Com S 631: Lower bounds and Separation Results Lecture 6 Scribe: Aaron Sterling Summary: Com S 631: Lower bounds and Separation Results Lecture 6 Scribe: Aaron Sterling 1. Circuits The underlying hardware of a machine is a circuit. Circuits perform the actual computa- tion. So, given a function, can a circuit that is not too large compute it? That is the main question of Circuit Complexity. We will consider Boolean functions f : {0, 1}n {0, 1}. A circuit is a DAG G such that each node is an -node, a -node, a ¬-node or an Input or Output node. The - and -nodes have indegree 2. The ¬-node has indegree 1. The Input node has indegree 0. The Output nodes have outdegree 0. Each Input node is labeled with a variable xi. Definition 1. Given a Boolean function f : {0, 1}n {0, 1}, and a circuit C, we say C computes f if C has n input nodes labeled x1, x2, . . . , xn and for every x = x1x2 . . . xn n , C(x1 . . . xn) = f(x1, . . . , xn). The size of the circuit is the number of nodes in it. Definition 2. Given a function f, CSize(f) = min{|C| | C computes f}. This is the circuit size of f. Definition 3. Given a circuit C, the depth of C, d(C), is the length of the longest path from the length of an input node to an output node. Collections: Computer Technologies and Information Sciences