This chapter provides an introduction to Description Logics as a formal language for representing knowledge and reasoning about it. It first gives a short overview of the ideas underlying Description Logics. Then it introduces syntax and semantics, covering the basic constructors that are used in systems or have been introduced in the literature, and the way these constructors can be used to build knowledge bases. Finally, it defines the typical inference problems, shows how they are interrelated, and describes different approaches for effectively solving these problems. Some of the topics that are only briefly mentioned in this chapter will be treated in more detail in subsequent chapters. 47 48 F. Baader, W. Nutt Proposition 2.5 Let T be a terminology, I be an interpretation, and J be the restriction of I to the base symbols of T . Then I is a model of T if, and only if, I is a fixpoint of T J .According to the preceding proposition, a terminology T is definitorial iff every base interpretation J has a unique extension that is a fixpoint of T J .Example 2.6 To get a feel for why cyclic terminologies are not definitorial, we discuss as an example the terminology T Momo that consists only of Axiom (2.4). Consider the base interpretation J defined by
A basic feature of Terminological Knowledge Representation Systems is to represent knowledge by means of taxonomies, here called terminologies, and to provide a specialized reasoning engine to do inferences on these structures. The taxonomy is built through a representation language called a concept language (or description logic), which is given a well-de ned set-theoretic semantics. The e ciency of reasoning has often been advocated as a primary motivation for the use of such systems. The main contributions of the paper are: (1) a complexity analysis of concept satis ability and subsumption for a wide class of concept languages; (2) the algorithms for these inferences that comply with the worst-case complexity of the reasoning task they perform. This is an extended and revised version of a paper presented at the 2nd Int. Conf. on Principles of Knowledge Representation and Reasoning, Cambridge, MA, 1991.
We i n v estigate the problem of rewriting queries with aggregate operators using views that may o r m a y not contain aggregate operators. A rewriting of a query is a second query that uses view predicates such that evaluating rst the views and then the rewriting yields the same result as evaluating the original query. In this sense, the original query and the rewriting are equivalent modulo the view de nitions. The queries and views we consider correspond to unnested SQL queries, possibly with union, that employ the operators min, max, count, and sum.Our approach is based on syntactic characterizations of the equivalence of aggregate queries. One contribution of this paper are characterizations of the equivalence of disjunctive aggregate queries, which generalize our previous results for the conjunctive case.For each operator , w e i n troduce several types of queries using views as candidates for rewritings. We unfold such a candidate by replacing each occurrence of a view predicate with its de nition, thus obtaining a regular aggregate query. The candidates have a di erent, usually more complex operator than . We prove that unfolding the candidate, however, results in a regular aggregate query that is equivalent to the candidate modulo the view de nitions. This property justi es considering these types of queries as natural candidates for rewritings. In this way, we reduce the problem of whether there exist rewritings of a particular type to a problem involving equivalence.We distinguish between partial rewritings that contain at least one view predicate and complete rewritings that contain only view predicates. In contrast to previous work on this topic, we not only give su cient, but also necessary conditions for a rewriting to exist. More precisely, w e show for each t ype of candidate that the existence of both, partial and complete rewritings is decidable, and we provide upper and lower complexity bounds. IntroductionRewriting queries using views is a fundamental problem in databases, which has attracted considerable attention. View usability techniques have applications in a number of areas. In query optimization, the execution of a query can be accelerated if results from previous queries can be used to compute answers YL87, CR94, CKPS95 . In designing information systems over which a h uge number of a priori known queries are posed periodically, it can be bene cial to store such intermediate results beforehand that are useful for as many queries as possible LFS97, RSS96 . Integrating heterogeneous information sources is another problem which may be reduced to the view usability problem LSK95 .While the focus of this work was for a long time on queries without aggregation, interest in aggregate queries has been motivated recently by the surge of data warehousing and decision support applications, where queries of this kind typically occur. Optimization based on the reuse of previously computed results is particularly promising for aggregate queries, since often huge numbers of data items are p...
Abstract. We study the problem of evolution for Knowledge Bases (KBs) expressed in Description Logics (DLs) of the DL-Lite family. DL-Lite is at the basis of OWL 2 QL, one of the tractable fragments of OWL 2, the recently proposed revision of the Web Ontology Language. We propose some fundamental principles that KB evolution should respect. We review known model and formula-based approaches for evolution of propositional theories. We exhibit limitations of a number of model-based approaches: besides the fact that they are either not expressible in DL-Lite or hard to compute, they intrinsically ignore the structural properties of KBs, which leads to undesired properties of KBs resulting from such an evolution. We also examine proposals on update and revision of DL KBs that adopt the model-based approaches and discuss their drawbacks. We show that known formula-based approaches are also not appropriate for DL-Lite evolution, either due to high complexity of computation, or because the result of such an action of evolution is not expressible in DL-Lite. Building upon the insights gained, we propose two novel formula-based approaches that respect our principles and for which evolution is expressible in DL-Lite. For our approaches we also developed polynomial time algorithms to compute evolution of DL-Lite KBs.
Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators min, max, count, count-distinct, and sum. Essentially, this class contains all unnested SQL queries with the above aggregate operators, with a WHERE clause consisting of a conjunction of comparisons, and without a HAVING clause. The comparisons can be interpreted over either a dense order e.g., over the rationals or a discrete order e.g., over the integers. Generally, h o w ever, di erent techniques and characterizations are needed in each of these two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a su cient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. F or conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case i.e., both comparisons and constants, the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable with polynomial space. Finally, it is shown that all the characterizations for min-, max-, count-, and sum-queries yield polynomial-time algorithms for linear queries, i.e., queries with no repeated predicates in their bodies.
Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count, count-distinct, min, max, and sum. Essentially, this class contains unnested SQL queries with the above aggregate operators, with a where clause consisting of a conjunction of comparisons, and without a having clause. The comparisons are either interpreted over a domain with a dense order (like the rationals) or with a discrete order (like the integers). Characterizations of equivalence differ for the two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and S. 2 S. COHEN ET AL. constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable in polynomial space.
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