23 research outputs found
Dualities and dual pairs in Heyting algebras
We extract the abstract core of finite homomorphism dualities using the
techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction
Dualities and Dual Pairs in Heyting Algebras
We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categorie
On the Relationship between Consistent Query Answering and Constraint Satisfaction Problems
Recently, Fontaine has pointed out a connection between consistent query answering (CQA) and constraint satisfaction problems (CSP) [Fontaine, LICS 2013]. We investigate this connection more closely, identifying classes of CQA problems based on denial constraints and GAV constraints that correspond exactly to CSPs in the sense that a complexity classification of the CQA problems in each class is equivalent (up to FO-reductions) to classifying the complexity of all CSPs. We obtain these classes by admitting only monadic relations and only a single variable in denial constraints/GAVs and restricting queries to hypertree UCQs. We also observe that dropping the requirement of UCQs to be hypertrees corresponds to transitioning from CSP to its logical generalization MMSNP and identify a further relaxation that corresponds to transitioning from MMSNP to GMSNP (also know as MMSNP_2). Moreover, we use the CSP connection to carry over decidability of FO-rewritability and Datalog-rewritability to some of the identified classes of CQA problems
When do homomorphism counts help in query algorithms?
A query algorithm based on homomorphism counts is a procedure for determining
whether a given instance satisfies a property by counting homomorphisms between
the given instance and finitely many predetermined instances. In a left query
algorithm, we count homomorphisms from the predetermined instances to the given
instance, while in a right query algorithm we count homomorphisms from the
given instance to the predetermined instances. Homomorphisms are usually
counted over the semiring N of non-negative integers; it is also meaningful,
however, to count homomorphisms over the Boolean semiring B, in which case the
homomorphism count indicates whether or not a homomorphism exists. We first
characterize the properties that admit a left query algorithm over B by showing
that these are precisely the properties that are both first-order definable and
closed under homomorphic equivalence. After this, we turn attention to a
comparison between left query algorithms over B and left query algorithms over
N. In general, there are properties that admit a left query algorithm over N
but not over B. The main result of this paper asserts that if a property is
closed under homomorphic equivalence, then that property admits a left query
algorithm over B if and only if it admits a left query algorithm over N. In
other words and rather surprisingly, homomorphism counts over N do not help as
regards properties that are closed under homomorphic equivalence. Finally, we
characterize the properties that admit both a left query algorithm over B and a
right query algorithm over B.Comment: 24 page