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Unformatted text preview: CSci 5403, Spring 2010 Homework 3 due: March 2, 2010 1. PH lower bounds. Show that for every k > 0, PH contains languages whose circuit complexity is ( n k ). Hint: Recall that by the size hierarchy theorem, there exist languages with circuit complexity ( n k ). Write a PH sentence that ensures that a language is one of these. 2. Compressed circuits Prove that every language has circuits of size O (2 n /n ). Hint: The circuit for input size n might take a very long time to build. One approach to this problem treats the first lg n 1 bits separately from the rest. 3. Shallow circuits. Besides NC , two other important classes of circuits that are often discussed in complexity theory are AC and TC . A language is in AC i if it is decided by a polynomialsize, O ((log n ) i )depth circuit family, with unbounded fanin gates in the set { , , } ; and a language is in TC i if it is decided by a polynomial size, O ((log n ) i )depth circuit family using unbounded fanin threshold gates...
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This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.
 Spring '08
 Sturtivant,C

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