Com S 631: Lower bounds and Separation Results Lecture 9 Scribe: Sanghack Lee Summary: Com S 631: Lower bounds and Separation Results Lecture 9 Scribe: Sanghack Lee 1. complexity measure Definition 1. A formal complexity measure is a function k from Bn to N where Bn is a set of all boolean functions over n such that (1) k (xi) = 1 for all 1 i n, (2) k (f) = k (¬f) for f Bn, and (3) k (f g) k (f) + k (g) for f, g Bn. By the definition and using the rules of deMorgan k (f g) = k ¯f ¯g k (f) + k (g) also holds. Since Fsize satisfies above conditions, it is also a formal complexity measure. In addition, Fsize is the largest formal complexity measure as in the following theorem. Theorem 1. Let k be a formal complexity measure, then for every function f Bn Fsize (f) k (f) . Proof. By induction on = Fsize (f), let's start with the case = 1, if = 1, f (x1, . . . , xn) = xi, Fsize(f) = 1 = k(f) by definition. Let = Fsize (f) > 1 and let F be an optimal formula for f. Consider the formula tree of F. Without loss of generality, the last gate of F is an gate, otherwise the rule of deMorgan can be considered. Let G and H be the subformulas that feed into this gate, and let g and h be the functions computed by them. Thus f = gh. Since F is optimal, G and H are optimal formulas for g and h. Thus FSize(g) = Size(G) Collections: Computer Technologies and Information Sciences