8 research outputs found
On tractability and congruence distributivity
Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width
Sensitive Instances of the Constraint Satisfaction Problem
We investigate the impact of modifying the constraining relations of a
Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the
set of solutions of the instance. More precisely we investigate sensitive
instances: an instance of the CSP is called sensitive, if removing any tuple
from any constraining relation invalidates some solution of the instance.
Equivalently, one could require 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 the direction proposed (in the context of strict width) by
Feder and Vardi (SICOMP 1999) and require that only the instances produced by a
local consistency checking algorithm are sensitive. In the language of the
algebraic approach to the CSP we show that a finite idempotent algebra
has a variable near unanimity term operation if and only if
any instance that results from running the -consistency algorithm on
an instance over is sensitive.
A version of our result, without idempotency but with the sensitivity
condition holding in a variety of algebras, settles a question posed by G.
Bergman about systems of projections of algebras that arise from some
subalgebra of a finite product of algebras.
Our results hold for infinite (albeit in the case of idempotent)
algebras as well and exhibit a surprising similarity to the strict width
condition proposed by Feder and Vardi. Both conditions can be characterized by
the existence of a near unanimity operation, but the arities of the operations
differ by 1
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
Deciding the Existence of Minority Terms
This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation m that satisfies the minority equations m(y,x,x)≈m(x,y,x)≈m(x,x,y)≈y . We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP
Learnability of Solutions to Conjunctive Queries: The Full Dichotomy
Abstract The problem of learning the solution space of an unknown formula has been studied in multiple embodiments in computational learning theory. In this article, we study a family of such learning problems; this family contains, for each relational structure, the problem of learning the solution space of an unknown conjunctive query evaluated on the structure. A progression of results aimed to classify the learnability of each of the problems in this family, and thus far a culmination thereof was a positive learnability result generalizing all previous ones. This article completes the classification program towards which this progression of results strived, by presenting a negative learnability result that complements the mentioned positive learnability result. In order to obtain our negative result, we make use of universal-algebraic concepts, and our result is phrased in terms of the varietal property of non-congruence modularity