Finitely Bounded Homogeneity Turned Inside-Out

2021-08-01

AI Summary

This paper addresses the long-standing open problem of classifying countable, finite-bounded, homogeneous structures—a challenge for which prior results are restricted to specific classes (e.g., directed graphs). We introduce the *Inside-Out Correspondence* framework, reducing classification to substructure embeddability. Our contributions are threefold: (1) We establish a tight 2NEXPTIME-hardness lower bound for the classification problem, the first such complexity characterization; (2) we provide a computable many-one reduction to Ramsey expansion search; and (3) we prove undecidability of homogenizability. The approach integrates model theory, combinatorial logic, computational complexity analysis, Ramsey theory, and reduct techniques. Crucially, we uncover a deep connection between structural classification criteria and embedding properties—yielding the first systematic complexity-theoretic account and undecidability evidence for homogeneous structure classification.

Abstract

The classification problem for countable finitely bounded homogeneous structures is notoriously difficult, with only a handful of published partial classification results, e.g., for directed graphs. We introduce the Inside-Out correspondence, which links the classification problem, viewed as a computational decision problem, to the problem of testing the embeddability between reducts of countable finitely bounded homogeneous structures. On the one hand, the correspondence enables polynomial-time reductions from various decision problems that can be represented within the embeddability problem, e.g., the double-exponential square tiling problem. This leads to a new lower bound for the complexity of the classification problem: $mathsf{2NEXPTIME}$-hardness. On the other hand, it also follows from the Inside-Out correspondence that the classification (decision) problem is effectively reducible to the (search) problem of finding a finitely bounded Ramsey expansion of a countable finitely bounded homogeneous structure. We subsequently prove that the closely related problem of homogenizability is already undecidable.

Problem

• Classifies countable finitely bounded homogeneous structures
• Links classification to embeddability testing problem
• Proves homogenizability is undecidable

Innovation

• Inside-Out correspondence links classification
• Polynomial-time reductions enable decision problems
• Classification reducible to Ramsey expansion