  This paper focuses on the algebraic theory underlying the study of the
complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We
unify, simplify, and extend parts of the three approaches that have been
developed to study the CSP over finite templates - absorption theory that was
used to characterize CSPs solvable by local consistency methods (JACM'14), and
Bulatov's and Zhuk's theories that were used for two independent proofs of the
CSP Dichotomy Theorem (FOCS'17, JACM'20).
  As the first contribution we present an elementary theorem about primitive
positive definability and use it to obtain the starting points of Bulatov's and
Zhuk's proofs as corollaries. As the second contribution we propose and
initiate a systematic study of minimal Taylor algebras. This class of algebras
is broad enough so that it suffices to verify the CSP Dichotomy Theorem on this
class only, but still is unusually well behaved. In particular, many concepts
from the three approaches coincide in the class, which is in striking contrast
with the general setting.
  We believe that the theory initiated in this paper will eventually result in
a simple and more natural proof of the Dichotomy Theorem that employs a simpler
and more efficient algorithm, and will help in attacking complexity questions
in other CSP-related problems.

Computational Complexity;Logic in Computer Science