从块正交性到复权计数CSP的可判定性
From Block Orthogonality to Decidability in Complex-Weighted Counting CSP
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中文总结 AI 辅助
本文解决了复权计数CSP二分性中无穷条件的可判定性问题，证明块正交性可简化三条件刻画，给出了判定算法并拓展至度倍数计数CSP。
中文摘要 AI 辅助
在一篇获得2021年哥德尔奖的里程碑式JACM论文中，Cai和Chen针对任意有限域上带代数复权的计数约束满足问题（counting CSPs）建立了完整的复杂性二分性。其多项式时间可解的一侧由三个条件刻画——块正交性（Block Orthogonality）、类型划分（Type Partition）以及由公共Mal'tsev运算保持——这些条件是通过从任意#CSP(𝒫)实例经部分求和生成的可数无限族W_𝒫来量化的。他们提出问题：仅从有限语言𝒫出发，这些无穷条件是否可判定？等价地，这种完整固定语言分类的多项式时间侧是否可统一识别？我们解决了该问题：对每个非空有限域D和每个有限精确编码的代数复语言𝒫，给出一个总精确算法，可判定整个无界族W_𝒫上的所有三个条件。除可判定性外，我们还证明，仅块正交性就同时强制类型划分的成立，以及存在一个单一Mal'tsev运算保持所有生成的支撑和行等价关系。因此，三条件刻画可简化为块正交性，且有限输入(D,𝒫)决定二分性的哪一侧适用。该同一框架还可判定Lin证明的度倍数计数CSP二分性定理中的对应条件。
英文摘要
In a landmark JACM paper recognized with the 2021 G{ö}del Prize, Cai and Chen established a complete complexity dichotomy for counting CSPs over arbitrary finite domains with algebraic complex weights. Its polynomial-time side is characterized by three conditions---Block Orthogonality, Type Partition, and preservation by a common Mal'tsev operation---quantified over the countably infinite family $W_{\mathcal{F}}$ generated from arbitrary $\#\mathrm{CSP}(\mathcal{F})$ instances by partial summation. They asked whether these infinitary conditions are decidable from the finite language $\mathcal{F}$ alone---equivalently, whether the polynomial-time side of this complete fixed-language classification is uniformly recognizable. We settle this problem by giving, for every nonempty finite domain $D$ and every finite exactly encoded algebraic-complex language $\mathcal{F}$, a total exact algorithm that decides all three conditions on the full unbounded family $W_{\mathcal{F}}$. Beyond decidability, we prove that Block Orthogonality alone forces both Type Partition and the existence of a single Mal'tsev operation preserving all generated support and row-equivalence relations. Thus the three-condition characterization collapses to Block Orthogonality, and the finite input $(D,\mathcal{F})$ determines which side of the dichotomy applies. The same framework decides the corresponding conditions in the dichotomy theorem for degree-multiple counting CSP proved by Lin.
