可判定性与可解释性

AI总结：

研究证明pp-可解释性等价关系在特定条件下可判定，并在描述集合论中具有最低复杂性，同时提供构造性方法证明模型核心的存在。

摘要：

Bodirsky-Pinsker猜想提出了一种关于约束满足问题（CSPs）计算复杂度的P vs NP完全二元性，这些CSPs属于一阶有限约束的同构结构。值得注意的是，如果两个结构pp-可解释性等价，则它们的CSPs在log空间中等价，或者等价于它们的多形克隆在拓扑上同构。后者导致了代数方法，将具有拓扑同构多形克隆的结构视为等价，并试图在拓扑克隆中识别硬度或可解性的结构原因。我们证明了这一方法背后的pp-可解释性等价关系是合理的：一方面，我们证明在温和的模板条件下，该等价关系是可判定的；这改进了Bodirsky、Pinsker和Tsankov（LICS'11）关于多形克隆等价可判定性的定理。另一方面，我们证明在更广泛的类中，即在无代数性且ω-范畴的传递结构中，该等价关系在描述集合论中具有最低的复杂性：即，它平滑的，即Borel-归约到实数上的等价。在证明第一个结果的过程中，我们证明了具有有限约束Ramsey扩展的结构的模型完全核心是可计算的，从而提供了对之前非构造性证明的构造性替代。

英文摘要：

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.

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