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hwsol3.1

# hwsol3.1 - Ted Kaminski CSci 5403 Computational Complexity...

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Ted Kaminski CSci 5403 - Computational Complexity Homework 3 Problem 1 Solution 1 Problem statement Show that for every k > 0, PH contains languages whose circuit complexity is Ω( n k ). Hint: Recall that by the size hierarchy theorem, there exist languages with circuit complexity Ω( n k ). Write a PH sentence that ensures that a language is one of these. 2 Solution Recall the size hierarchy theorem, which states: For every T 1 , T 2 , with 2 n /n > T 1 ( n ) > 10 T 2 ( n ) > n , SIZE ( T 2 ( n )) SIZE ( T 1 ( n )). Our goal is to describe a language L k PH that, given some k , (deterministically) finds a unique circuit (guaranteed to exist by the above theorem) of minimum size n k , and evaluates the input on that circuit. We will, in fact, do this with a language that is in Σ p 3 . The trick is relatively simple: we can use quantifiers to range over all possible circuits, and simply select one with the desired properties. Given some constant k , define the language L k = { x |∃ C 1 C 2 , C 3 Y 1 , Y 2 .R ( x, C 1 , C 2 , C 3 , Y 1 , Y 2 ) } Where C 1 and C 3

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hwsol3.1 - Ted Kaminski CSci 5403 Computational Complexity...

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