Fredkin Gates for Finite-valued Reversible and Conservative Logics

The basic principles and results of Conservative Logic introduced by Fredkin and Toffoli on the basis of a seminal paper of Landauer are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram

Study on Rough Sets and Fuzzy Sets in Constructing Intelligent Information System

Since human being is not an omniscient and omnipotent being, we are actually living in an uncertain world. Uncertainty was involved and connected to every aspect of human life as a quotation from Albert Einstein said: �As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.� The most fundamental aspect of this connection is obviously shown in human communication. Naturally, human communication is built on the perception1-based information instead of measurement-based information in which perceptions play a central role in human cognition [Zadeh, 2000]. For example, it is naturally said in our communication that �My house is far from here.� rather than let say �My house is 12,355 m from here�. Perception-based information is a generalization of measurement-based information, where perception-based information such as �John is excellent.� is hard to represent by measurement-based version. Perceptions express human subjective view. Consequently, they tend to lead up to misunderstanding. Measurements then are needed such as defining units of length, time, etc., to provide objectivity as a means to overcome misunderstanding. Many measurers were invented along with their methods and theories of measurement. Hence, human cannot communicate with measurers including computer as a product of measurement era unless he uses measurement-based information. Perceptions are intrinsic aspect in uncertainty-based information. In this case, information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. 1In psychology, perception is understood as a process of translating sensory stimulation into an organized experience Generally, these various information deficiencies may express different types of uncertainty. It is necessary to construct a computer-based information system called intelligent information system that can process uncertainty-based information. In the future, computers are expected to be able to make communication with human in the level of perception. Many theories were proposed to express and process the types of uncertainty such as probability, possibility, fuzzy sets, rough sets, chaos theory and so on. This book extends and generalizes existing theory of rough set, fuzzy sets and granular computing for the purpose of constructing intelligent information system. The structure of this book is the following: In Chapter 2, types of uncertainty in the relation to fuzziness, probability and evidence theory (belief and plausibility measures) are briefly discussed. Rough set regarded as another generalization of crisp set is considered to represent rough event in the connection to the probability theory. Special attention will be given to formulation of fuzzy conditional probability relation generated by property of conditional probability of fuzzy event. Fuzzy conditional probability relation then is used to represent similarity degree of two fuzzy labels. Generalization of rough set induced by fuzzy conditional probability relation in terms of covering of the universe is given in Chapter 3. In the relation to fuzzy conditional probability relation, it is necessary to consider an interesting mathematical relation called weak fuzzy similarity relation as a generalization of fuzzy similarity relation proposed by Zadeh [1995]. Fuzzy rough set and generalized fuzzy rough set are proposed along with the generalization of rough membership function. Their properties are examined. Some applications of these methods in information system such as α-redundancy of object and dependency of domain attributes are discussed. In addition, multi rough sets based on multi-context of attributes in the presence of multi-contexts information system is defined and proposed in Chapter 4. In the real application, depending on the context, a given object may have different values of attributes. In other words, set of attributes might be represented based on different context, where they may provide different values for a given object. Context can be viewed as background or situation in which somehow it is necessary to group some attributes as a subset of attributes and consider the subset as a context. Finally, Chapter 5 summarizes all discussed in this book and puts forward some future topics of research

Geometric lattice structure of covering and its application to attribute reduction through matroids

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice to study the attribute reduction issues of information systems

Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets