ps5 - CS 151 Complexity Theory Spring 2011 Problem Set 5...

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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 write-up 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 3-CNF 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 3-DNF form if it is the disjunction of terms , with each term being the conjunction of at most 3 literals. Describe a polynomial-time computable function that is given as input a fan-in two ( , , )- circuit C ( x 1 ,x 2 ,...,x n ), and produces a 3-CNF 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 polynomial-time computable function that is given as input a fan-in two ( , , )-circuit C ( x 1 ,x 2 ,...,x n ), and produces a 3-DNF 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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ps5 - CS 151 Complexity Theory Spring 2011 Problem Set 5...

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