The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
We prove that an o categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations a,b, s satisfying the identity as(x, y, x, z, y, z) bs(y, x, z, x, z, y).This establishes an algebraic criterion equivalent to the conjectured borderline between P and NP complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case.Our theorem is also of independent mathematical interest, characterizing a topological property of any o categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above)
2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)