problem set 2

problem set 2 - CSE 105: Automata and Computability Theory...

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Unformatted text preview: CSE 105: Automata and Computability Theory Winter 2012 Problem Set #2 Due: Friday, February 10th, 2012 Problem 1 For a string x, let xR denote the symbol-by-symbol reverse of x. In other words, if x = x1 x2 . . . xn then xR = xn xn-1 . . . x1 . For a language L, let LR denote xR x L . Prove that if L is regular then so is LR . Hint: Given a DFA M accepting L construct an NFA M R accepting LR . Problem 2 Suppose that L1 and L2 are nonregular languages over some alphabet . Is L1 L2 always nonregular? Is L1 L2 always nonregular? Explain. Problem 3 Consider the language of palindromes over = {0, 1}. (Palindromes are strings that remain the same if reversed. For example, "bob" or, ignoring spaces, "lisa bonet ate no basil".) Using the character-by-character operator wR introduced in Problem 1, above, we can write Lbob = {w | w = wR }. a. Prove that the language Lbob is not regular. b. Give a context-free grammar whose language is Lbob . Problem 4 Let L1 be a regular language and L2 be a context-free language over some alphabet . Show that L1 L2 is context-free. Hint: You'll want to work with PDAs here, not CFGs. 1 ...
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