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# hw07 - CS 154 Intro to Automata and Complexity Theory...

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CS 154 Intro. to Automata and Complexity Theory Handout 29 Autumn 2009 David Dill November 17, 2009 Problem Set 7 Due: December 1, 2009 Homework: (Total 100 points) Do the following exercises. Problem 1. [15 points] (Derived from Exercise 7.3.2 in the textbook.) Consider the following two languages: L 1 = { a n b 2 n c m | n,m 0 } L 2 = { a n b m c 2 m | n,m 0 } a) Show that each language is context-free by writing a context-free gram- mar for it. b) Show that L 1 L 2 is not context-free using closure properties and lan- guages from the lecture or textbook that are known not to be context- free (don’t use the context-free pumping lemma). Problem 2. [15 points] (From Exercise 7.4.3 (b,c) in the textbook.) Using the grammar G of Example 7.34, use the CYK algorithm to de- termine whether each of the following is in L ( G ): baaab aabab The grammar G is: S AB | BC A BA | a B CC | b C AB | a Problem 3. [15 points] (Derived from Exercise 11.1.1 in the textbook.) Below are some problems. For each, tell whether it is in NP and whether it is in co-NP. Describe the complement of each problem. If either the problem

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