The Feder-Vardi dichotomy conjecture for Constraint Satisfaction Problems (CSPs) with finite templates, confirmed independently by Bulatov and Zhuk, has an extension to certain well-behaved infinite templates due to Bodirsky and Pinsker which remains wide open. We formulate three fundamental questions on the scope of the Bodirsky-Pinsker conjecture and provide positive answers to them. Our first two main results provide two simplifications of this scope, one of structural, and the other one of algebraic nature. The former simplification implies that the conjecture is equivalent to its restriction to templates without algebraicity, a crucial assumption in the most powerful classification methods. The latter yields that the higher-arity invariants of any template within its scope can be assumed to be essentially injective, and any algebraic condition characterizing any complexity class within the conjecture closed under Datalog reductions must be satisfiable by injections, thus lifting the mystery of the better applicability of certain algebraic conditions over others. Our third main result uses the first one to show that any non-trivially tractable template within the scope serves, up to a Datalog-computable modification of it, as the witness of the tractability of a non-finitely tractable finite-domain Promise Constraint Satisfaction Problem (PCSP) by the so-called sandwich method. This provides a particularly strong connection between the Bodirsky-Pinsker conjecture and finite-domain PCSPs. In the light of the third main result, we initiate a new case study-of phylogeny CSPs-which we investigate from the perspective of descriptive complexity. Within this study, we show that there exists a tractable phylogeny CSP that pp-constructs a finite-domain PCSP inexpressible in fixed-point logic with counting but does not pp-construct any finite-domain CSP with this property.

翻译：有限模板约束满足问题(CSPs)的Feder-Vardi二分性猜想（由Bulatov和Zhuk独立证实）已由Bodirsky和Pinsker推广至某些性质良好的无限模板，但该推广至今仍悬而未决。我们针对Bodirsky-Pinsker猜想的适用范围提出了三个基本问题，并给出了肯定回答。前两个主要成果提供了该适用范围的两类简化形式：其一为结构性简化，其二为代数性质简化。结构性简化表明该猜想等价于其对无代数性模板的限制——这一假设在最具威力的分类方法中至关重要。代数性质简化则证明，适用范围中任意模板的高元不变性可假定为本质单射的，且任何在猜想中刻画复杂度类（在Datalog归约下封闭）的代数条件均可由单射满足，从而揭示了某些代数条件具有更优适用性的奥秘。第三个主要成果利用第一个结论证明：适用范围中任意非平凡可解模板，通过所谓的夹层方法，可视为某有限域承诺约束满足问题(PCSP)可解性的见证（后者需经Datalog可计算修正）。这建立了Bodirsky-Pinsker猜想与有限域PCSP之间尤为紧密的联系。基于第三项成果，我们启动了新的案例研究——系统发育CSPs——从描述复杂性视角展开分析。研究表明：存在一个可解系统发育CSP，其能pp-构造一个无法用带计数不动点逻辑表达的有限域PCSP，却不能pp-构造任何具有该性质的有限域CSP。