Keyun Qin

Lattice-Valued Logic

I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.

Generalized rough sets based on reflexive and transitive relations

In this paper, we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite. It is proved that there exists a one-to-one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition.

Extracting complex linguistic data summaries from personnel database via simple linguistic aggregations

A linguistic data summary of a given data set is desirable and human consistent for any personnel department. To extract complex linguistic data summaries, the LOWA operator is used from fuzzy logic and some numerical examples are also provided in this paper. To obtain a complex linguistic data summary with a higher truth degree, genetic algorithms are applied to optimize the number and membership functions of linguistic terms and to select a part of truth degrees for aggregations, in which linguistic terms are represented by the 2-tuple linguistic representation model.

Interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference

Information systems, which contain only crisp data, precise and unique attribute values for all objects, have been widely investigated. Due to the fact that in realworld applications imprecise data are abundant, uncertainty is inherent in real information systems. In this paper, information systems are called fuzzy information systems, and formalized by (objects; attributes; f), in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values. To interpret and extract fuzzy decision rules from fuzzy information systems, the meta-theory based on modal logic proposed by Resconi et al. is modified. The modified meta-theory not only expresses uncertainty between objects and their attributes, but also uncertainty in the process of recognizing fuzzy information systems. In addition, according to perception computing (proposed by Zadeh), granules of fuzzy information systems can be represented by fuzzy decision rules, so that, fuzzy inference methods can be used to obtain the decision attribute of a new object. Finally, a novel way of combining evidences based on the modified meta-theory is introduced, which extends the concept of combining evidences based on Dempster-Shafer theory.

Approximation operators on complete completely distributive lattices

Rough set, a tool for data mining, deals with the vagueness and granularity in information systems. In 2006, Chen et al. initiated the study of rough approximations on a complete completely distributive lattice (CCD lattice for short) and brought generalizations of rough sets into a unified framework. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed. Basic properties of lower and upper approximation operators are examined. Furthermore, the relationships among these operators are analyzed.

Dissimilarity functions and divergence measures between fuzzy sets

In this paper we propose two approaches to constructing divergence measures. The construction is based on the use of dissimilarity functions and fuzzy equivalencies. Firstly, we introduce some ways of generating dissimilarity functions. Then, we present several formulae of divergence measures. Finally, we examine the properties of divergence measures as a whole.

Combination of interval set and soft set

Abstract Soft set theory and interval set theory are all mathematical tools for dealing with uncertainties. This paper is devoted to the discussion of soft interval set and its application. The notion of soft interval sets is introduced by combining soft set and interval set. Several operations on soft interval sets are presented in a manner parallel to that used in defining operations on soft sets and the lattice structures of soft interval sets are established. In addition, a soft interval set based decision making problem is analyzed.

Relations among similarity measure, subsethood measure and fuzzy entropy

In this paper we study the relations among similarity measure, subsethood measure and fuzzy entropy and present several propositions that similarity measure, subsethood measure and fuzzy entropy can be transformed by each other based on their axiomatic definitions. Some new formulae to calculate similarity measure, subsethood measure and fuzzy entropy are proposed.

Properties of Raha's similarity-based approximate reasoning method

A similarity-based approximate reasoning methodology that requires the construction of fuzzy relations was proposed by Raha et al. [26]. This paper investigates properties of four inferred conclusions calculated by Rahas similarity-based approximate reasoning method. The relationships among the four inferred conclusions are examined. The monotonicity and the approximation property for similarity-based approximate reasoning methods are defined. Then the monotonicity and the approximation property of Rahas similarity-based approximate reasoning method are studied.

Robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures

Abstract This paper discusses the robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures. First of all, the concept of DF-metric is proposed. Secondly, several DF-metrics are introduced as well as their properties and some inequalities about them. Then a formula of divergence measure composed by DF-metric is presented. Finally, based on the proposed divergence measures, the concept of perturbations of fuzzy sets is extended. According to the extended concept, the perturbation parameters raised by various fuzzy connectives are studied and the perturbations of fuzzy reasoning are also investigated.