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# Automata - 6.045J/18.400J Automata Computability and...

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6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch Practice Quiz 1-Solutions Elena Grigorescu Please write your name in the upper corner of each page. PQ1-1

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Problem 1 : True or False (20 points) Full credit will be given for correct answers. If you include justification for your answers, you may obtain partial credit for incorrect answers. In all parts of this question, the alphabet Σ is { 0 , 1 } . 1. True or False: A DFA with n states must accept at least one string of length greater than n . False. The DFA might accept no strings at all. 2. True or False: A DFA with n states that accepts an infinite language must accept at least one string x such that 2 n < | x | ≤ 3 n . True. Follows by a pumping lemma argument. The DFA has to accept at least one string longer than 3 n . Now, pump this down. In each step, we pump down by at most n , and at least 1 . It follows that if one does this sufficiently many times, one gets a string with length in the range [2n. . . 3n]. 3. If R is a regular language and L is some language, and L R is a regular language, then L must be a regular language. False. Let R = Σ and L be some non-regular language. 4. If F is a finite language and L is some language, and L - F is a regular language, then L must be a regular language. True. L = ( L - F ) F , where F is some subset of F , and is therefore finite. The statement now follows from the closure under union of regular languages. 5. True or False: Define
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Automata - 6.045J/18.400J Automata Computability and...

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