Finitely Bounded Homogeneity Turned Inside-Out

TL;DR

This paper investigates the computational complexity of the amalgamation property for finitely bounded classes, establishing 2NEXPTIME-hardness and exploring decidability under certain conditions, with implications for classifying homogeneous structures.

Contribution

It links the amalgamation decision problem to reduct containment, providing complexity bounds and decidability results, and shows testing homogenizability is undecidable.

Findings

Amalgamation decision problem is 2NEXPTIME-hard.

Decidability depends on computable Ramsey expansions.

Testing homogenizability is undecidable.

Abstract

Deciding the amalgamation property for a given class of finite structures is an important subroutine in classifying countable finitely homogeneous structures. We study the computational complexity of the amalgamation decision problem for finitely bounded classes, i.e., classes specified by a finite set of forbidden finite substructures, or equivalently by a finite set of universal axioms. We link the amalgamation decision problem to the problem of testing the containment between the reducts of two given finitely bounded amalgamation classes to a given common subset of their signatures. On the one hand, this link enables polynomial-time reductions from various decision problems that can be represented within the reduct containment problem for finitely bounded amalgamation classes, e.g., the 2-exponential square tiling problem, leading to a new lower bound for the complexity of the amalgamation decision problem: 2NEXPTIME-hardness. On the other hand, the link also allows us to show that the amalgamation decision problem is decidable under the assumption that every finitely bounded strong amalgamation class has a computable finitely bounded Ramsey expansion. The runtime of our conditional decision procedure depends 2-exponentially on the size of a minimal Ramsey expansion. We subsequently prove that the closely related problem of testing homogenizability is already undecidable, by a polynomial-time reduction from the regularity of context-free languages. Our results indicate that the relationship between finitely bounded amalgamation classes and arbitrary finitely bounded classes shares similarities with the relationship between regular grammars and context-free grammars. A key difference is that the regularity of context-free grammars can be tested in linear time, while the problem of testing the amalgamation property for finitely bounded classes is 2NEXPTIME-hard.