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Unformatted text preview: CS 154  Introduction to Automata and Complexity Theory Autumn Quarter, 20092010 SAMPLE FINAL SOLUTIONS Note: This final was from CS154 in the Autumn quarter of 20089. In retrospect, it was unusually difficult, so we intend to make the upcoming final a little less challenging. However, there was new material in CS154 this quarter that wasn’t covered last quarter, so some of that will be on the upcoming final although it does not appear here. Look at the questions on the last homework for an idea of what might appear. Instructions: Answer all 6 questions in the space provided. You have a total time of 3 hours and the maximum possible score is 150. The point allocation is roughly proportional to the time required to solve a problem, i.e., 1 point for each minute, with a halfhour left over to reduce time pressure. We strongly recommend that you use a few minutes to go over the exam and plan your strategy. It is important that you be brief in your answers and ensure that it fits in the space provided. You will lose points for a complicated solution, even if it is correct. This exam is an openbook and opennotes exam, i.e., you are allowed to con sult both the textbook, your class notes and homeworks, and any of the handouts from us. You are not permitted to refer to any other material during the exam (including, of course, online material). Write proofs on scratch paper first but do not turn those in. You may use the reverse side of each page for extra space. Please print your name and sign the pledge before you start. Name: Pledge: I acknowledge and accept the honor code. Problem Points Score Problem Points Score 1 30 4 25 2 20 5 25 3 25 6 25 TOTAL 150 1 Problem 1. [30 points] Decide if the following statements are TRUE or FALSE, and let us know by circling the appropriate box. Also write one line of justification after each answer in the space provided. You will receive 3 points for each correct answer. There are no negative points. Think carefully before answering. (a) T F If L is a regular language, the minimumstate DFA for L R always has exactly the same number of states as the minimumstate DFA for L . False. The DFA for L R might have exponentially more (or fewer) states. (b) T F For every language L in P , there is a polynomialtime reduction from L to the language { ǫ } (i.e., problem instances are strings w , and the decision problem is: “Is w = ǫ ?”). True. The reduction is: Solve the problem in PTIME. If the answer is “yes,” let x = ǫ , and if the answer is “no,” output x = 0 . The reduced problem will have x = ǫ iff the original answer is “yes.” (c) T F If L is not a contextfree language, then L R cannot be contextfree....
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 '08
 Motwani,R
 Formal language, Recursively Enumerable Languages

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