On the computational power of Boolean decision lists

Abstract.

We study the computational power of decision lists over AND-functions versus $\hbox{threshold-}\oplus$ circuits. AND-decision lists are a natural generalization of formulas in disjunctive or conjunctive normal form. We show that, in contrast to CNF- and DNF-formulas, there are functions with small AND-decision lists which need exponential size unbounded weight $\hbox{threshold-}\oplus$ circuits. Consequently, it is questionable if the polynomial learning algorithm for DNFs of Jackson (1994), which is based on the efficient simulation of DNFs by polynomial weight $\hbox{threshold-}\oplus$ circuits (Krause & Pudlák 1994), can be successfully applied to functions with small AND-decision lists. A further result is that for all k ≥ 1 the complexity class defined by polynomial length AC ⁰ _k -decision lists lies strictly between AC ⁰ _k+1 and AC ⁰ _k+2.