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Theory of Computation 18.404/6.840 Fall, 2006 Michael Sipser Sample MidTerm Examination Questions 1. (a) Let � = {a, b, c} and let A = {ai bj ck | i, j, k � 0, and i = j or i = k}. Describe (in English) a pushdown automaton that recognizes A. (b) Let R be the regular expression �� 1100�� where � = {0, 1}. Let D = L(R) and let E = D, the complement of D. Give the state diagram of a DFA with at most 5 states that recognizes E. 2. Let � = {(, )} and let P be the language consisting of all strings of properly nested parentheses. For example, P contains “()()”, “((()))”, “(()((()(()))))” and “�”, but not “)(” and “((((”. (a) Give a CFG that generates P . (b) Show that P is not a regular language. 3. (a) Let A = {ai bj ci | i � j � 2i}. Prove that A is not a context-free language. (b) Let B = {ai bj | i � j � 2i}. Give an unambiguous context-free grammar generating B. 4. Let D = {≥M ∩| M is a TM that accepts the input string 101}. (a) Show that D is undecidable. (Do not use Rice’s theorem. If you don’t know Rice’s theorem, ignore this comment.) (b) Show that the complement of D is not Turing-recognizable. 5. A 2-way pushdown automaton (2WAY-PDA) is a nondeterministic pushdown automaton that has a single stack and that can move its input head in both directions on the input tape. In addition we assume that a 2WAY-PDA is capable of detecting when its input head is at either end of its input tape. A 2WAY-PDA accepts its input by entering an accept state. (a) Show that a 2WAY-PDA can recognize the language {am bm cm | m � 0}. (b) Let E2WAY-PDA = {≥P ∩| P is a 2WAY-PDA which recognizes the empty language}. Show that E2WAY-PDA is not decidable. 6. Consider the inﬁnite two-dimensional grid, G = {(m, n)| m and n are integers}. Every point in G has 4 neighbors, North, South, East, and West, obtained by varying m or n by ±1. Starting at the origin (0, 0), a string of commands N, S, E, W, generates a path in G. For example, the string NESW, generates a path clockwise around a unit square touching the origin. Say that a path is closed if it starts at the origin and ends at the origin. Let C be the collection of all strings over � = {N, S, E, W} that generate a closed path. (a) Give a clear mathematical description of C as a language. (b) Describe in English two CFLs, A and B, such that C = A � B. Give a CFG that generates A. (c) Prove that C is not context-free. 7. Let � = {0, 1}. Consider the problem of testing whether a PDA accepts some string of the form {w| w ≤ 0� 1� } . Is this problem decidable? Prove your answer.