A.1. PROOF OF PROPOSITION 4.3 .(a) From the definition of Rep(t) and Proposition 4.2, it follows that:Rep ,A (t) = {T | T is a tree of vocabulary τ ,A } ∩ {T | there exists a homomorphism h from re (t) to T } Hence, by observing that a homomorphism from an incomplete relational structure to a complete relational structure is a valuation, it easily follows that re (t) represents t.(b) Suppose that D is such that for every D in Rep(D), D is not a tree. It follows that Rep(D)∩TREES = ∅. Hence, the proposition trivially holds, since D represents any inconsistent incomplete tree.Suppose now that D is such that there exists D in Rep(D) such that D is a tree. We next show how to build an incomplete tree t D starting from D, that represents t D .Intuitively, t D can be built starting from the nodes occurring in D and then, for each of them, defining the forests of its children and descendants as the union of "connected components" of nodes among which at least one node is respectively its child or descendant in D.

The term naïve evaluation refers to evaluating queries over incomplete databases as if nulls were usual data values, that is, to using the standard database query evaluation engine. Since the semantics of query answering over incomplete databases is that of certain answers, we would like to know when naïve evaluation computes them, that is, when certain answers can be found without inventing new specialized algorithms. For relational databases it is well known that unions of conjunctive queries possess this desirable property, and results on preservation of formulae under homomorphisms tell us that, within relational calculus, this class cannot be extended under the open-world assumption.Our goal here is twofold. First, we develop a general framework that allows us to determine, for a given semantics of incompleteness, classes of queries for which naïve evaluation computes certain answers. Second, we apply this approach to a variety of semantics, showing that for many classes of queries beyond unions of conjunctive queries, naïve evaluation makes perfect sense under assumptions different from open world. Our key observations are: (1) naïve evaluation is equivalent to monotonicity of queries with respect to a semanticsinduced ordering, and (2) for most reasonable semantics of incompleteness, such monotonicity is captured by preservation under various types of homomorphisms. Using these results we find classes of queries for which naïve evaluation works, for example, positive first-order formulae for the closed-world semantics. Even more, we introduce a general relation-based framework for defining semantics of incompleteness, show how it can be used to capture many known semantics and to introduce new ones, and describe classes of first-order queries for which naïve evaluation works under such semantics. ACM Reference Format:Amélie Gheerbrant, Leonid Libkin, and Cristina Sirangelo. 2014. Naïve evaluation of queries over incomplete databases.

The paper presents a declarative semantics for the maintenance of integrity constraints expressed by means of production rules. A production rule is a special form of active rule, called active integrity constraint, whose body contains an integrity constraint (conjunction of literals which must be false) and whose head contains a disjunction of update atoms, i.e. actions to be performed if the corresponding constraints are not satisfied (i.e. are true). The paper introduces i) a formal declarative semantics allowing the computation of founded repairs, that is repairs whose actions are specified and supported by active integrity constraint, ii) an equivalent semantics obtained by rewriting production rules into disjunctive logic rules, so that repairs can be derived from the answer sets of the logic program, iii) a characterization of production rules allowing a methodology for integrity maintenance.