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One of the central problems in the study of parametrized constraintsatisfaction problems is the Dichotomy Conjecture by T. Feder and M. Vardistating that the constraint satisfaction problem (CSP) over a fixed, finiteconstraint language is either solvable in polynomial time or\textsc{NP}-complete. The conjecture was verified in certain special cases(domains with a relatively small number of elements, constraint languagescontaining all unary relations, etc.) In this article, we present a proof ofthe Dichotomy Conjecture via local consistency and AF- consistency checks. Infact, we show that, for every Taylor domain, which is$(2\lceil\frac{K}{2}\rceil,3\lceil\frac{K}{2}\rceil)$-consistent, where $K$ isthe largest arity of a relation in the constraint language, we can definepolynomially many proper subinstances such that, the original instance of theCSP is solvable if, and only if, the problem has a solution in one of thosesubinstances.. Finally, a solution is constructed using the combination of SLAC(Singleton Linear Arc Consistency), introduced by M. Kozik, and AF-consistency.

Constraint Satisfaction Problem Dichotomy for Finite Templates: a Proof Via Consistency Checks