# Two-element structures modulo primitive positive constructability

## TL;DR

This paper characterizes the structure of pp-constructability among two-element relational structures and uses this to clarify the complexity landscape of Boolean constraint satisfaction problems.

## Contribution

It provides a complete description of the pp-constructability poset restricted to two-element structures and proves it forms a lattice, linking it to Boolean CSP complexity regimes.

## Key findings

- The poset of two-element structures under pp-constructability is a lattice.
- The structure helps explain the complexity regimes of Boolean CSPs.
- The poset is countably infinite, but its detailed structure was previously unknown.

## 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.