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381-2007fa-final-solutions

381-2007fa-final-solutions - CS 381 Fall 2007 Final Exam...

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CS 381 Final Exam Thursday Dec 13 Fall 2007 Closed Book 7:00 pm to 9:30 pm Hollister B14 Partial credit will depend on clarity and conciseness of your answers. Please do not put down correct but irrelevant information. 1. (a) Write a regular expression for the set of all strings of a ’s and b ’s in which each a is immediately preceded and immediately followed by a b . (b) Write a regular expression for all strings of a ’s and b ’s which do not contain the substring aab . Solution : (a) ( ) * * * b ba bb ε + (b) ( ) * * * * b abb a 2. Prove that the class of regular sets is closed under inverse homomorphisms. Two or three sentences should suffice. Solution : Construct a finite automaton ' M that applies the homomorphism to each input symbol and feeds the resulting string to the finite automata M accepting the regular set. Then ( ) ( ) ( ) 1 ' L M h L M = . 3. Write a context-free grammar for the set { } | 0 and i-j is even i j L a b i j = . Solution : S aSb S T T aaT T ε . 4. Is the set ( ) { } | * R L ww w w a b = + a regular set, a context-free language that is not a regular set, or not a context-free language. Prove your answer using a construction, closure properties, and/or the pumping lemma. You cannot use the fact that any language is known not to be regular or context-free.

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