6 research outputs found
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
Sensitive instances of the constraint satisfaction problem
We investigate the impact of modifying the constraining relations of a Constraint SatisfactionProblem (CSP) instance, with a fixed template, on the set of solutions of the instance. More preciselywe investigate sensitive instances: an instance of theCSPis called sensitive, if removing any tuplefrom any constraining relation invalidates some solution of the instance. Equivalently, one couldrequire that every tuple from any one of its constraints extends to a solution of the instance.Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow thedirection proposed (in the context of strict width) by Feder and Vardi in [13] and require that onlythe instances produced by a local consistency checking algorithm are sensitive. In the languageof the algebraic approach to theCSPwe show that a finite idempotent algebraAhas ak+ 2variable near unanimity term operation if and only if any instance that results from running the(k, k+ 1)-consistency algorithm on an instance overA2is sensitive.A version of our result, without idempotency but with the sensitivity condition holding in avariety of algebras, settles a question posed by G. Bergman about systems of projections of algebrasthat arise from some subalgebra of a finite product of algebras.Our results hold for infinite (albeit in the case ofAidempotent) algebras as well and exhibit asurprising similarity to the strict widthkcondition proposed by Feder and Vardi. Both conditionscan be characterized by the existence of a near unanimity operation, but the arities of the operationsdiffer by1