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CS5371 Theory of Computation

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CS5371 Theory of Computation

Homework 2

Due: 2:00 pm, October 31, 2006 (before class)

  1. Let G be a CFG in Chomsky normal form that contains b variables. Show that, if G generates some string with a derivation having at least 2b steps, L(G) is infinite.

  1. Give a formal description and the corresponding state diagram of a PDA that recognizes the language A = {w | 2#a(w) ≠ 3#b(w), w∈{a,b}*}, where #c(w) denotes the number of character c occurring in the word w.

  1. Let C = {xy | x, y ∈ {0,1}*, |x| = |y|, and x ≠ y}. Show that C is a context-free language.

  1. Let A = {wtwR | w,t ∈{0,1}* and |w| = |t|}. Prove that A is not a context-free language.

  1. (Ogden’s Lemma.) There is a stronger version of the CFL pumping lemma known as Ogden’s lemma. It differs from the pumping lemma by allowing us to focus on any p “distinguished” positions of a string z and guaranteeing that the strings to be pumped have between 1 and p distinguished positions. The formal statement of Ogden’s lemma is: Let L be a context-free language. Then there is a constant p such that for any string z in L with at least p characters, we can mark any p or more positions in z to be distinguished, and then z can be written as z = uvwxy, satisfying the following conditions:

        1. vwx has at most p distinguished positions.

        2. vx has at least one distinguished position.

        3. For all i  0, uviwxiy is in L.

Prove Ogden’s lemma. (Hint: The proof is really the same as that of the pumping lemma if we pretend that the non-distinguished positions of z are not present as we select a long path in the parse tree for z.)

  1. (Bonus Question) In this question, we apply Ogden’s lemma and show that the language L = {aibjck | i = j or j = k where i, j, k≥0} is inherently ambiguous.

          1. Suppose that G = (V, T, , S) is a CFG for L, and let p be the constant specified for G in Ogden’s lemma. Assume that p > 3. (We can have this assumption, because Ogden’s lemma holds for p implies that it holds for any q with q > p.) Consider the string z = apbpcp+p! in L. Suppose we mark all the positions of ‘a’ as distinguished. Let u, v, w, x, y be the five parts of z as specified in the Ogden’s lemma. Show that v = at and x = bt for some t.

          1. Following the above step, show that there exists a variable A in V such that S * uAy * uvAxy * ap+p!bp+p!cp+p!.

          1. In a similar manner, find another derivation for ap+p!bp+p!cp+p!, and show that this derivation corresponds to a distinct parse tree from the derivation in Part B. Conclude that L is inherently ambiguous.

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