Worst-case Optimal Query Answering for Greedy Sets of Existential Rules and Their Subclasses

The need for an ontological layer on top of data, associated with advanced reasoning mechanisms able to exploit the semantics encoded in ontologies, has been acknowledged both in the database and knowledge representation communities. We focus in this paper on the ontological query answering problem, which consists of querying data while taking ontological knowledge into account. More specifically, we establish complexities of the conjunctive query entailment problem for classes of existential rules (also called tuple-generating dependencies, Datalog+/- rules, or forall-exists-rules. Our contribution is twofold. First, we introduce the class of greedy bounded-treewidth sets (gbts) of rules, which covers guarded rules, and their most well-known generalizations. We provide a generic algorithm for query entailment under gbts, which is worst-case optimal for combined complexity with or without bounded predicate arity, as well as for data complexity and query complexity. Secondly, we classify several gbts classes, whose complexity was unknown, with respect to combined complexity (with both unbounded and bounded predicate arity) and data complexity to obtain a comprehensive picture of the complexity of existential rule fragments that are based on diverse guardedness notions. Upper bounds are provided by showing that the proposed algorithm is optimal for all of them

On the Relation between Conceptual Graphs and Description Logics

Aus der Einleitung: 'Conceptual graphs (CGs) are an expressive formalism for representing knowledge about an application domain in a graphical way. Since CGs can express all of first-order predicate logic (FO), they can also be seen as a graphical notation for FO formulae. In knowledge representation, one is usually not only interested in representing knowledge, one also wants to reason about the represented knowledge. For CGs, one is, for example, interested in validity of a given graph, and in the question whether one graph subsumes another one. Because of the expressiveness of the CG formalism, these reasoning problems are undecidable for general CGs. In the literature [Sow84, Wer95, KS97] one can find complete calculi for validity of CGs, but implementations of these calculi have the same problems as theorem provers for FO: they may not terminate for formulae that are not valid, and they are very ineficient. To overcome this problem, one can either employ incomplete reasoners, or try to find decidable (or even tractable) fragments of the formalism. This paper investigates the second alternative. The most prominent decidable fragment of CGs is the class of simple conceptual graphs (SGs), which corresponds to the conjunctive, positive, and existential fragment of FO (i.e., existentially quantified conjunctions of atoms). Even for this simple fragment, however, subsumption is still an NP-complete problem [CM92]. SGs that are trees provide for a tractable fragment of SGs, i.e., a class of simple conceptual graphs for which subsumption can be decided in polynomial time [MC93]. In this report, we will identify a tractable fragment of SGs that is larger than the class of trees. Instead of trying to prove new decidability or tractability results for CGs from scratch, our idea was to transfer decidability results from description logics [DLNN97, DLNS96] to CGs. The goal was to obtain a \natural' sub-class of the class of all CGs in the sense that, on the one hand, this sub-class is defined directly by syntactic restrictions on the graphs, and not by conditions on the first-order formulae obtained by translating CGs into FO, and, on the other hand, is in some sense equivalent to a more or less expressive description logic. Although description logics (DLs) and CGs are employed in very similar applications (e.g., for representing the semantics of natural language sentences), it turned out that these two formalisms are quite different for several reasons: (1) conceptual graphs are interpreted as closed FO formulae, whereas DL concept descriptions are interpreted by formulae with one free variable; (2) DLs do not allow for relations of arity > 2 ; (3) SGs are interpreted by existential sentences, whereas almost all DLs considered in the literature allow for universal quantification; (4) because DLs use a variable-free syntax, certain identifications of variables expressed by cycles in SGs and by co-reference links in CGs cannot be expressed in DLs. As a consequence of these differences, we could not identify a natural fragment of CGs corresponding to an expressive DL whose decidability was already shown in the literature. We could, however, obtain a new tractability result for a DL corresponding to SGs that are trees. This correspondence result strictly extends the one in [CF98]. In addition, we have extended the tractability result from SGs that are trees to SGs that can be transformed into trees using a certain \cycle-cutting' operation. The report is structured as follows. We first introduce the description logic for which we will identify a subclass of equivalent SGs. In Section 3, we recall basic definitions and results on SGs. Thereafter, we introduce a syntactical variant of SGs which allows for directly encoding the support into the graphs (Section 4.1). In order to formalize the equivalence between DLs and SGs, we have to consider SGs with one distinguished node called root (Section 4.2). In Section 5, we finally identify a class of SGs corresponding to a DL that is a strict extension of the DL considered in [CF98]

Reasoning with Forest Logic Programs and f-hybrid Knowledge Bases

Open Answer Set Programming (OASP) is an undecidable framework for integrating ontologies and rules. Although several decidable fragments of OASP have been identified, few reasoning procedures exist. In this article, we provide a sound, complete, and terminating algorithm for satisfiability checking w.r.t. Forest Logic Programs (FoLPs), a fragment of OASP where rules have a tree shape and allow for inequality atoms and constants. The algorithm establishes a decidability result for FoLPs. Although believed to be decidable, so far only the decidability for two small subsets of FoLPs, local FoLPs and acyclic FoLPs, has been shown. We further introduce f-hybrid knowledge bases, a hybrid framework where \SHOQ{} knowledge bases and forest logic programs co-exist, and we show that reasoning with such knowledge bases can be reduced to reasoning with forest logic programs only. We note that f-hybrid knowledge bases do not require the usual (weakly) DL-safety of the rule component, providing thus a genuine alternative approach to current integration approaches of ontologies and rules

The Logic of Counting Query Answers

We consider the problem of counting the number of answers to a first-order formula on a finite structure. We present and study an extension of first-order logic in which algorithms for this counting problem can be naturally and conveniently expressed, in senses that are made precise and that are motivated by the wish to understand tractable cases of the counting problem