Feder-Vardi conjecture, which proposed that every finite-domain Constraint Satisfaction Problem (CSP) is either in P or it is NP-complete, has been solved independently by Bulatov and Zhuk almost ten years ago. Bodirsky-Pinsker conjecture which states a similar dichotomy for countably infinite first-order reducts of finitely bounded homogeneous structures is wide open. In this paper, we prove that CSPs over first-order expansions of finitely bounded homogeneous model-complete cores are either first-order definable (and hence in non-uniform AC$^0$) or L-hard under first-order reduction. It is arguably the most general complexity dichotomy when it comes to the scope of structures within Bodirsky-Pinsker conjecture. Our strategy is that we first give a new proof of Larose-Tesson theorem, which provides a similar dichotomy over finite structures, and then generalize that new proof to infinite structures.

翻译：Feder-Vardi猜想提出每个有限域约束满足问题（CSP）要么属于P类，要么是NP完全的，该猜想约十年前已分别由Bulatov与Zhuk独立解决。而Bodirsky-Pinsker猜想针对有限有界齐次结构的一阶可数无限归约提出了类似的二分性，至今仍悬而未决。本文证明了有限有界齐次模型完全核的一阶扩张上的CSP问题，要么是一阶可定义的（因而属于非均匀AC$^0$类），要么在FO归约下是L-硬的。这堪称Bodirsky-Pinsker猜想所涵盖结构范围内最广义的复杂度二分性结果。我们的研究策略是：首先给出Larose-Tesson定理的新证明（该定理在有限结构上提供了类似的二分性），随后将该新证明推广至无限结构。