A New Rank Technique for Formula Size Lower Bounds

We exactly determine the formula size of the parity function. If n = 2 + k, where 0 ≤ k < 2 , then the formula size of parity on n bits is 2 (2 + 3k) = n2 + k2 − k2 . Khraphchenko 1971 had previously shown a n2 lower bound on the formula size of parity—our result shows that when n is not a power of two parity requires larger formulas, and in fact that limn→∞ sup of the formula size of parity is (9/8)n2 . To obtain this result, we introduce a new technique for proving formula size lower bounds based on matrix rank. This result cannot be proven by any of the lower bound techniques of Khrapchenko, Neˇciporuk, Koutsoupias, or the quantum adversary method, which are all limited by n2 .

• Type: Conference
• Year: 2007
• Where: STACS
• Authors: Troy Lee
• Keywords: Formula Size, Lower Bound, Parity Function, STACS 2007, Theoretical Computer Science