 
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)
