The Complexity of Satisfiability Problems: Refining Schaefer's Theorem

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P 6= NP). We show that if one considers AC isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP,P,⊕L,NL, and L are all distinct).