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 prove the Inside-Out correspondence, which states that the classification problem, viewed as a computational decision problem, is polynomial-time equivalent to the problem of testing the containment between finitely bounded amalgamation classes up to taking reducts to a given subset of their signatures. The correspondence enables polynomial-time reductions from various decision problems that can be represented within such containment, e.g., the double-exponential square tiling problem. This leads to a new lower bound for the complexity of the classification problem: $\textsf{2NEXPTIME}$-hardness. On the other hand, it also follows from the correspondence that the classification (decision) problem is effectively reducible to the (search) problem of finding a finitely bounded Ramsey expansion of a finitely bounded strong amalgamation class. We subsequently prove that the closely related problem of homogenizability is already undecidable.

翻译：可数有限有界齐性结构的分类问题因其难度而著称，目前仅发表了少量部分分类结果，例如针对有向图的分类。我们证明了内外对应关系：该分类问题作为计算决策问题，在多项式时间等价于检验有限有界融合类之间的包含关系（需考虑对签名给定子集进行约化）。该对应关系使得多种可表示为此类包含关系的决策问题（例如双指数平方铺砖问题）能够进行多项式时间归约。由此得到分类问题复杂性的新下界：$\textsf{2NEXPTIME}$困难性。另一方面，根据对应关系可推知，分类（决策）问题可有效归约为寻找有限有界强融合类的有限有界拉姆齐扩张这一（搜索）问题。我们进一步证明，与之密切相关的可齐化问题本身是不可判定的。