Two-element structures modulo primitive positive constructability

Abstract

Primitive positive constructions have been introduced in recent work of Barto, Opr\v{s}al, and Pinsker to study the computational complexity of constraint satisfaction problems. Let $\mathfrak P_{\operatorname{fin}}$ be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable. In this paper, we give a complete description of the restriction $\mathfrak P_{\operatorname{Boole}}$ of $\mathfrak P_{\operatorname{fin}}$ to relational structures on a two-element set; in particular, we prove that $\mathfrak P_{\operatorname{Boole}}$ is a lattice. Finally, we use $\mathfrak P_{\operatorname{Boole}}$ to present the various complexity regimes of Boolean constraint satisfaction problems that were described by Allender, Bauland, Immerman, Schnoor and Vollmer.