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problem set 1

# problem set 1 - CSE 105 Automata and Computability Theory...

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CSE 105: Automata and Computability Theory Winter 2012 Problem Set #1 Due: Monday, January 30th, 2012 Problem 1 Let M = ( Q, Σ , δ, q 0 , F ) be a DFA. Let w = w 1 w 2 · · · w n be a string accepted by M , for which let r 0 , r 1 , . . . , r n be the corresponding accepting path. (See the section titled “Formal Definition of Computation,” page 40, in Sipser.) Suppose that r i = r j for some i and j such that i < j . Prove that, in addition to accepting w , M also accepts some string w 0 that is shorter than w , i.e., such that | w 0 | < | w | . Problem 2 In class, we showed that swapping the accepting and nonaccepting states of a DFA whose language is L gives a DFA whose language is ¯ L = Σ * \ L . a. Show (by construction) that swapping the accepting and nonaccepting states of an NFA whose language is L does not necessarily give an NFA whose language is ¯ L . Hint: There are examples with a very small number of states. b. Explain, given an NFA whose language is L , how to construct another NFA whose language is ¯ L . Problem 3 In this problem, we consider a generalization of DFAs called second-order deter- ministic finite automata, or DFA
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