On the complexity of branching programs and decision trees for clique functions

Abstract

Because of the slow progress in proving lower bounds on the circuit complexity of Boolean functions one is interested in restricted models of Boolean circuits like depth restricted circuits, decision trees, branching programs, width-k branching programs and k-times-only branching programs. We prove here exponential lower bounds on the decision tree complexity of clique functions. For one-time-only branching programs we prove for k-clique functions large polynomial lower bounds if k is fixed and exponential lower bounds for k increasing with n. Finally we introduce the hierarchy of the classes BPk(P) of all sequences of Boolean functions which may be computed by k-times-only branching programs of polynomial size. We show constructively that BP1 (P) is a proper subset of BP2(P).