The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
Abstract
We prove that an $\omega$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 $\alpha$, $\beta$, $s$ satisfying the identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This establishes an algebraic criterion equivalent to the conjectured borderline between P and NPcomplete 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 $\omega$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).
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.04353
 arXiv:
 arXiv:1602.04353
 Bibcode:
 2016arXiv160204353B
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Computational Complexity;
 Mathematics  Logic
 EPrint:
 15 pages