Codd's relational model describes just one possible world. To better cope with incomplete information, extended database models allow several possible worlds. Vague tables are one such convenient extended model where attributes accept sets of possible values (e.g., the manager is either Jill or Bob). However, conceptual database design in such cases remains an open problem. In particular, there is no canonical definition of functional dependencies (FDs) over possible worlds (e.g., each employee has just one manager). We identify several desirable properties that the semantics of such FDs should meet including Armstrong's axioms, the independence from irrelevant attributes, seamless satisfaction and implied by strong satisfaction. We show that we can define FDs such that they have all our desirable properties over vague tables. However, we also show that no notion of FD can satisfy all our desirable properties over a more general model (disjunctive tables). Our work formalizes a trade-off between having a general model and having well-behaved FDs.
The second approach, based on possibilistic and probabilistic databases, can be considered an extension of the the possible worlds framework @cite_4 @cite_6 @cite_29 . However, Link and Prade @cite_23 assign possibilities to tuples, not to values. Based on these possibilistic tuples, possible worlds are generated as a nested chain: the smallest world is the one with only fully possible tuples; the largest world contains all tuples. Certainty degrees are attached to standard FDs, based on the possibility degree of the smallest world in which they are violated.
@cite_23 @cite_29 @cite_4 @cite_6
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We investigate the impact of uncertainty on relational data -base schema design. Uncertainty is modeled qualitatively by assigning to tuples a degree of possibility with which they occur, and assigning to functional dependencies a degree of certainty which says to which tuples they apply. A design theory is developed for possibilistic functional dependencies, including efficient axiomatic and algorithmic characterizations of their implication problem. Naturally, the possibility degrees of tuples result in a scale of different degrees of data redundancy. Scaled versions of the classical syntactic Boyce-Codd and Third Normal Forms are established and semantically justified in terms of avoiding data redundancy of different degrees. Classical decomposition and synthesis techniques are scaled as well. Therefore, possibilistic functional dependencies do not just enable designers to control the levels of data integrity and losslessness targeted but also to balance the classical trade-off between query and update efficiency. Extensive experiments confirm the efficiency of our framework and provide original insight into relational schema design. Probabilistic databases have received considerable attention recently due to the need for storing uncertain data produced by many real world applications. The widespread use of probabilistic databases is hampered by two limitations: (1) current probabilistic databases make simplistic assumptions about the data (e.g., complete independence among tuples) that make it difficult to use them in applications that naturally produce correlated data, and (2) most probabilistic databases can only answer a restricted subset of the queries that can be expressed using traditional query languages. We address both these limitations by proposing a framework that can represent not only probabilistic tuples, but also correlations that may be present among them. Our proposed framework naturally lends itself to the possible world semantics thus preserving the precise query semantics extant in current probabilistic databases. We develop an efficient strategy for query evaluation over such probabilistic databases by casting the query processing problem as an inference problem in an appropriately constructed probabilistic graphical model. We present several optimizations specific to probabilistic databases that enable efficient query evaluation. We validate our approach by presenting an experimental evaluation that illustrates the effectiveness of our techniques at answering various queries using real and synthetic datasets. The problem of maintaining consistency via functional dependencies (FDs) has been studied and analyzed extensively within traditional database settings. There have also been many probabilistic data models proposed in the past decades. However, the problem of maintaining consistency in probabilistic relations via FDs is still unclear. In this paper, we clarify the concept of FDs in probabilistic relations and present an efficient chase algorithm LPChase(r,F) for maintaining consistency of a probabilistic relation r with respect to an FD set F. LPChase(r,F) adopts a novel approach that uses Linear Programming (LP) method to modify the probability of data values in r. There are many benefits of our approach. First, LPChase(r,F) guarantees that the output result is always the minimal change to r. Second, assuming that the expected size of an active domain consisting data values with non-zero probability is fixed, we demonstrate the interesting result that the LP solving time in LPChase(r,F) decreases as the probabilistic data domains grow, and becomes negligible for large domain size. On the other hand, the I O time and modeling time become stable even when the domain size increases. This paper deals with the extraction of default rules from a database of examples. The proposed approach is based on a special kind of probability distributions, called "big-stepped probabilities", which are known to provide a semantics for non-monotonic reasoning. The rules which are learnt are genuine default rules, which could be used (under some conditions) in a non-monotonic reasoning system and can be encoded in possibilistic logic.