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 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. A new approach, to measure normalization completeness for conceptual model, is introduced using quantitative fuzzy functionality in this paper. We measure the normalization completeness of the conceptual model in two steps. In the first step, different normalization techniques are analyzed up to Boyce Codd Normal Form (BCNF) to find the current normal form of the relation. In the second step, fuzzy membership values are used to scale the normal form between 0 and 1. Case studies to explain schema transformation rules and measurements. Normalization completeness is measured by considering completeness attributes, preventing attributes of the functional dependencies and total number of attributes such as if the functional dependency is non-preventing then the attributes of that functional dependency are completeness attributes. The attributes of functional dependency which prevent to go to the next normal form are called preventing attributes. This paper is situated in the area of fuzzy databases, i.e., databases containing imprecise information. More precisely, it deals with the notion of a functional dependency when data involved in this type of property are possibly imprecise. Contrary to previous works, the idea is not to fuzzify the concept of a functional dependency. In the view suggested here, we rather consider regular functional dependencies and we study the impact of the presence of such FDs on the insertion and handling of imprecise data.   Abstract The paper contains an analysis of integrity constraints for -ary relations in fuzzy databases. Apart from dependencies between all attributes there may be also dependencies describing relationships of fewer attributes. However, there is no complete arbitrariness. Relationships comprising ( 1) attributes must not infringe integrity constraints of the -ary relation. The analysis is carried out using the theory of functional dependencies. In this paper, we assume that attribute values are represented by means of interval-valued possibility distributions. The notion of fuzzy functional dependency has been appropriately extended according to the representation of fuzzy data. The paper formulates the rules to which fuzzy functional dependencies between ( 1) attributes of the -ary relation must be subordinated. In this paper, we introduce a definition of the concept of a functional dependency (FD) in the context of databases containing ill-known attributes values represented by possibility distributions. Contrary to previous proposals, this definition is based on the possible worlds model and consists in viewing the satisfaction of an FD by a relation as an uncertain event whose possibility and necessity can be quantified. We give the principle of a method for incrementally computing the related possibility and necessity degrees and tackle the issue of tuple refinement in the presence of an FD.  Vague information is common in many database applications due to intensive data dissemination arising from different pervasive computing sources, such as the high volume data obtained from sensor networks and mobile communications. In this paper, we utilize functional dependencies (FDs) and inclusion dependencies (INDs), which are the most fundamental integrity constraints that arise in practice in relational databases, to maintain the consistency of a vague database. First, we tackle the problem, given a vague relation r and a set of FDs F, of how to obtain the ''best'' approximation of r with respect to F when taking into account the median membership (m) and the imprecision membership (i) thresholds. Using these two thresholds of a vague set, we define the notion of mi-overlap between vague sets and a merge operation on r. Second, we consider, given a vague database d and a set of INDs N, how to obtain the minimal possible change in value-precision for d. Satisfaction of an FD in r is defined in terms of values being mi-overlapping while satisfaction of an IND in d is defined in terms of value-precision. We show that Lien's and Atzeni's axiom system is sound and complete for FDs being satisfied in vague relations and that 's axiom system is sound and complete for INDs being satisfied in vague databases. Finally, we study the chase procedure VChase(d,[email protected]?N) as a means to maintain consistency of d with respect to F and N. Our main result is that the output of the procedure is the most object-precise approximation of r with respect to F and the minimum value-precision change of d with respect to N. The complexity of VChase(r,F) is polynomial time in the sizes of r and F whereas the complexity of VChase(d,[email protected]?N) is exponential. 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. Handling missing data is widely studied to make proper replacement and reduce uncertainty of data. Several approaches have been proposed for providing the most possible results. However, few studies provide solutions to the problem of missing data in extended possibility-based fuzzy relational (EPFR) databases. This type of problem in the context of EPFR databases is difficult to resolve because of the complexity of the data involved. In this paper, we propose an approach of filling missing data and query processing of the databases. To obtain the rational predict of the missing data, we adopt a concept and measurement of proximate equality of tuples to define data operation and fuzzy functional dependency (FFD). We provide a method to predict the missing data and replace the data based on our proposal. The results of the missing value process preserve those FFDs that hold in the original database instance. Abstract Incomplete relations are relations which contain null values, whose meaning is “value is at present unknown”. Such relations give rise to two types of functional dependency (FD). The first type, called the strong FD (SFD), is satisfied in an incomplete relation if for all possible worlds of this relation the FD is satisfied in the standard way. The second type, called the weak FD (WFD), is satisfied in an incomplete relation if there exists a possible world of this relation in which the FD is satisfied in the standard way. We exhibit a sound and complete axiom system for both strong and weak FDs, which takes into account the interaction between SFDs and WFDs. An interesting feature of the combined axiom system is that it is not k -ary for any natural number k ⩾ 0. We show that the combined implication problem for SFDs and WFDs can be solved in time polynomial in the size of the input set of FDs. Finally, we show that Armstrong relations exist for SFDs and WFDs.