The Bodirsky-Pinsker conjecture asserts a P vs. NP-complete dichotomy for the computational complexity of Constraint Satisfaction Problems (CSPs) of first-order reducts of finitely bounded homogeneous structures. Prominently, two structures in the scope of the conjecture have log-space equivalent CSPs if they are pp-bi-interpretable, or equivalently, if their polymorphism clones are topologically isomorphic. The latter gives rise to the algebraic approach which regards structures with topologically isomorphic polymorphism clones as equivalent and seeks to identify structural reasons for hardness or tractability in topological clones. We establish that the equivalence relation of pp-bi-interpretability underlying this approach is reasonable: On the one hand, we show that it is decidable under mild conditions on the templates; this improves a theorem of Bodirsky, Pinsker and Tsankov (LICS'11) on decidability of equality of polymorphism clones. On the other hand, we show that within the much larger class of transitive $ω$-categorical structures without algebraicity, the equivalence relation is of lowest possible complexity in terms of descriptive set theory: namely, it is smooth, i.e., Borel-reduces to equality on the real numbers. On our way to showing the first result, we establish that the model-complete core of a structure that has a finitely bounded Ramsey expansion (which might include all structures of the Bodirsky-Pinsker conjecture) is computable, thereby providing a constructive alternative to previous non-constructive proofs of its existence.

翻译：Bodirsky-Pinsker猜想断言，对于有限有界齐次结构的一阶归约的约束满足问题（CSP），其计算复杂度存在P与NP完全的二相性。该猜想范围内一个显著结论是：若两个结构是pp-双可解释的，或其多态克隆是拓扑同构的，则它们的CSP在对数空间意义下等价。后者催生了代数方法，该方法将具有拓扑同构多态克隆的结构视为等价，并试图在拓扑克隆中识别硬度或易处理性的结构性原因。我们证明了支撑该方法的pp-双可解释等价关系具有合理性：一方面，我们证明在模板满足温和条件时该关系是可判定的，这改进了Bodirsky、Pinsker和Tsankov（LICS'11）关于多态克隆相等性可判定性的定理。另一方面，我们证明在更大的无代数性的传递性$ω$-范畴结构类中，该等价关系在描述集合论意义下具有最低可能的复杂度：即它是光滑的，亦即可Borel归约到实数上的相等关系。在证明第一个结果的过程中，我们确立了具有有限有界Ramsey扩张的结构（可能包含Bodirsky-Pinsker猜想的所有结构）的模型完全核是可计算的，从而为其存在性提供了构造性证明，替代了以往的非构造性证明。