The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

• Published in: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science pp. 615 - 622
• Main Authors: Barto, Libor, Pinsker, Michael
• Format: Conference Proceeding
• Language: English
• Published: New York, NY, USA ACM 05.07.2016
• Series: ACM Conferences
• Subjects: Theory of computation > Logic; Theory of computation > Models of computation > Computability; continuous clone homomorphism; dichotomy conjecture; homogeneous structure; term condition; stabilizer; constraint satisfaction problem; polymorphism clone; ω-categorical structure
• ISBN: 9781450343916; 1450343910
• DOI: 10.1145/2933575.2934544

We prove that an ω-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 α, β, s satisfying the identity αs(x, y, x, z, y, z) ≈ βs(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 ω-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).