THE FUNDAMENTAL THEOREM OF ULTRAPRODUCT IN PAVELKA'S LOGIC

In [This Zeitschrift 25 (1979), 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order Pavelka's logic is preserved under some direct product of lattices of truth values.     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

EQ-algebras

We introduce a new class of algebras called EQ-algebras. An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. These algebras are intended to become algebras of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that until now, the truth values in FTT were assumed to form either an IMTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation (multiplication) and its residuum. The latter is a natural interpretation of implication in fuzzy logic; the equivalence is then interpreted by the biresiduum, a derived operation. The basic connective in FTT, however, is a fuzzy equality and, therefore, it is not natural to interpret it by a derived operation. This defect is expected to be removed by the class of EQ-algebras introduced and studied in this paper. From the algebraic point of view, the class of EQ-algebras generalizes, in a certain sense, the class of residuated lattices and so, they may become an interesting class of algebraic structures as such.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Fuzzy Horn logic I

The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of provability equals degree of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic.     

扰动模糊命题逻辑的代数结构及其广义重言式性质

着眼于扰动模糊命题逻辑的代数结构,为研究二维扰动模糊命题逻辑最大子代数I2R及其广义重言式提供了一些代数理论基础,最后研究了子代数间广义重言式的关系.     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

模糊Hopfield网络及其模糊聚类功能研究

本文提出了一种能进行模糊逻辑计算的Ｈｏｐｆｉｌｅｄ型人工神经元网络，ＦｕｚｚｙＨＮ的神经元对应模式集合中的元素，模式间的模糊相似关系作为联结神经元的权值矩阵被存储在ＦｕｚｚｙＨＮ中。本文对ＦｕｚｚｙＨＮ的稳定性及模糊聚类功能进行研究，获得了良好的理论分析结果和实验研究结果。     

建立模糊诊断矩阵的一种新方法

本文首先提出了一个新的模糊逻辑算子——模糊加权综合算子“#”,进而给出建立模糊诊断矩阵的新方法——网络法 ,并给出了相应的算法     

Feed-forward modelling and fuzzy logic based control strategy for powertrain efficiency improvement in a parallel hybrid electric vehicle

With the stricter limitations on both fuel consumption and air pollution, the advantages of a hybrid electric vehicle are becoming more evident than ever. In the present study, an energy management system for a hybrid electric vehicle is developed. Because the plant under consideration is nonlinear, multi-domain, time-varying, has multiple uncertainties and, in addition, the designed control strategy must be able to obey the driver's commands and achieve the par-internship for a new generation of vehicle regulations, the fuzzy logic approach is chosen. A feed-forward hybrid vehicle simulation model is used to demonstrate the validity and the convenience of the current approach and its results have been compared with the other parallel hybrid electric vehicle control strategies. Simulation results show considerable improvement in the efficiency of the internal combustion engine and, consequently, fuel consumption and acceleration performances.     

Continuous fuzzy Horn logic

The paper deals with fuzzy Horn logic (FHL) which is a fragment of predicate fuzzy logic with evaluated syntax. Formulas of FHL are of the form of simple implications between identities. We show that one can have Pavelka‐style completeness of FHL w.r.t. semantics over the unit interval [0, 1] with (residuated lattices given by) left‐continuous t‐norm and a residuated implication, provided that only certain fuzzy sets of formulas are considered. The model classes of fuzzy structures of FHL are characterized by closure properties. We also give comments on related topics proposed by N. Weaver. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formal verification of tail distribution bounds in the HOL theorem prover

Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher‐order‐logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety‐critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well‐known commercially used algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Theory of T-norms and fuzzy inference methods

In this paper, the theory of T-norm and T-conorm is reviewed and the T-norm, T-conorm and negation function are defined as a set of T-operators. Some typical T-operators and their mathematical properties are presented. Finally, the T-operators are extended to the conventional fuzzy reasoning methods which are based on the and operators. This extended fuzzy reasoning provides both a general and a flexible method for the design of fuzzy logic controllers and, more generally, for the modelling of any decision-making process.     

Normal forms for fuzzy logics: a proof-theoretic approach

A method is described for obtaining conjunctive normal forms for logics using Gentzen-style rules possessing a special kind of strong invertibility. This method is then applied to a number of prominent fuzzy logics using hypersequent rules adapted from calculi defined in the literature. In particular, a normal form with simple McNaughton functions as literals is generated for ?ukasiewicz logic, and normal forms with simple implicational formulas as literals are obtained for Gödel logic, Product logic, and Cancellative hoop logic.     

深入研究模糊逻辑的必要性与紧迫性

因为"取大取小"不是数学计算,所以基于"取大取小"的模糊逻辑不能为数值转换提供算法支撑,使得模糊理论面临无合适模型可用的被动境地.指出,模糊逻辑是逻辑的一个新的近似推理研究方向,它的量化方法是数值计算;目的是支撑隶属度转换,使得由指标隶属度确定的目标隶属度是"真值"在当前条件下的最优近似.模糊逻辑是在隶属度转换条件下对人类近似推理本领规范的一种方法.而进行规范的依据是区分权滤波的冗余理论,实质性计算是由冗余理论导出的、实现隶属度转换的非线性去冗算法;相应的隶属度转换模型是非线性数学模型.     

模糊集的取大取小算法的不合理性

许多作者对模糊综合评判的取大取小算法的不合理性给予充分重视,并提出许多改进方法,从模糊命题逻辑人手,指出这些不合理性的根本所在.     

Modelling the clogging of gas turbine filter houses in heavy-duty power generation systems

ABSTRACT

A prognostic approach based on a MISO (multiple inputs and single output) fuzzy logic model was introduced to estimate the pressure difference across a gas turbine (GT) filter house in a heavy-duty power generation system. For modelling and simulation of clogging of the GT filter house, nine real-time process variables (ambient temperature, humidity, ambient pressure, GT produced load, inlet guide vane position, airflow rate, wind speed, wind direction and PM10 dust concentration) were fuzzified using a graphical user interface within the framework of an artificial intelligence-based methodology. The results revealed that the proposed fuzzy logic model produced very small deviations and showed a superior predictive performance than the conventional multiple regression methodology, with a very high determination coefficient of 0.974. A complicated dynamic process, such as clogging phenomenonin heavy-duty GT system, was successfully modelled due to high capability of the fuzzy logic-based prognostic approach in capturing the nonlinear interactions.     

Lusin's theorem on fuzzy measure spaces

In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin's theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoroff's theorem for the fuzzy measure defined on metric spaces is given. An application of Lusin's theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented.