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Unformatted text preview: CSci 5403, Spring 2010 Stefan NelsonLindall Homework 3 Example solution 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 — gates with a specified threshold k that output true iff at least k of their inputs are true — and ¬ gates. (a) Prove that for every i , AC i ⊆ TC i . Pf: An unbounded fanin gate ∨ is equivalent to an unbounded fanin threshold gate with k = 1. Similarly, an unbounded fanin gate ∧ may be expressed using an unbounded fanin threshold gate with k = m , where m is the number of inputs to the gate. Any circuit in AC is equivalent to a circuit in TC , without increasing the depth, hence AC i ⊆ TC i . (b) Show that in order to implement arbitrary threshold gates, it is sufficient to have ma jority gates, that output true iff (strictly) more than half of their inputs are true....
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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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