lecture9 - A CIRCUIT CSci 5403 COMPLEXITY THEORY LECTURE IX...

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1 COMPLEXITY THEORY CSci 5403 LECTURE IX: CIRCUITOUS COMPUTING A CIRCUIT Each node has a label 2 {x 1 ,…,x n , Æ , Ç , ¬ } ¬ x 0 x 1 x 2 A circuit C with n inputs and m outputs computes a function ƒ C : {0,1} n ! {0,1} m . Is a directed acyclic graph, where: Each node has in-degree either 0 (inputs), 1 ( ¬ ), or m ( Æ , Ç ) Nodes with out-degree 0 are the outputs of the circuit. Definition. A circuit family is a sequence of circuits C 1 , C 2 , C 3 , C 4 , … where for each n, C n has n inputs. We denote the family as {C n } n 2 N . The language of a circuit family is the set L, where x 2 L , C |x| (x) = 1. The size of a circuit, |C|, is the number of gates. The size of a circuit family is the function s(n) = |C n | Definition. A language is in SIZE(t(n)) if it is decided by a circuit family of size O(t(n)). P/poly = c 2 N SIZE(n c ) Theorem. P µ P/poly.
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2 Given TM M and time bound t, we create a circuit that takes n input bits and runs up to t steps of M.
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