In this paper, we present a generalization of the relational data model based on interval neutrosophic set. Our data model is capable of manipulating incomplete as well as inconsistent information. Fuzzy relation or intuitionistic fuzzy relation can only handle incomplete information. Associated with each relation are two membership functions one is called truth-membership function T which keeps track of the extent to which we believe the tuple is in the relation, another is called falsity-membership function F which keeps track of the extent to which we believe that it is not in the relation. A neutrosophic relation is inconsistent if there exists one tuple a such that T(a) + F(a) > 1 . In order to handle inconsistent situation, we propose an operator called "split" to transform inconsistent neutrosophic relations into pseudo-consistent neutrosophic relations and do the set-theoretic and relation-theoretic operations on them and finally use another operator called "combine" to transform the result back to neutrosophic relation. For this data model, we define algebraic operators that are generalizations of the usual operators such as intersection, union, selection, join on fuzzy relations. Our data model can underlie any database and knowledge-base management system that deals with incomplete and inconsistent information.
Relational data model was proposed by Ted Codd's pioneering paper *2+. Since then, relational database systems have been extensively studied and a lot of commercial relational database systems are currently available [3,4]. This data model usually takes care of only welldefined and unambiguous data. However, imperfect information is ubiquitous -almost all the information that we have about the real world is not certain, complete and precise [5]. Imperfect information can be classified as: incompleteness, imprecision, uncertainty, and inconsistency. Incompleteness arises from the absence of a value, imprecision from the existence of a value which cannot be measured with suitable precision, uncertainty from the fact that a person has given a subjective opinion about the truth of a fact which he/she does not know for certain, and inconsistency from the fact that there are two or more conflicting values for a variable.
In order to represent and manipulate various forms of incomplete information in relational databases, several extensions of the classical relational model have been proposed [6, 7, 8, 9, 10, 11+. In some of these extensions, a variety of “null values” have been introduced to model unknown or not-applicable data values. Attempts have also been made to generalize operators of relational algebra to manipulate such extended data models [6,8,11,12,13]. The fuzzy set theory and fuzzy logic proposed by Zadeh [14] provide a requisite mathematical framework for dealing with incomplete and imprecise information. Later on, the concept of interval-valued fuzzy sets was proposed to capture the fuzziness of grade of membership itself [15]. In 1986, Atanassov introduced the intuitionistic fuzzy set [16] which is a generalization of fuzzy set and provably equivalent to interval-valued fuzzy set. The intuitionistic fuzzy sets consider both truth-membership T and falsity-membership
Because of the restriction, the fuzzy set, interval-valued fuzzy set, and intuitionistic fuzzy set cannot handle inconsistent information. Some authors [17,18,19,20,21,22,23] have studied relational databases in the light of fuzzy set theory with an objective to accommodate a wider range of real-world requirements and to provide closer man-machine interactions. Probability, possibility, and Dempster-Shafer theory have been proposed to deal with uncertainty.
Possibility theory [24] is built upon the idea of a fuzzy restriction. That means a variable could only take its value from some fuzzy set of values and any value within that set is a possible value for the variable. Because values have different degrees of membership in the set, they are possible to different degrees. Prade and Testemale [25] initially suggested using possibility theory to deal with incomplete and uncertain information in database. Their work is extended in [26] to cover multivalued attributes. Wong [27] proposes a method that quantifies the uncertainty in a database using probabilities. His method maybe is the simplest one which attached a probability to every member of a relation, and to use these values to provide the probability that a particular value is the correct answer to a particular query. Carvallo and Pittarelli [28] also use probability theory to model uncertainty in relational databases systems. Their method augmented projection and join operations with probability measures.
However, unlike incomplete, imprecise, and uncertain information, inconsistent information has not enjoyed enough research attention. In fact, inconsistent information exists in a lot of applications. For example, in data warehousing application, inconsistency will appear when trying to integrate the data from many different sources. Another example is that in the expert system, there exist facts which are inconsistent with each other. Generally, two basic approaches have been followed in solving the inconsistency problem in knowledge base: belief revision and paraconsistent logic. The goal of the first approach is to make an inconsistent theory consistent, either by revising it or by representing it by a consistent semantics. On the other hand, the paraconsistent approach allows reasoning in the presence of inconsistency, and contradictory information can be derived or introduced without trivialization [29]. Bagai and Sunderraman [30,31] proposed a paraconsistent realational data model to deal with incomplete and inconsistent information. The data model has been applied to compute the well-founded and fitting model of logic programming [32,33]. This data model is based on paraconsistent logics which were studied in detail by de Costa [34] and Belnap [35].
In this paper, we present a new relational data model -neutrosophic relational data model (NRDM). Our model is based on the neutrosophic set theory which is an extension of intuitionistic fuzzy set theory [36] and is capable of manipulating incomplete as well as inconsistent information. We use both truth-membership function
