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problem set 4

problem set 4 - CSE 105 Automata and Computability Theory...

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Unformatted text preview: CSE 105: Automata and Computability Theory Winter 2012 Problem Set #4 Due: Wednesday, March 14th, 2012 Problem 1 Show that the following language is decidable: L= A A is a DFA and L(A) includes at least one non-palindrome . (Recall: a palindrome is a string x such that x = xR ; a non-palindrome is a string x such that x = xR .) Hint: You may use, without needing to prove, the result of Problem 4 on PS#2: The intersection of a regular language with a context-free language is context-free. Problem 2 NOTSTATE be the language M, w, q M is a Turing machine, w is a string, and q is a state; and M , when run on input w, never enters the state q. . Show that NOSTATE is undecidable. Hint: Assume that NOSTATE is decidable, and use a decider for NOSTATE to decide the acceptance problem ATM , yielding a contradiction. Problem 3 Give a mapping function f showing that ETM m L, where ETM is the emptyness problem for Turing machines and L is the language M, w M is a Turing machines and L(M ) = {w} . Explain why your choice of f gives a correct mapping reduction. Problem 4 Give an algorithm that recognizes EQCFG . (In other words, give a constructive proof that EQCFG is co-Turing-recognizable aka co-R.E.) Note: Give the algorithm for the recognizer. You may use deciders for any of the decidable languages we covered in class as subroutines without recalling the details of those deciders. 1 ...
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