A Dichotomy for First-Order Reducts of Unary Structures

TL;DR

This paper establishes a polynomial-time reduction from infinite-domain CSPs to finite-domain CSPs, introduces new tractability conditions based on topological polymorphisms, and proves a major conjecture for a broad class of structures.

Contribution

It provides a general reduction technique and new tractability conditions for CSPs of reducts of finitely bounded homogeneous structures, extending the class of known tractable problems.

Findings

Polynomial-time reduction from infinite to finite domain CSPs.

New tractability conditions based on topological polymorphism clones.

Proof of the tractability conjecture for reducts of finitely bounded homogeneous structures within class .

Abstract

Many natural decision problems can be formulated as constraint satisfaction problems for reducts $\mathbb{A}$ of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of the topological polymorphism clone of $\mathbb{A}$. Moreover, we study the subclass $\mathcal{C}$ of CSPs for structures $\mathbb{A}$ that are reducts of a structure with a unary language. Also this class $\mathcal{C}$ properly extends the class of all finite-domain CSPs. We apply our new tractability conditions to prove the general tractability conjecture of Bodirsky and Pinsker for reducts of finitely bounded homogeneous structures for the class $\mathcal{C}$.