# hw03 - CS 154 Intro to Automata and Complexity Theory...

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CS 154 Intro. to Automata and Complexity Theory Handout 10 Autumn 2006 David Dill October 17, 2006 Problem Set 3 Due: October 24, 2006 Homework: (Total 100 points) Do the following exercises. Problem 1. [15 points] (Exercise 4.2.6(c) on page 147 of the second edition of the textbook) Show that the regular languages are closed under the following operation: init ( L ) = { w | for some x, wx is in L } . Hint : It is easiest to start with a DFA for L and perform a construction to get the desired language. Problem 2. [40 points] Give an algorithm to solve the following deci- sion problems. In each case you can assume that you are given a concrete representation of the language in the form of DFA. You must justify the correctness of the algorithm to receive full credit. a). Suppose L is a regular language over the alphabet Σ = { 0 , 1 } . Give an algorithm to decide whether | L | ≥ 100. b). Suppose L is a regular language over the alphabet Σ = { 0 , 1 } . Give an algorithm to decide whether L contains any string w such that w has zero occurrences of the patterns 00 and 11. Problem 3. [10 points] Prove that { 0 n 1 m 2 n - m | n m 0 } is not a regular language. Problem 4. [10 points]

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