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# hw3 - CSci 5403 Spring 2010 Homework 3 due March 2 2010 1...

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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 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. 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 polynomial-size, O ((log n ) i )-depth circuit family, with unbounded fan-in 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 fan-in threshold gates — gates with a specified threshold k

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• Spring '08
• Sturtivant,C
• Computational complexity theory, Tci, threshold gates, unbounded fan-in gates, fan-in threshold gates, arbitrary threshold gates

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hw3 - CSci 5403 Spring 2010 Homework 3 due March 2 2010 1...

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