On the Applicability of Post's Lattice

TL;DR

This paper investigates the reducibility of decision problems over Boolean formulas and circuits within Post's lattice, showing specific reductions between classes defined by different gate sets and their computational complexities.

Contribution

It establishes new reductions between classes of Boolean formula decision problems, extending known results from circuits to formulas and involving weaker complexity classes.

Findings

P(B) NC2 many-one reduces to P(B' union {and,or})

P(B) NC2 many-one reduces to P(B' union {false,true})

Reductions hold for all finite sets B and B' with definability conditions

Abstract

For decision problems P defined over Boolean circuits from a restricted set of gates, we have that P(B) AC0 many-one reduces to P(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 paper, we show that a weaker version of this statement holds for decision problems defined over Boolean formulae, namely that P(B) NC2 many-one reduces to P(B' union {and,or}) and that P(B) NC2 many-one reduces to P(B' union {false,true}), for all finite sets B and B' of Boolean functions such that all f in B can be defined in B'.