A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints

We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan (1995) and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko (1971). Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy (2006)

A new rank technique for formula size lower bounds

We introduce a new technique for proving formula size lower bounds based on matrix rank. A simple form of this technique gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an��lower bound on the parity function. Applying our method to the parity function, we are able to give an exact expression for the formula size of parity: if������, where������, then the formula size of parity on �bits is exactly��������������� �  ��. Such a bound cannot be proven by any of the lower bound techniques of Khrapchenko, Nečiporuk, Koutsoupias, or the quantum adversary method, which are limited by��.

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..