hwsol5.4 - Ted Kaminski CSci 5403 Computational Complexity...

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Unformatted text preview: Ted Kaminski CSci 5403 - Computational Complexity Homework 5 Problem 4 Solution Describe an interactive proof system ( P,V ) for set upper bounds: given a circuit C S that samples uniform elements of the set S , (i.e., for any string r , C S ( R ) S and for all s S , Pr r [ C S ( r ) = s ] = 1 / | S | .) a circuit C V that recognizes elements of S , and a claimed bound K on | S | , | S | > 2 K Pr [ V accepts ( K,C S ,C V )] 1 3 and | S | < K Pr [ V accepts ( K,C S ,C V )] 2 3 . Hint: Use hashing: how many elements of S hash to a given output? The correct protocol does not depend on the fact that C V and C S are circuits, except to bound the running time, and should call C S exactly once. Solution Idea: We will use the circuit C S backwards and find the preimage of some random element of S , and lower bound the size of this resulting set, which will correspond to a upper bound on the size of the set S ....
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