Symmetric approximation arguments for monotone lower bounds without sunflowers

Abstract.

We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sun ower Lemma that has been essential for the method of approximation. The new approximation argument corresponds to Haken's recent method for proving lower bounds for monotone circuit complexity (counting bottlenecks) in a natural way.¶We provide lower bounds for the BMS problem introduced by Haken, Andreev's polynomial problem, and for Clique. The exponential bounds obtained are the same as those previously best known for the respective problems.