On convex complexity measures

Khrapchenko's classical lower bound n^2 on the formula size of the parity function f can be interpreted as designing a suitable measure of sub-rectangles of the combinatorial rectangle f^-^1(0)xf^-^1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n^2) and show that several measures considered for proving lower bounds on the formula size are convex. We also prove quadratic upper bounds on a class of measures that are not necessarily convex.

[1]  Avi Wigderson,et al.  Monotone Circuits for Connectivity Require Super-Logarithmic Depth , 1990, SIAM J. Discret. Math..

[2]  Troy Lee.  A New Rank Technique for Formula Size Lower Bounds , 2007, STACS.

[3]  V. M. Khrapchenko.  Method of determining lower bounds for the complexity of P-schemes , 1971 .

[4]  Johan Hå stad.  The Shrinkage Exponent of de Morgan Formulas is 2 , 1998 .

[5]  Eyal Kushilevitz,et al.  Fractional Covers and Communication Complexity , 1995, SIAM J. Discret. Math..

[6]  Alfred V. Aho,et al.  On notions of information transfer in VLSI circuits , 1983, STOC.

[7]  Nathan Linial,et al.  Lower bounds in communication complexity based on factorization norms , 2007, STOC '07.

[8]  Michael E. Saks,et al.  Quantum query complexity and semi-definite programming , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[9]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[10]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[11]  Troy Lee,et al.  THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[12]  Mauricio Karchmer,et al.  Communication complexity - a new approach to circuit depth , 1989 .

[13]  Johan Håstad.  The shrinkage exponent is 2 , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[14]  Troy Lee,et al.  Negative weights make adversaries stronger , 2007, STOC '07.

[15]  Alexander A. Razborov,et al.  Applications of matrix methods to the theory of lower bounds in computational complexity , 1990, Comb..

[16]  Elias Koutsoupias,et al.  Improvements on Khrapchenko's theorem , 1993, Theor. Comput. Sci..

[17]  A. Razborov.  On submodular complexity measures , 1992 .

[18]  Mario Szegedy,et al.  Quantum Decision Trees and Semidefinite Programming. , 2001 .