On the size of probabilistic formulae

Abstract

We investigate size complexity for different models of probabilistic formulae. Relating formula-size to one-way communication complexity we devise a lower bound method for probabilistic formulae based on the VC-dimension and the Nečiporuk lower bound and give for the first time lower bounds on the size of 2-sided bounded error, Monte Carlo, and Las Vegas formulae over an arbitrary basis. We show that for the Boolean matrix product Monte Carlo probabilistic formulae are smaller by a factor of Ω/trn than Las Vegas formulae and prove an analogous gap between two sided bounded error and Monte Carlo formulae. This is the maximal gap between probabilistic and deterministic formulae provable using the Nečiporuk method. We also consider a function for which two-sided-error probabilism does not help. Furthermore we investigate the effect of restricted access to randomness on formula size, and study generalizations of matrix products which may be candidates for proving larger gaps between probabilism and determinism.