SIAM Journal on Computing, Volume 50, Issue 4, Page 1359-1409, January 2021.
The logic MMSNP is a restricted fragment of existential second-order logic which can express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi, who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs that does not rely on the results of Kun. The new universal-algebraic proof allows us to obtain a stronger statement and to verify the more general Bodirsky--Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite $\omega$-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property.

中文翻译：

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单调一元 SNP 代数可追踪性猜想的证明

SIAM Journal on Computing，第 50 卷，第 4 期，第 1359-1409 页，2021 年 1 月。
逻辑 MMSNP 是存在二阶逻辑的受限片段，可以表达图论和有限模型论中的许多有趣的查询。该逻辑是由 Feder 和 Vardi 引入的，他们表明每个 MMSNP 句子在计算上等效于有限域约束满足问题 (CSP)；涉及的概率减少由 Kun 使用扩展器结构的显式构造进行去随机化。我们提出了一个不依赖于 Kun 的结果的有限域 CSP 减少的新证明。新的通用代数证明使我们能够获得更强的陈述并验证更一般的 Bodirsky-Pinsker 二分法猜想对于 MMSNP 中的 CSP。我们的方法使用了这样一个事实，即每个 MMSNP 句子都描述了可数无限 $\omega$-categorical 结构的 CSP 的有限联合；