Three Fundamental Questions in Modern Infinite-Domain Constraint Satisfaction
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.