We present a complete logic for reasoning with functional dependencies (FDs) with semantics defined over classes of commutative integral partially ordered monoids and complete residuated lattices. The dependencies allow us to express stronger relationships between attribute values than the ordinary FDs. In our setting, the dependencies not only express that certain values are determined by others but also express that similar values of attributes imply similar values of other attributes. We show complete axiomatization using a system of Armstrong-like rules, comment on related computational issues, and the relational vs. propositional semantics of the dependencies. A logic for reasoning with similarity-preserving dependencies is proposed.Its semantics is defined using commutative integral partially ordered monoids.The logic is complete and decidable.A closure-like algorithm for non-contracting sets of dependencies is presented.Two kinds of semantics are discussed: a propositional and a relational one.
A wide range of applications have recently emerged that need to manage large, imprecise data sets. The reasons for imprecision in data are as diverse as the applications themselves: in sensor and RFID data, imprecision is due to measurement errors [15, 34]; in information extraction, imprecision comes from the inherent ambiguity in natural-language text [20, 26]; and in business intelligence, imprecision is tolerated because of the high cost of data cleaning [5]. In some applications, such as privacy, it is a requirement that the data be less precise. For example, imprecision is purposely inserted to hide sensitive attributes of individuals so that the data may be published [30]. Imprecise data has no place in traditional, precise database applications like payroll and inventory, and so, current database management systems are not prepared to deal with it. In contrast, the newly emerging applications offer value precisely because they query, search, and aggregate large volumes of imprecise data to find the “diamonds in the dirt”. This wide-variety of new applications points to the need for generic tools to manage imprecise data. In this paper, we survey the state of the art of techniques that handle imprecise data, by modeling it as probabilistic data [2–4,7,12,15,23,27,36]. A probabilistic database management system, or ProbDMS, is a system that stores large volumes of probabilistic data and supports complex queries. A ProbDMS may also need to perform some additional tasks, such as updates or recovery, but these do not differ from those in conventional database management systems and will not be discussed here. The major challenge in a ProbDMS is that it needs both to scale to large data volumes, a core competence of database management systems, and to do probabilistic inference, which is a problem studied in AI. While many scalable data management systems exists, probabilistic inference is a hard problem [35], and current systems do not scale to the same extent as data management systems do. To address this challenge, researchers have focused on the specific  Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks. References. Index.