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# 20111025s - Theory of Computation Homework 2 Problem 1...

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Theory of Computation Homework 2 Problem 1. Given a Boolean expression ϕ = (( a b ) ( c ( d e ))) ( a f ) . (a) Turn ϕ into a CNF. (b) Illustrate a Boolean circuit for CNF. Ans: (a) By implication, ϕ 1 ϕ 2 = ¬ ϕ 1 ϕ 2 , ϕ = ( ¬ ( a b ) ( c ( d e ))) ( a f ) = ( ¬ ( a b ) ( c ( ¬ d e ))) ( a f ) = ( ¬ ( a b ) ( c ( ¬ d e ))) ( ¬ a f ) . By De Morgan’s laws, ¬ ( ϕ 1 ϕ 2 ) ( ¬ ϕ 1 ∨ ¬ ϕ 2 ), ϕ = ( ¬ ( a b ) ( c ( ¬ d e ))) ( ¬ a f ) = ( ¬ a ∨ ¬ b ( c ( ¬ d e ))) ( ¬ a f ) . Finally, the CNF of ϕ is ϕ = ( ¬ a ∨ ¬ b c ∨ ¬ d e ) ( ¬ a f ) . (b) A Boolean circuit is as follows:

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Problem 2. If f ( n ) and g ( n ) are proper complexity functions, sketch proofs that show the following items are proper complexity functions: (a) f ( g ), (b) f + g , (c) f · g , (d) 2 g . Proof. Assume that f and g are computed by TMs M f and M g , respectively. (a) Simulate M g , storing the “output” on a work tape, and then simulate M f (using a different set of tapes), using that work tape as input. Note that f ( n ) n has to be satisfied. (b) Simulate M f , then simulate M g . The outputs will be concatenated to- gether, and so the output will be of length
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20111025s - Theory of Computation Homework 2 Problem 1...

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