A Dichotomy Theorem for the Inverse Satisfiability Problem

Victor Lagerkvist, Biman Roy, Marc Herbstritt
2018 Foundations of Software Technology and Theoretical Computer Science  
The inverse satisfiability problem over a set of Boolean relations Γ (Inv-SAT(Γ)) is the computational decision problem of, given a relation R, deciding whether there exists a SAT(Γ) instance with R as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite Γ containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that Inv-SAT(Γ) is always either tractable or co-NP-complete for
more » ... all finite sets of relations Γ, thus solving a problem open since 1998. Very few of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when Γ is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite Γ such that Inv-SAT(Γ) is tractable even though there exists finite ∆ ⊂ Γ such that Inv-SAT(∆) is co-NP-complete.
doi:10.4230/lipics.fsttcs.2017.39 dblp:conf/fsttcs/LagerkvistR17 fatcat:uzdeleecbzcttpqqhg3lwdvsey