A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively, including formal concepts in formal concept analysis (FCA), rough concepts arising from rough set theory (RST), and semiconcepts and protoconcepts for dealing with negation. While all these kinds of concepts are associated with lattices, semiconcepts and protoconcepts additionally yield an ordered algebraic structure, called double Boolean algebras. As the name suggests, a double Boolean algebra contains two underlying Boolean algebras. In this article, we investigate logical and algebraic aspects of the representation and reasoning about different concepts with respect to formal contexts. We first review our previous work on two-sorted modal logic systems KB and KF for the representation and reasoning of rough concepts and formal concepts, respectively. Then, in order to represent and reason about both formal and rough concepts in a single framework, these two logics are unified into a two-sorted Boolean modal logic BM , in which semiconcepts and protoconcepts are also expressible. Based on the logical representation of semiconcepts and protoconcepts, we prove the characterization of double Boolean algebras in terms of their underlying Boolean algebras. Finally, we also discuss the possibilities of extending our logical systems for the representation and reasoning of more fine-grained quantitative information in formal contexts.

We present variable precision generalized rough set approach to characterize incomplete decision tables. We show how to determine the discernibility threshold for a reflexive relational decision system in the variable precision generalized rough set model. We also point out some properties of positive regions and prove a statement of the necessary condition for weak consistency of an incomplete decision table. We present two examples to illustrate the results obtained in this paper.

In this paper, we provide a definition of α-fuzzified lower and upper approximations for fuzzy sets based on the α-cut of fuzzy binary relations. We show that the definition is a proper generalization of the previous one for approximations of crisp sets and compare it with an existing definition in the context of fuzzy tolerance relation.

In this article, we propose a logic for reasoning about belief based on fusion of uncertain information. The resultant reason-maintenance possibilistic belief logic can represent both implicit and explicit uncertain beliefs of an agent. While implicit beliefs stipulate what are believable, explicit beliefs can trace the process of belief formation by information fusion. To set up the formal framework, we start with developing a basic reason-maintenance belief logic, present its syntax and semantics, and investigate its axiomatization and properties. Then, we extend the basic logic to accommodate the possibilistic uncertainty of information and beliefs, provide a complete axiomatization of the extended logic, and show that it can address the reason-maintenance issue of partially inconsistent beliefs. We also demonstrate the applicability of our formalisms by using several examples in realistic scenarios.

In reasoning about games, we can understand players' behaviors according to their belief, action, and preference. While modal logic can be easily used to represent and reason about agents' beliefs and knowledge if we adopt an epistemic reading of modal operators, reasoning about action requires the extension of modalities. Dynamic logic is one of the earliest attempt along this direction. The original motivation of dynamic logic is to reason about program. However, it can be applied to any structural set of actions. In this paper, we propose a graded propositional dynamic logic (gPDL) for possibilistic reasoning about regular program.

The notions of approximation and definability in classical rough set theory and their generalizations have received much attention. In this paper, we study such generalizations from the perspective of neighborhood systems. We introduce four different types of definability, called interior definability, closure definability, interior-closure (IC) definability, and weak IC definability respectively. We also point out the relationship between IC definability and other types of definability for some special kinds of neighborhood systems. Several examples are presented to illustrate the concepts introduced in this paper.

In agent-based systems, an agent generally forms her belief based on evidence from multiple sources, such as messages from other agents or perception of the external environment. In this paper, we present a logic for reasoning about evidence and belief. Our framework not only takes advantage of the source-tracking capability of justification logic, but also allows the distinction between the actual observation and simply potential admissibility of evidence. We present the axiomatization for the basic logic and its dynamic extension, investigate its properties, and use a running example to show its applicability to information fusion for autonomous agents.

In this paper, we present the connection between the concepts of Variable Precision Generalized Rough Set model (VPGRS-model) and Neighborhood Systems through binary relations. We provide characterizations of lower and upper approximations for VPGRS-model by introducing minimal neighborhood systems. Furthermore, we explore generalizations by investigating variable parameters which are limited by variable precision. We also prove some properties of lower and upper approximations for VPGRS-model.

Justification logic originated from the study of the logic of proofs. However, in a more general setting, it may be regarded as a kind of explicit epistemic logic. In such logic, the reasons a fact is believed are explicitly represented as justification terms. Traditionally, the modeling of uncertain beliefs is crucially important for epistemic reasoning. Graded modal logics interpreted with possibility theory semantics have been successfully applied to the representation and reasoning of uncertain beliefs; however, they cannot keep track of the reasons an agent believes a fact. This article is aimed at extending the graded modal logics with explicit justifications. We introduce a possibilistic justification logic, present its syntax and semantics, and investigate its metaproperties, such as soundness, completeness, and realizability.

Motivated by successful applications of rough set theory to symbolic knowledge discovery from data tables, we would like to extend the approach to description logic (DL)-based mining of social networks. Unlike classical rough set theory, in which the attribute values of objects fully determine the indiscernibility relation, the rough set analysis of social networks must account for the social relationships between objects as well as their attributes. In this paper, the indiscernibility relation is defined by using the notions of positional equivalences in social network analysis. The indiscernibility relation can partition the universe of a social network into elementary sets which are used to define the lower and upper approximations of an arbitrary concept as in classical rough set theory. To induce concept definitions from such approximations, we use DL to represent knowledge discovered from social networks and present a constructive procedure to find a characterizing DL concept terms for each elementary set. Because the lower and upper approximations of a target concept are unions of elementary sets, we can use the disjunction of such characterizing concept terms to describe the definition of the target concept. This leads to a complete process of DL-based concept mining from social networks.

Social network analysis is a methodology used extensively in social sciences. While classical social network can only represent the qualitative relationships between actors, weighted social networks can describe the degrees of connection between actors. In classical social network, regular equivalence is used to capture the similarity between actors based on their linking patterns with other actors. Specifically, two actors are regularly equivalent if they are equally related to equivalent others. The definition of regular equivalence have been extended to weighted social networks in two ways. The first definition, called regular similarity, is to consider regular equivalence as an equivalence relation that commutes with the underlying graph edges. The second definition, called generalized regular equivalence, is based on the notion of role assignment. A role assignment is a mapping from the set of actors to a set of roles. The mapping is regular if actors assigned to the same role have the same roles in their neighborhoods. Recently, it was shown that social positions based on regular equivalence can be syntactically expressed as well-formed formulas in a kind of modal logic. Thus, actors occupying the same social position based on regular equivalence will satisfy the same set of modal formulas. In this paper, we will present analogous results for regular similarity and generalized regular equivalence based on many-valued modal logics.

A large variety of data mining and machine learning techniques are applied to a wide range of applications today. There- fore, there is a real need to develop technologies that allows data analysis while preserving the confidentiality of the data. Secure multi-party computation (SMC) protocols allows participants to cooperate on various computations while retaining the privacy of their own input data, which is an ideal solution to this issue. Although there is a number of frameworks developed in SMC to meet this challenge, but they are either tailored to perform only on specific tasks or provide very limited precision. In this paper, we have developed protocols for floating point arithmetic based on secure scalar product protocols, which is re- quired in many real world applications. Our protocols follow most of the IEEE-754 standard, supporting the four fundamental arithmetic operations, namely addition, subtraction, multiplication, and division. We will demonstrate the practicality of these protocols through performing various statistical calculations that is widely used in most data analysis tasks. Our experiments show the performance of our framework is both practical and promising.

In data mining problems, data is usually provided in the form of data tables. To represent knowledge discovered from data tables, a decision logic (DL) is proposed in rough set theory. DL is an instance of propositional logic, but we can use other logical formalisms to describe data tables. In this paper, we propose two descriptions of data tables based on first-order data logic (FODL) and attribute value-sorted logic (AVSL) respectively. In the context of FODL, we show that explicit definability and implicit definability in classical logic implies the notion of definability in rough set theory. We also show that AVSL is particularly useful for the representation of properties of many-valued data tables.

Attribute reduction is very important in rough set-based data analysis (RSDA) because it can be used to simplify the induced decision rules without reducing the classification accuracy. The notion of reduct plays a key role in rough set-based attribute reduction. In rough set theory, a reduct is generally defined as a minimal subset of attributes that can classify the same domain of objects as unambiguously as the original set of attributes. Nevertheless, from a relational perspective, RSDA relies on a kind of dependency constraint. That is, the relationship between the class labels of a pair of objects depends on the componentwise comparison of their condition attributes. The larger the number of condition attributes compared, the greater the probability that the constraint will hold. Thus, elimination of condition attributes may cause more object pairs to violate the constraint. Based on this observation, a reduct can be defined alternatively as a minimal subset of attributes that does not increase the number of objects violating the constraint. While the alternative definition coincides with the original one in ordinary RSDA, it is more easily generalized to cases of fuzzy RSDA and relational data analysis.

Collaborative recommender systems use various types of information to help customers find products of personalized interest. To increase the usefulness of collaborative recommender systems in certain circumstances, it could be desirable to merge recommender system databases between companies, thus expanding the data pool. This can lead to privacy disclosure hazards during the merging process. This paper addresses how to avoid privacy disclosure in collaborative recommender systems by comparing with major cryptology approaches and constructing a more efficient privacy-preserving collaborative recommender system based on the scalar product protocol.

There is a fair amount of research about privacy, but few empirical studies about its cost have been conducted. In the area of secure multiparty computation, the scalar product has long been reckoned as one of the most promising alternatives to classic logic gates. The reason for this is that the scalar product is not only complete, which is as good as logic gates, but also much more efficient than logic gates. As a result, we set out to study the computation and communication resources needed for some of the most well-known and frequently referenced secure scalar product protocols, including the composite residuosity, the invertible matrix, the polynomial sharing, and the commodity-based approaches. In addition to the implementation details of these approaches, we analyze and compare their execution time, computation time, and memory and random number consumption. Moreover, Fairplay, the benchmark approach that implements Yao's circuit evaluation protocol, is also included in our experiments in order to demonstrate the potential for the scalar products to replace logic gates.

Rough set theory is an effective tool for data mining. According to the theory, a concept is definable if it can be written as a Boolean combination of equivalence classes induced from classification attributes. On the other hand, definability in logic has been explicated by Beth's theorem. In this paper, we propose two data representation formalisms, called first-order data logic (FODL) and attribute value-sorted logic (AVSL), respectively. Based on these logics, we explore the relationship between logical definability and rough set definability.

Fuzzy interpolative reasoning is an inference technique for dealing with the sparse rules problem in sparse fuzzy-rule-based systems. In this paper, we present a new fuzzy interpolative reasoning method for sparse fuzzy-rule-based systems based on the areas of fuzzy sets. The proposed method uses the weighted average method to infer the fuzzy interpolative reasoning results and has the following advantages: 1) it holds the normality and the convexity of the fuzzy interpolative reasoning result, 2) it can deal with fuzzy interpolative reasoning with complicated membership functions, 3) it can deal with fuzzy interpolative reasoning when the fuzzy sets of the antecedents and the consequents of the fuzzy rules have different kinds of membership functions, 4) it can handle fuzzy interpolative reasoning with multiple antecedent variables, 5) it can handle fuzzy interpolative reasoning with multiple fuzzy rules, and 6) it can handle fuzzy interpolative reasoning with logically consistent properties with respect to the ratios of fuzziness. We use some examples to compare the fuzzy interpolative reasoning results of the proposed method with those of the existing fuzzy interpolative reasoning methods. In terms of the six evaluation indices, the experimental results show that the proposed method performs more reasonably than the existing methods. The proposed method provides us a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy-rule-based systems.

Fuzzy interpolative reasoning is an important inference technique for sparse fuzzy rule-based systems. In this paper, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the areas of fuzzy sets. The proposed method can guarantee the normality and the convexity of the conclusion and can deal with fuzzy interpolative reasoning with complicated membership functions, such as hexagons, polygons and Gaussians. Moreover, the proposed method can deal with the situation when the antecedents and the consequents of fuzzy rules belong to different kinds of membership functions. We use several examples to compare the fuzzy interpolative reasoning results of the proposed method with the ones of the existing methods. The experimental results show that the proposed method is more suitable to deal with fuzzy interpolative reasoning than the existing methods. The proposed method provides a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy rule-based systems.

Social network analysis is an important methodology in sociological research. Though social network data is very useful to researchers and policy makers, releasing such data to the public may cause an invasion of privacy. We generalize the techniques for protecting personal privacy in tabulated data, and propose some metrics of anonymity for assessing the risk of breaching confidentiality by disclosing social network data. We assume a situation of data publication, where data is released to the general public. We adopt description logic as the underlying knowledge representation formalism, and consider the metrics of anonymity in open world and closed world contexts respectively.

In rough set theory, objects are partitioned into equivalence classes based on their attribute values, which are essentially functional information associated with the objects. Therefore, rough set theory can be viewed as a theory of functional granulation. In contrast, relational information systems (RIS) specify the relationships between objects, instead of the properties of objects. In this paper, we present a theory of granulation based on hybrid information systems (HIS), which combine functional information systems (FIS) and RIS. We study the relationship between FIS and RIS. We also define the indiscernibility relation based on relational information, and use it to develop a theory of granulation based on HIS.

This article provides a modal logic framework for reasoning about multi-agent belief and its fusion. We propose logics for reasoning about cautiously merged agent beliefs that have different degrees of reliability. These logics are obtained by combining the multi-agent epistemic logic and multi-source reasoning systems. The fusion is cautious in the sense that if an agent's belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first, calledlevel cutting fusion, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously. In the second, calledlevel skipping fusion, only the level at which the inconsistency occurs is skipped. We present the formal semantics and axiomatic systems for these two strategies and discuss some applications of the proposed logical systems. We also develop a tableau proof system for the logics and prove the complexity result for the satisfiability and validity problems of these logics.

A query on the distribution of a sensitive field within a selected population in a database can be submitted to the data center, and the answer to this query can leak private information, even though no identification information is provided. Inspired by decision theory, we present a quantitative model of the privacy protection problem in such a database query environment. In our model, the user information states are defined as classes of probability distributions on the set of possible confidential values. These states can be modified and refined by knowledge acquisition actions. The data confidentiality is guaranteed by ensuring that misusing private information is more costly than any possible gain. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A generalized decision logic (GDL) language is proposed for granular computing (GrC) in Tarski's style through the notions of a model and satisfiability. The model is an information table consisting of a finite set of objects described by a finite set Of attributes. A concept or a granule is characterized by a pair consisting of the intension of the concept, a formula of the language GDL, and the extension of the concept, a subset of the universe. We discuss the application of GDL in formal concepts and decision rules. The former deals with the description and interpretation of granules, and the latter deals with the relationships between granules.

In this paper the qualitative and quantitative semantics for rules in data tables are investigated from a logical viewpoint. In modern data analysis, knowledge can be discovered from data tables and is usually represented by some rules. However the knowledge is useful for a human user only when he can understand the meaning of the rules. This is called the interpretability problem of intelligent data analysis. The solution of the problem depends on the selection of the rule representation language. A good representation language should have clear semantics so that a rule can be effectively validated with respect to the given data tables. In this regard, logic is one of the best choices. Starting from reviewing the decision logic for data tables, we subsequently generalize it to fuzzy and possibilistic decision logics. The rules are then viewed as the implications between well-formed formulas of these logics and their semantics with respect to precise or uncertain data tables are presented. The validity, support, and confidence of a rule are also rigorously defined in the framework.

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

In this paper, we will develop a class of logics for reasoning about qualitative and quantitative uncertainty. The semantics of the logics is uniformly based on possibility theory. Each logic in the class is parameterized by a t-norm operation on [0,1], and we express the degree of implication between the possibilities of two formulas explicitly by using residuated implication with respect to the t-norm. The logics are then shown to be applicable to possibilistic reasoning, approximate reasoning, and nonmonotonic reasoning.

In this paper, we would like to investigate the relationship between evidential structures (ES)—the basic qualitative structures of Dempster-Shafer theory, and the data table based knowledge representation systems(KRS) subject to rough set analysis. It is shown that an ES has a natural representation as a data table and from a given data table and two of its attributes, an ES can be extracted. We also show that some important operations on ES's can be realized in relational algebra. The results are then generalized to the fuzzy case. Consequently, we further clarify the connection between evidence theory and rough set theory.

The concept of fuzzy equality and its related contents to the first order predicate calculus are discussed. It is proved that, in the viewpoint of computational logic, resolution and paramodulation mechanisms are complete and sound for fuzzy logic with equality. Term rewriting system, that is the set of left to right directional equations, provides an essential computational paradigm for word problems in universal algebra. We embody the fuzzy equality to the theory of this computation system and give an algorithmic solution to the word problems in fuzzy algebra.