence of such states. Below is a possible trajectory consisting of 9 transitions.

trajectory:          ga  -->  ag  -->  gg  -->  gc  -->  cg  -->  ga  -->  ag  -->  ga  -->  aa  -->  ag
probability for a:       1/5      3/5      1/3       0        1       1/5      3/5      1/5      1/2
probability for c:        0        0       1/3       0        0        0        0        0        0
probability for g:       4/5      2/5      1/3       1        0       4/5      2/5      4/5      1/2

To generate random text from a Markov model of order k, set the initial state to the first k characters in the input text. Then, simulate a trajectory through the Markov chain by performing T − k transitions, printing the random character selected at each step. For example, if k = 2 and T = 11, then the following is a possible trajectory, leading to the output gaggcgagaag.

trajectory:          ga  -->  ag  -->  gg  -->  gc  -->  cg  -->  ga  -->  ag  -->  ga  -->  aa  -->  ag
output:              ga        g        g        c        g        a        g        a        a        g

Write a client program TextGenerator that takes two integer command-line arguments k and T; reads the input text from standard input; builds a Markov model of order k from the input text; then, starting with the k-gram consisting of the first k characters of the input text, prints T characters generated by simulating a trajectory through the corresponding Markov chain.

% more input17.txt
gagggagaggcgagaaa

% java-introcs TextGenerator 2 11 < input17.txt 
gaggcgagaag

% java-introcs TextGenerator 2 11 < input17.txt 
gaaaaaaagag

Corner cases. You may assume that the length of the text is at least k and T ≥ k.

Performance requirement. If k is a fixed constant, then the running time of TextGenerator should be proportional to n log n + T log n, where n is the number of characters in the input text and T is the number of characters in the output.

Experimentation. Once you get the program working, test it on different inputs of different sizes and different orders. Does increasing the order have the effect you expect? Try your model on something that you have written or some other text you know well. Make sure to test both long inputs (we provide several) and long outputs.

Files provided. The file markov.zip contains sample test files, the readme.txt template, the abbreviated partner readme.txt template, and ST.java.

Deliverables. Submit MarkovModel.java, TextGenerator.java, and readme.txt. If working in a pair, one student should submit these, and the other should only submit the abbreviated partner readme.txt. Include in your readme.txt two of the most entertaining language-modeling fragments that you discover.

Challenge for the bored 1. Extend your solution to handle Unicode text, not just ASCII.

Challenge for the bored 2. Imagine you receive a message where some of the characters have been corrupted by noise. We represent unknown characters by the ~ symbol (and assume the character ~ does not appear in the original text). Devise a scheme based on the Markov model to determine the most likely value for each corrupted character. Assume unknown characters are at least k characters apart (and appear at least k characters away from the start and end of the message). Test your new method by writing a client program FixCorrupted.java that takes as arguments the model order and the noisy string. The program should print the most likely original string:

Original:  it was the best of times, it was the worst of times.
Noisy:     it w~s th~ bes~ of tim~s, i~ was ~he wo~st of~times.

% java-introcs FixCorrupted 4 "it w~s th~ bes~ of tim~s, i~ was ~he wo~st of~times." < wiki_100k.txt 
it was the best of times, it was the worst of times.

	[Enter DUKE SENIOR, AMIENS, and two or three Lords,
	like foresters]

DUKE SENIOR	Now, my co-mates and brothers in exile,
	Hath not old custom made this life more sweet
	Than that of painted pomp? Are not these woods
	More free from peril than the envious court?
	Here feel we but the penalty of Adam,
	The seasons' difference, as the icy fang
	And churlish chiding of the winter's wind,
	Which, when it bites and blows upon my body,
	Even till I shrink with cold, I smile and say
	'This is no flattery: these are counsellors
	That feelingly persuade me what I am.'
	Sweet are the uses of adversity,
	Which, like the toad, ugly and venomous,
	Wears yet a precious jewel in his head;
	And this our life exempt from public haunt
	Finds tongues in trees, books in the running brooks,
	Sermons in stones and good in every thing.
	I would not change it.

AMIENS	Happy is your grace,
	That can translate the stubbornness of fortune
	Into so quiet and so sweet a style.

DUKE SENIOR	Come, shall we go and kill us venison?
	And yet it irks me the poor dappled fools,
	Being native burghers of this desert city,
	Should in their own confines with forked heads
	Have their round haunches gored.

DUKE SENIOR	Now, my co-mates and thus bolden'd, man, how now, monsieur Jaques,
	Unclaim'd of his absence, as the holly!
	Though in the slightest for the fashion of his absence, as the only wear.

TOUCHSTONE	I care not for meed!
	This I must woo yours: your request than your father: the time,
	That ever love I broke
	my sword upon some kind of men
	Then, heigh-ho! sing, heigh-ho! sing, heigh-ho! sing, heigh-ho! unto the needless stream;
	'Poor deer,' quoth he,
	'Call me not so keen,
	Because thou the creeping hours of the sun,
	As man's feasts and women merely players:
	Thus we may rest ourselves and neglect the cottage, pasture?

	[Exit]

	[Enter DUKE FREDERICK	Can in his time in my heartily,
	And have me go with your fortune
	In all this fruit
	Till than bear
	the arm's end: I will through
	Cleanse the uses of the way to look you.
	Know you not, master,
	Sighing like upon a stone another down his bravery is not so with his effigies with my food:
	To speak my mind, and inquisition
	And unregarded age in corners throat,
	He will come hither:
	He dies that hath engender'd:
	And you to
	the bed untreasured of the brutish sting it.