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.

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.

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.

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

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.

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.

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.

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.

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.