On the applicability of Postʼs lattice

For decision problems Π(B)Π(B) defined over Boolean circuits using gates from a restricted set B   only, we have Π(B)⩽mAC0Π(B′) for all finite sets B   and B′B′ of gates such that all gates from B   can be computed by circuits over gates from B′B′. In this note, we show that a weaker version of this statement holds for decision problems defined over Boolean formulae, namely that Π(B)⩽mNC2Π(B′∪{∧,∨}) and Π(B)⩽mNC2Π(B′∪{0,1}) for all finite sets B   and B′B′ of Boolean functions such that all f∈Bf∈B can be defined in B′B′.

► Base-independence results concerning decision problems defined over Boolean formulae;
► We classify when such problems are independent of the set of available connectives;
► Results are obtained with respect to NC2NC2 many-one reductions.