Consider the following definition of bigram language models

Consider the following definition of bigram language models (it is very similar to the definition of trigram language models seen in class): Definition 1 (Bigram Language Model) A bigram language model consists of a finite set nu, and a parameter q(w\\v) for each bigram upsilon, w such that w nu U {STOP}, and upsilon nu U {*}. The value for q(w\\upsilon) can be interpreted as the probability of seeing the word w immediately after the word upsilon. For any sentence x_1 ... x_n where x_i nu for i = 1... (n - 1), and x_n = STOP, the probability of the sentence under the bigram language model is p(x_1 ...x_n) = Product_i=1^n q(x_i|x_i-1) where we define x_0 = *. Now assume that our vocabulary nu = {the}, that is, the vocabulary has a single word the. We would like to define the parameters of a bigram language model such that p(STOP) = 0 p(the STOP) = 0.4 p(the the STOP) = 0.4 times 0.6 p(the the the STOP) = 0.4 times 0.6^2 ... (In general the probability of a sentence which has the word the n times, for n greaterthanorequalto 1, is 0.4 times 0.6^n-1.) Write down the parameters of the language model such that it gives the above distribution over sentences (i.e., p(x) = 0.4 times 0.6^n-1 if x is a sentence of n consecutive these. followed by the STOP symbol). Write down a PCFG such that: Any sentence consisting of the word the n times in a row, where n greaterthanorequalto 1, has probability 0.4 times 0.6^n-1 Any other sentence has probability 0. (I.e., this is the same distribution as in the last question)

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In modular arithmetic the set of congruence classes relatively prime to the modulus number, say n, form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. (Units refers to elements with a multiplicative inverse.)