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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Problem Set 5 Out: May 5 Due: May 12 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own writeup and note with whom you worked. You may consult the course notes and the text (Papadimitriou). Please attempt all problems. To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. 1. CNFs and DNFs. Recall that a Boolean formula is said to be in 3CNF form if it is the conjunction of clauses , with each clause being the disjunction of at most 3 literals. A Boolean formula is said to be in 3DNF form if it is the disjunction of terms , with each term being the conjunction of at most 3 literals. Describe a polynomialtime computable function that is given as input a fanin two ( , , ) circuit C ( x 1 ,x 2 ,...,x n ), and produces a 3CNF Boolean formula on variables x 1 ,x 2 ,...,x n and additional variables z 1 ,z 2 ,...,z m for which for every setting of the x variables z 1 ,z 2 ,...,z m ( x 1 ,x 2 ,...,x n ,z 1 ,z 2 ,...,z m ) = 1 C ( x 1 ,x 2 ,...,x n ) = 1 . Also, describe a polynomialtime computable function that is given as input a fanin two ( , , )circuit C ( x 1 ,x 2 ,...,x n ), and produces a 3DNF Boolean formula on variables x 1 ,x 2 ,...,x n and additional variables z 1 ,z 2 ,...,z m for which for every setting of the x vari ables z 1 ,z 2 ,...,z m ( x 1 ,x 2 ,...,x n ,z 1 ,z 2 ,...,z m ) = 1 C ( x 1 ,x 2 ,...,x n ) = 1 . Hint: identify the z variables with the gates of C ....
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 Fall '09

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