A Power Algebra for Theory Change

Various representation results have been established for logics of belief revision, in terms of remainder sets, epistemic entrenchment, systems of spheres and so on. In this paper I present another representation for logics of belief revision, as an algebra of theories. I show that an algebra of theories, enriched with a set of rejection operations, provides a suitable algebraic framework to characterize the theory change operations of systems of belief revision. The theory change operations arise as power operations of the conjunction and disjunction connectives of the underlying logic.     

Complex fuzzy logic

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.     

Characterization of Interval Fuzzy Logic Systems of Connectives by Group Transformations

The global structure of various systems of logic connectives is investigated by looking at abstract group properties of the group of transformations of these. Such characterizations of fuzzy interval logics are examined in Sections 4–9. The paper starts by introducing readers to the Checklist Paradigm semantics of fuzzy interval logics (Sections 2 and 3). In the Appendix we present some basic notions of fuzzy logics, sets and many-valued logics in order to make the paper accessible to readers not familiar with fuzzy sets.     

Disjunction property and complexity of substructural logics

We systematically identify a large class of substructural logics that satisfy the disjunction property (DP), and show that every consistent substructural logic with the DP is PSPACE-hard. Our results are obtained by using algebraic techniques. PSPACE-completeness for many of these logics is furthermore established by proof theoretic arguments.     

Fuzzy logic controller design for spacecraft proximity operations

A Genetic Algorithms (GAs) based method is presented in this paper for concurrent design of rule sets and membership functions for a fuzzy logic controllers to be used in spacecraft proximity operations. The heuristic nature of fuzzy logic makes GAs a natural candidate for logic design in which both rule sets and membership functions are optimized simultaneously. The employment of GAs natural genetic operations provides a means to search in a complex system space that is difficult to described mathematically. A one-dimensional controller for spacecraft proximity operations is implemented for examination in detail. The expension of the algorithm for a 6 DOP controller is discussed.     

命题对象的空间逻辑运算模型

在现有逻辑系统中,各连接词的运算模型都可以归结为某些“代数算子”,其共同特征是仅考虑了命题所描述集合的代数测度大小,而没有考虑它们在几何空间中的位置关系。文章以“空间位置相关性”为中心,提出了“摸天花板问题”,分析了逻辑运算中存在的几何位置相关性。在“命题对象”、“真位向量”、“空间图像”等概念的基础上,提出了命题对象的空间逻辑运算模型,并结合格分维理论给出了在几何图像中的具体应用形式。本文工作拓展了泛逻辑学中广义相关性的含义,为连接词的运算形式提供了一种新的模型。     

基于一阶模态逻辑的模糊推理

研究基于可信度的模糊一阶模态逻辑,给出了基于常域的模糊一阶模态逻辑语义以及推理形式系统描述.为有效进行模糊断言间的推理,考虑了模糊约束的概念.模糊约束是一个表达式,其中既有语法成分又包含意义信息.模糊推理形式系统中的基本对象是模糊约束,针对模糊约束引进可满足性概念,研究模糊约束可满足性相关性质.利用模糊约束的概念,模糊断言间的推理可以直接在语义环境下加以考虑,因此,以模糊约束为基本元素的模糊推理形式系统随之建立.主要分析新产生断言有效性与模糊约束集可满足性之间的关系,并在此基础上给出了模糊推理形式系统的推理规则.进一步的工作可探讨模糊推理形式系统的可靠性与完全性,建立推理过程的能行机制.研究结果可在人工智能和计算机科学等领域得以应用.     

A polynomial space construction of tree-like models for logics with local chains of modal connectives

The LA-logics (“logics with Local Agreement”) are polymodal logics defined semantically such that at any world of a model, the sets of successors for the different accessibility relations can be linearly ordered and the accessibility relations are equivalence relations. In a previous work, we have shown that every LA-logic defined with a finite set of modal indices has an NP-complete satisfiability problem. In this paper, we introduce a class of LA-logics with a countably infinite set of modal indices and we show that the satisfiability problem is PSPACE-complete for every logic of such a class. The upper bound is shown by exhibiting a tree structure of the models. This allows us to establish a surprising correspondence between the modal depth of formulae and the number of occurrences of distinct modal connectives. More importantly, as a consequence, we can show the PSPACE-completeness of Gargov's logic DALLA and Nakamura's logic LGM restricted to modal indices that are rational numbers, for which the computational complexity characterization has been open until now. These logics are known to belong to the class of information logics and fuzzy modal logics, respectively.     

Fuzzy Logic: Misconceptions and Clarifications

Some commonly accepted statements concerning the basic fuzzy logicproposed by Lotfi Zadeh in 1965, have led to suggestions that fuzzy logicis not a logic in the same sense as classical bivalent logic. Thoseconsidered herein are: fuzzy logic generates results that contradictclassical logic, fuzzy logic collapses to classical logic, there can be no prooftheory for fuzzy logic, fuzzy logic is inconsistent, fuzzy logic producesresults that no human can accept, fuzzy logic is not proof-theoreticcomplete, fuzzy logic is too complex for practical use, and, finally, fuzzylogic is not needed. It is either proved or argued herein that all of the thesestatements are false and are, hence, misconceptions. A fuzzy logic withtruth values specified as subintervals of the real unit interval [0.0, 1.0] isintroduced. Proofs of the correctness, consistency, and proof theoreticcompleteness of the truth interval fuzzy logic are either summarized orcited. It is concluded that fuzzy logics deserve the accolade of logic tothe same degree that the term applies to classical logics.     

Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making

Intuitionistic fuzzy sets are useful for modeling uncertain data of realistic problems. In this paper, we generalize and expand the utility of complex intuitionistic fuzzy sets using the space of quaternion numbers. The proposed representation can capture composite features and convey multi-dimensional fuzzy information via the functions of real membership, imaginary membership, real non-membership, and imaginary non-membership. We analyze the order relations and logic operations of the complex intuitionistic fuzzy set theory and introduce new operations based on quaternion numbers. We also present two quaternion distance measures in algebraic and polar forms and analyze their properties. We apply the quaternion representations and measures to decision-making models. The proposed model is experimentally validated in medical diagnosis, which is an emerging application for tackling patient’s symptoms and attributes of diseases.     

A resolution-like strategy based on a lattice-valued logic

As the use of nonclassical logics becomes increasingly important in computer science, artificial intelligence and logic programming, the development of efficient automated theorem proving based on nonclassical logic is currently an active area of research. This paper aims at the resolution principle for the Pavelka type fuzzy logic (1979). Pavelka showed that the only natural way of formalizing fuzzy logic for truth-values in the unit interval [0, 1] is by using the Lukasiewicz's implication operator a/spl rarr/b=min{1,1-a+b} or some isomorphic forms of it. Hence, we first focus on the resolution principle for the Lukasiewicz logic L/sub /spl aleph// with [0, 1] as the truth-valued set. Some limitations of classical resolution and resolution procedures for fuzzy logic with Kleene implication are analyzed. Then some preliminary ideals about combining resolution procedure with the implication connectives in L/sub /spl aleph// are given. Moreover, a resolution-like principle in L/sub /spl aleph// is proposed and the soundness theorem of this resolution procedure is also proved. Second, we use this resolution-like principle to Horn clauses with truth-values in an enriched residuated lattice and consider the L-type fuzzy Prolog.     

Expressivity of coalgebraic modal logic: The limits and beyond

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modelled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioural equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.     

A new time invariant fuzzy time series forecasting method based on particle swarm optimization

In the analysis of time invariant fuzzy time series, fuzzy logic group relationships tables have been generally preferred for determination of fuzzy logic relationships. The reason of this is that it is not need to perform complex matrix operations when these tables are used. On the other hand, when fuzzy logic group relationships tables are exploited, membership values of fuzzy sets are ignored. Thus, in defiance of fuzzy set theory, fuzzy sets’ elements with the highest membership value are only considered. This situation causes information loss and decrease in the explanation power of the model. To deal with these problems, a novel time invariant fuzzy time series forecasting approach is proposed in this study. In the proposed method, membership values in the fuzzy relationship matrix are computed by using particle swarm optimization technique. The method suggested in this study is the first method proposed in the literature in which particle swarm optimization algorithm is used to determine fuzzy relations. In addition, in order to increase forecasting accuracy and make the proposed approach more systematic, the fuzzy c-means clustering method is used for fuzzification of time series in the proposed method. The proposed method is applied to well-known time series to show the forecasting performance of the method. These time series are also analyzed by using some other forecasting methods available in the literature. Then, the results obtained from the proposed method are compared to those produced by the other methods. It is observed that the proposed method gives the most accurate forecasts.     

Geometric Type-1 and Type-2 Fuzzy Logic Systems

This paper presents a novel approach to the representation of type-1 and type-2 fuzzy sets utilising computational geometry. To achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic. We provide new algorithms for various operations on type-1 and type-2 fuzzy sets and for defuzzification. Results of experiments indicate that this approach reduces the execution speed of these operations     

Implication operators in fuzzy logic

The choice of fuzzy implication as well as other connectives is an important problem in the theoretical development of fuzzy logic, and at the same time, it is significant for the performance of the systems in which fuzzy logic technique is employed. There are mainly two ways in fuzzy logic to define implication operators: (1) an implication operator is considered as the residuation of conjunction operator; and (2) it is directly defined in terms of negation, conjunction, and disjunction operators. The purpose of this paper is to determine the number of implication operators defined in the second way for some usual negation, conjunction and disjunction operators in fuzzy logic     

An introduction to fuzzy answer set programming

In this paper we show how the concepts of answer set programming and fuzzy logic can be successfully combined into the single framework of fuzzy answer set programming (FASP). The framework offers the best of both worlds: from the answer set semantics, it inherits the truly declarative non-monotonic reasoning capabilities while, on the other hand, the notions from fuzzy logic in the framework allow it to step away from the sharp principles used in classical logic, e.g., that something is either completely true or completely false. As fuzzy logic gives the user great flexibility regarding the choice for the interpretation of the notions of negation, conjunction, disjunction and implication, the FASP framework is highly configurable and can, e.g., be tailored to any specific area of application. Finally, the presented framework turns out to be a proper extension of classical answer set programming, as we show, in contrast to other proposals in the literature, that there are only minor restrictions one has to demand on the fuzzy operations used, in order to be able to retrieve the classical semantics using FASP.     

复合模糊命题运算中的弱范数研究

在模糊诊断和分析问题中,一个复合命题往往由多个子命题组成,子命题之间除有合取、析取及加权平均运算关系外,还存在一种非常重要的弱逻辑关系。对基于单一数值表达的模糊命题间的合取、析取关系的最一般运算形式是三角范数和三角余范数。该文给出基于单一数值表示模糊命题的弱逻辑关系的最一般运算形式f范数和基于区间值表示的模糊命题弱逻辑关系最一般运算形式f-范数,具体给出相关算子,讨论了它们的性质及与传统的一些算子之间的联系,通过引入强f范的概念,给出了构造f－范数算子的一般方法,对具有弱逻辑关系复合模糊命题的真值运算可采用本文介绍的概念和方法。