One source of partial information in databases is the need to combine information from several databases. Even if each database is complete for some "world", the combined databases will not be, and answers to queries against such combined databases can only be approximated. In this paper we describe various situations in which a precise answer cannot be obtained for a query asked against multiple databases. Based on an analysis of these situations, we propose a classification of constructs that can be used to model approximations. A major goal is to obtain universality properties for these models of approximations. Universality properties suggest syntax for languages with approximations based on the operations which are naturally associated with them. We prove universality properties for most of the approximation constructs. Then we use them to design languages built around datatypes given by the approximation constructs. A straightforward approach results in langauges that have a numb...

We present a new principle for the development of database query languages that the primitive operations should be organized around types. Viewing a relational database as consisting of sets of records, this principle dictates that we should investigate separately operations for records and sets. There are two immediate advantages of this approach, which is partly inspired by basic ideas from category theory. First, it provides a language for structures in which record and set types may be freely combined: nested relations or complex objects. Second, the fundamental operations for sets are closely related to those for other "collection types" such as bags or lists, and this suggests how database languages may be uniformly extended to these new types. The most general operation on sets, that of structural recursion, is one in which not all programs are welldefined. In looking for limited forms of this operation that always give rise to well-defined operations, we find a number of close ...

Abstract. We discuss, compare and relate some old and some new models for incomplete and probabilistic databases. We characterize the expressive power of c-tables over infinite domains and we introduce a new kind of result, algebraic completion, for studying less expressive models. By viewing probabilistic models as incompleteness models with additional probability information, we define completeness and closure under query languages of general probabilistic database models and we introduce a new such model, probabilistic c-tables, that is shown to be complete and closed under the relational algebra. 1

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Or-sets were introduced by Imielinski, Naqvi and Vadaparty for dealing with limited forms of disjunctive information in database queries. Independently, Rounds used a similar notion for representing disjunctive and conjunctive information in the context of situation theory. In this paper we formulate a query language with adequate expressive power for or-sets. Using the notion of normalization of or-sets, queries at the "structural" and "conceptual" levels are distinguished. Losslessness of normalization is established for a large class of queries. We have obtained upper bounds for the cost of normalization. An approach related to that of Rounds is used to provide semantics for or-sets. We also treat or-sets in the context of partial information in databases.

This paper derives a new and surprisingly low complexity result for inference in a new form of Reiter's propositional default logic [18]. The problem studied here is the default inference problem whose fundamental importance was pointed out by Kraus, Lehmann, and Magidor [12]. We prove that "normal" default inference, in propositional logic, is a problem complete for co-NP(3), the third level of the Boolean hierarchy [4]. Our result (by changing the underlying semantics) contrasts favorably with a similar result of Gottlob [7], who proves that standard default inference is \Pi P 2 -complete. Our inference relation also obeys all of the laws for preferential consequence relations set forth by Kraus, Lehmann, and Magidor. In particular, we get the property of being able to reason by cases and the law of cautious monotony. Both of these laws fail for standard propositional default logic. The key technique for our results is the use of Scott's domain theory to integrate defaults into pa...

We extend the relational data model to incorporate partial orderings into data domains, which we call the ordered relational model. Within the extended model, we define the Partially Ordered Relational Algebra (the PORA) by allowing the ordering predicate ⊑ to be used in formulae of the selection operator (σ). The PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphism of databases. This result characterises the expressiveness of the PORA and justifies the development of Ordered SQL (OSQL) as a query language for ordered databases. OSQL provides users with the capability of capturing the semantics of ordered data in many advanced applications, such as those having temporal or incomplete information. Ordered Functional Dependencies (OFDs) on ordered databases are studied, based on two possible extensions of domain orderings: (1) pointwise-ordering and (2) lexicographical ordering. We present a sound and complete axiom system for OFDs in the first case and establish a set of sound and complete chase rules for OFDs in the second. Our results suggest that the implication problems for both cases of OFDs are decidable and that the enforcement of OFDs in ordered relations are practically feasible. In a wider perspective, the proposed model explores an important area of object-relational databases, since ordered domains can be viewed as a general kind of data type. Categories and Subject Descriptors: H.2.1 [Database Management]: Logical Design—data

Abstract | We extend the relational data model to incorporate linear orderings into data domains, which we call the ordered relational model. The conventional Functional Dependencies (FDs) are examined in the context of ordered relational databases by using the notion of System Ordering Independence (SOI), which refers to the desirable scenario that the ordering of tuples in a relation is independent of the implementation of the underlying DBMS. We also extend Armstrong's axiom system for FDs to object relations, which are a subclass of ordered relations that allow us to view tuples as objects. We formally de ne Ordered Functional Dependencies (OFDs) for the extended model by means of two possible extensions of domains, pointwise-orderings and lexicographical orderings. We rst present a sound and complete axiom system for OFDs in the case of pointwise-orderings and then establish a sound and complete set of chase rules for OFDs in the case of lexicographical orderings. Our main result shows that the implication problems for both cases of OFDs are decidable, and that

. Most of work on partial information in databases asks which operations of standard languages, like relational algebra, can still be performed correctly in the presence of nulls. In this paper a different point of view is advocated. We believe that the semantics of partiality must be clearly understood and it should give us new design principles for languages for databases with partial information. There are different sources of partial information, such as missing information and conflicts that occur when different databases are merged. In this paper, we develop a common semantic framework for them which can be applied in a context more general than the flat relational model. This ordered semantics, which is based on ideas used in the semantics of programming languages, cleanly intergrates all kinds of partial information and serves as a tool to establish connections between them. Analyzing properties of semantic domains of types suitable for representing partial inform...

In this paper we explore the computational aspects of Propositional Power Default Reasoning (PDR), a form of non-monotonic reasoning in which the underlying logic is Kleene's 3-valued propositional logic. PDR leads to a concise meaning of the problem of skeptical entailment which has better complexity characteristics than the usual formalisms (co-NP(3)-Complete instead of \Pi P 2 -Complete). We take advantage of this in an implementation called powdef to encode and solve hard graph problems and explore randomly generated instances of skeptical entailment. Introduction 1 Many forms of propositional default reasoning have unexpectedly intractable complexity problems. For example, deciding skeptical entailment in normal propositional default logic (Reiter 1980) is \Pi P 2 -complete, a result due to Gottlob (Gottlob 1992). Even special cases stay this way -- they tend to be NP-hard when we might expect them to be polynomial (Kautz and Selman 1989). This somewhat paradoxical situation...