On The Applicability Of Post'S Lattice

For decision problems Pi(B) defined over Boolean circuits using gates from a restricted set B only, we have Pi(B) <=(AC0)(m) Pi(B') for all finite sets B and B' of gates such that all gates from B can be computed by circuits over gates from B'. In this note, we show that a weaker version of this statement holds for decision problems defined over Boolean formulae, namely that Pi(B) <=(NnC2)(m) Pi(B'boolean OR{boolean AND, boolean OR}) and Pi(B) <=(NC2)(m) Pi(B'boolean OR{0,1}) for all finite sets B and B' of Boolean functions such that all f is an element of B can be defined in B'. (C) 2012 Elsevier B.V. All rights reserved.