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hwsol3.3 - CSci 5403 Spring 2010 Stefan Nelson-Lindall...

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CSci 5403, Spring 2010 Stefan Nelson-Lindall 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 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 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 fan-in gate is equivalent to an unbounded fan-in threshold gate with k = 1. Similarly, an unbounded fan-in gate may be expressed using an unbounded fan-in 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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