Com S 631: Lower bounds and Separation Results Lecture 7 Scribe: Brian Patterson Summary: Com S 631: Lower bounds and Separation Results Lecture 7 Scribe: Brian Patterson Last time, we discussed how a formula can be treated as a directed binary tree with variables or negated variables as the leaves. Given a circuit c, it has size size(c) and depth depth(c). Given a formula F, it has size size(F) and depth depth(F). Given a function f, csize(f) is the size of the smallest circuit that computes f and cdepth(f) is the depth of the circuit with the lowest depth that computes f. Similarly define Fsize(f) and Fdepth(f) for formulas. A key difference between circuits and formulas is that a gate in a circuit can have multiple outputs a gate in a formula can have only one output wire. The parity function illustrates the difference between formulas and circuits. Consider the parity function f(x1, x2, . . . , xn) = x1 x2 . . . xn. As a warm-up, f(x1, x2) = (x1 ¯x2)( ¯x1 x2) is a formula for f with n = 2 attributes. A general circuit Cn for the parity function is given in Figure 1. Cn/2 and Cn/2 represent a circuit for the parity function over the first and last n 2 literals (respectively). This circuit shows that the size(Cn) = 2size(Cn/2) + 5 so size(Cn) = O(n). However, a similar binary tree for a formula would require more copies of Fn/2, where Fn/2 is a formula tree for the parity function over the first and last n Collections: Computer Technologies and Information Sciences