A Note on the Simplification of Fuzzy Switching Functions

Abstract

This note is concerned with the modification of the definition of fuzzy consensus. The modified version is used to find the set of all fuzzy prime implicants of a fuzzy switching function as defined in [1]. The fuzzy algebra used to derive these functions satisfies the set of axioms of a distributive lattice with unique identities under the operators of maximum and minimum as described in [5], [4], and [3].     

Calculation of the paired Haar transform through shared binary decision diagrams

A concept of paired Haar transform (PHT) for representation and efficient optimization of systems of incompletely Boolean functions has recently been introduced. In this article, a method to calculate PHT for incompletely specified switching functions through shared binary decision diagrams (SBDDs) is presented. The algorithm converts switching functions in the form of SBDDs into their paired Haar spectra and can operate on functions with many variables.     

加权关联规则挖掘算法的研究

讨论了加权关联规则的挖掘算法，对布尔型属性，在挖掘算法MINWAL（O）和MINWAL（W）的基础上给出一种改进的加权关联规则挖掘算法，此算法能有效地考虑布尔型属必的重要性和规则中所含属性的个数，对数量型属性，应用竞争聚集算法将数量型属性划分成若干个模糊集，产系统地提出加权模糊关联规则的挖掘算法，此算法能有效地考虑数量型属性的重要性和规则中所含属性的个数，并适用于大型数据库。     

Fuzzy Sets and Neural Networks

Abstract

It is possible that a better model for the behavior of a nerve cell may be provided by what might be called a fuzzy neuron, which is a generalization of the McCulloch-Pitts model. The concept of a fuzzy neuron employs some of the concepts and techniques of the theory of fuzzy sets which was introduced by Zadeh [2, 3] and applied to the theory of automaton by Wee and Fu [6], Tanaka et al. [7], Santo [8] and others. In effect, the introduction of fuzziness into the model of a neuron makes it better adapted to the study of the behavior of systems which are imprecisely defined by virtue of their high degree of complexity. Many of the biological systems, economic systems, urban systems and more generally, large-scale systems fall into this category.

In the nearly three decades since its publication, the pioneering work of McCulloch and Pitts [1], has had a profound influence on the development of the theory of neural nets, in addition to stimulating much of the early work in automata theory and regular events.

Although the McCulloch-Pitts model of a neuron has contributed a great deal to the understanding of the behavior of neural-like systems, it fails to reflect the fact that the behavior of even the simplest type of nerve cell exhibits not only randomness but, more importantly, a type of imprecision which is associated with the lack of sharp transition from the occurrence of an event to its non-occurrence.

In this paper, some basic properties of fuzzy neural networks as well as their applications to the synthesis of fuzzy automata are investigated. It is shown that any n-state minimal fuzzy automaton can be realized by a network of m fuzzy neurons, where ┌log₂ n┐ ? m ? 2n. Examples are given to illustrate the procedure. As an example of application, a realization of λ-fuzzy language recognizer using a fuzzy neural network is presented. The techniques described in this paper may be of use in the study of neural networks as well as in formal languages, pattern recognition, and learning.     

A NOTE ON “DECOMPOSITION PROBLEM OF FUZZY RELATIONS”

The decomposition problem of a fuzzy relation R ∈ F(X × X) can be stated as: “Given a fuzzy relation R∈F(X ×.X), to determine whether there exists a fuzzy relation Z∈f(X × X) such that R = Z [Odot] Z, where X is a finite set and “[Odot]” is the max-min composition of two fuzzy relations.” In particular, if R is a Boolean matrix, then this problem becomes to find the square root of a Boolean matrix, which is a well-known unsolved problem. In 1985, Di Nola et al. (A. Di Nola, S. Sessa and W. Pedrycz, Int J. General Systems,. 10, 1985, 123?133) had solved it in theory, and proposed a numerical algorithm, illustrated by a flowchart In this note, we first point out that the flowchart proposed by Di Nola et al. is in error and give a correct flowchart. Then we give a numerical example, which is also a counterexample of the flowchart given by Di Nola el al., to explain our flowchart.     

Algebraic Methods Versus Map Methods Of Solving Boolean Equations

We illustrate the efficiency of algebraic methods of solution for Boolean equations by resuming an equation solved in Ref. [9] by tabular methods and solving it algebraically.     

Logic Minimization as an Efficient Means of Fuzzy Structure Discovery

Established methods of Boolean minimization have previously unseen potential as an efficient and unrestricted means of fuzzy structure discovery, becoming particularly useful within a design methodology for the automatic development of fuzzy models. Traditionally used in digital systems design, logic minimization tools allow us to exploit the fundamental links between binary (two-valued) and fuzzy (multivalued) logic. In this paper, we show how logic optimization plays an integral role in a two-phase fuzzy model design process. Adaptive logic processing is realized as the discovered Boolean structures are augmented with fuzzy granules and then refined by adjusting connections of fuzzy neurons, helping to further capture the numeric details of the target systems behavior. Accurate and highly interpretable fuzzy models are the result of the entire development process.     

A Survey on Fuzzy Implication Functions

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.     

Genetic-fuzzy approach to the Boolean satisfiability problem

This study is concerned with the Boolean satisfiability (SAT) problem and its solution in setting a hybrid computational intelligence environment of genetic and fuzzy computing. In this framework, fuzzy sets realize an embedding principle meaning that original two-valued (Boolean) functions under investigation are extended to their continuous counterparts resulting in the form of fuzzy (multivalued) functions. In the sequel, the SAT problem is reformulated for the fuzzy functions and solved using a genetic algorithm (GA). It is shown that a GA, especially its recursive version, is an efficient tool for handling multivariable SAT problems. Thorough experiments revealed that the recursive version of the GA can solve SAT problems with more than 1000 variables     

Information systems in modeling interactive computations on granules

In this paper, we discuss the importance of information systems in modeling interactive computations performed on (complex) granules and we propose a formal approach to interactive computations based on generalized information systems and rough sets which can be combined with other soft computing paradigms such as fuzzy sets or evolutionary computing, but also with machine learning and data mining techniques. Information systems are treated as dynamic granules used for representing the results of the interaction of attributes with the environment. Two kinds of attributes are distinguished, namely, the perception attributes, including sensory attributes, and the action attributes. Sensory attributes are the basic perception attributes, other perception attributes are constructed on the basis of the sensory ones. Actions are activated when their guards, being often complex and vague concepts, are satisfied to a satisfactory degree. The guards can be approximated on the basis of measurements performed by sensory attributes rather than defined exactly. Satisfiability degrees for guards are results of reasoning called the adaptive judgment. The approximations are induced using hierarchical modeling. We show that information systems can be used for modeling more advanced forms of interactions in hierarchical modeling. The role of hierarchical interactions is emphasized in the modeling of interactive computations. Some illustrative examples of interactions used in the ACT-R 6.0 system are reported. ACT-R 6.0 is based on a cognitive architecture and can be treated as an example of a highly interactive complex granule which can be involved in hierarchical interactions. For modeling of interactive computations, we propose much more general information systems than the studied dynamic information systems (see, e.g., Ciucci (2010) [8] and Pa?asiński and Pancerz (2010) [32]). For example, the dynamic information systems are making it possible to consider incremental changes in information systems. However, they do not contain the perception and action attributes necessary for modeling interactive computations, in particular for modeling intrastep interactions.     

Complexity Management in Fuzzy Systems: A Rule Base Compression Approach [Book Review]

This book consists of ten chapters and covers some of the following material: fuzzy logic systems; rule structures; base reduction methods; Boolean matrices; binary relations; and transformations.     

Boolean vector algebraic structures: properties and applications to digital logic simulation

Boolean algebra is the basic mathematic used in the analysis and synthesis of binary systems. Technological advances have led to an increasing interest in multivalued logic systems where more than two logical values are used.In the application of multivalued logic, each logical value could often be represented by a Boolean vector, ie a vector with binary components (0 or 1). Therefore, it is quite important to have a thorough understanding of properties embedded in the algebraic structures using Boolean vectors as basic operands. This gives the motivation for this study.Boolean vector operations are introduced and two major modes associated with the complement are characterized. Then we define a Boolean vector E-algebra and its major features are given. Finally, some applications are discussed and illustrated.     

Quasi-time-dependent stabilisation for 2-D switched systems with persistent dwell-time

ABSTRACT

This paper is concerned with the stabilisation problem for a class of discrete-time two-dimensional (2-D) switched systems with persistent dwell-time (PDT). The systems are described by the well-known Fornasini-Marchesini local state space (FMLSS) model. The concept of PDT switching signals is introduced herein, and each stage consists of a dwell-time period in which no switching occurs and a persistent period an arbitrary switching occurs. Based on a proper Lyapunov function suitable to the PDT switching, which is both quasi-time-dependent (QTD) and mode-dependent, the QTD state feedback controller is designed to ensure the closed-loop system is exponentially stable. Compared with time-independent criteria, new results are more general and flexible, and have less conservativeness. Finally, two examples are provided to show the effectiveness and potential of our proposed methods.     

Cooperative games with multiple attributes

ABSTRACT

In this paper, we introduce the notion of a cooperative game with multiple attributes where players can provide partial participations in multiple attributes and form coalitions. The power or influence of the players due to their multiple attributes is evaluated based on their memberships in the coalitions. Our game therefore, extends the notion of cooperative games with fuzzy coalitions. The Shapley function for this class of games is proposed as a rational and fair solution concept. Every fuzzy game stems out of a specific crisp game under the assumption that the players provide partial memberships in forming multiple coalitions simultaneously. We adopt similar techniques to obtain the cooperative games with multiple attributes from their crisp counterparts and subsequently determine their Shapley functions.     

A new lower bound on the monotone network complexity of Boolean sums

Summary Neciporuk [3], Lamagna/Savage [1] and Tarjan [6] determined the monotone network complexity of a set of Boolean sums if each two sums have at most one variable in common. By this result they could define explicitely a set of n Boolean sums which depend on n variables and whose monotone complexity is of order n ^3/2. In the main theorem of this paper we prove a more general lower bound on the monotone network complexity of Boolean sums. Our lower bound is for many Boolean sums the first nontrivial lower bound. On the other side we can prove that the best lower bound which the main theorem yields is the n ^3/2-bound cited above. For the proof we use the technical trick of assuming that certain functions are given for free.     

On the Q.R. algorithm with shift

Fuzzy grammars on Boolean lattices (B-fuzzy grammars) are newly defined and their basic properties are investigated. B-fuzzy grammars are defined as the extension of fuzzy grammars by Lee and Zadeh, where the grades of the application of rewriting rules of B-fuzzy grammars are the elements of Boolean lattice rather than the elements of unit interval [0,1].

It is shown that type 2 B-fuzzy grammars can generate type 1 languages though type 2 fuzzy grammars cannot generate type 2 languages. And the closure properties of B-fuzzy grammars are also studied.     

区间属性值信息系统的不确定性度量

以区间属性值信息系统为研究对象,建立了其中的自反模糊关系、模糊信息粒以及模糊粗糙集.定义了两种新的粗糙熵对区间属性值信息系统中的不确定性因素进行度量,并进行了其单调性的讨论.     

A decomposition of fuzzy relations

This study is concerned with a decomposition of fuzzy relations, that is their representation with the aid of a certain number of fuzzy sets. We say that some fuzzy sets decompose an original fuzzy refraction if the sum of their Cartesian products approximate the given fuzzy relation. The theoretical underpinnings of the problem are presented along with some linkages with Boolean matrices (such as a Schein rank). Subsequently, we reformulate the decomposition of fuzzy relations as a problem of numeric optimizing and propose a detailed learning scheme leading to a collection of decomposing fuzzy sets. The role of the decomposition in a general class of data compression problems (including those of image compression and rule-based system condensation) is formulated and discussed in detail.     

THE PHENOMENON OF SCIENCE,by Valentin F. Turchin,Columbia University Press,New York, 1977, 348 pages. THE INERTIA OF FEAR AND THE SCIENTIFIC WORLDVIEW,by Valentin F. Turchin,Columbia University Press,New York, 1981, 300 pages.

Interaction concepts are the most basic concepts in (complex) systems theory. The concepts are defined very differently depending on their frameworks. An usual procedure to treat the interactions is to construct a specific model for each of them and analyse the models independently. We already know that there are many different complex systems representations which give the same global input-output relation [Takahara and Nakano 1981]. We have the freedom of choosing the interaction variables. However, the meanings of interactions are different depending on their formulations and representations. In this paper we discuss the relationships between the concepts used in the literatures [Nakano (to appear)], [Takahara and Nakano 1981] and [Mesarovic and Takahara 1975], which formulate the concepts from the general system theoretic viewpoint.

This paper shows that the interaction concepts used in the literature [Mesarovic and Takahara 1975] are included in the ones used in the literature [Nakano [1994], which are also included in the ones used in the literature [Takahara and Nakano 1981].     

非线性布尔网络系统模糊建模与动态性能分析

针对非线性系统难以精确建模与动态性能分析的基本控制问题,基于模糊动态模型把布尔网络系统理论推广到非线性布尔网络系统,建立了模糊动态布尔网络控制系统的模型。引入模糊动态模型,对非线性布尔网络进行模糊建模,分别建立了非线性布尔网络系统的局部模型和全局模型。从系统的局部意义和全局意义上,对系统进行了能控性、能观性、稳定性等动态性能分析。最后,以多输入多输出的非线性布尔网络系统实例为具体研究对象,建立了系统的局部模型和全局模型,并对动态性能进行了仿真分析,得到了实验结果。实验结果表明,模糊动态布尔网络控制系统对非线性布尔网络系统的建模是有效的,动态性能分析是合理的,对模糊动态布尔网络控制系统的进一步分析有重要意义。