Fuzzy constructive logic

We introduce a logical system in which the principles of fuzzy logic are interpreted from the point of view of the constructive approach The language of predicate formulas without functional symbols and symbols of constants is considered. The notion of identically trae predicate formula in the framework of the introduced logic is defined; two variants of this definition are given. Theorems concerning identically true predicate formulas are proved. Some connections between the introduced logic and the constructive (intuitionistic) predicate calculus are established. Bibliography: 40 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 130–152.     

Fuzzy inequational logic

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if–then rules which is obtained as particular case of the general result.     

Fuzzy equational logic

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established. Received: 15 March 2000 / Revised version: 20 September 2000 Published online: 12 December 2001     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

Fuzzy Horn logic II

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.     

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.     

Fuzzy logic, continuity and effectiveness

 It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general). Received: 15 February 2001 / Revised version: 31 May 2001 / Published online: 12 July 2002     

Fuzzy modelling through logic optimization

This study proposes a new logic-driven approach to the development of fuzzy models. We introduce a two-phase design process realizing adaptive logic processing in the form of structural and parametric optimization. By recognizing the fundamental links between binary (two-valued) and fuzzy (multi-valued) logic, effective structural learning is achieved through the use of well-established methods of Boolean minimization encountered in digital systems. This blueprint structure is then refined by adjusting connections of fuzzy neurons, helping to capture the numeric details of the target system’s behavior. The introduced structure along with the learning mechanisms helps achieve high accuracy and interpretability (transparency) of the resulting model.     

Fuzzy logic based intersection delay estimation

A fuzzy logic based delay estimation system is proposed and modelled. Conventional method of delay study involves solving static engineering equations in which only technical factors (traffic demand, roadway geometry, and signal control, etc.) are considered and the affect of nontechnical factors (such as weather or visibility) cannot be analyzed since they do not follow a predefined process. The fuzzy logic based delay estimation combines the complex technical and nontechnical factors and is adaptive to the changing driving environment. The rule base of the delay estimation system is constructed either following a mathematical model or from real-time traffic operational data. Simulation and field test of the fuzzy system have shown that fuzzy logic based modelling is a promising approach to improving intersection delay estimation.     

模糊逻辑系统公理真度分析

分别对Lukasiewicz逻辑系统中的公理在R0系统和G(o)del系统中的真度大小、R0系统逻辑系统中的公理在Lukasiewicz系统和中G(o)del系统的真度大小和G(o)del逻辑系统中的公理在R0系统和Lukasiewicz系统中的真度大小进行了计算和分析,从真度方面研究和分析了常用逻辑系统之间的关系.     

Fuzzy logic in commercial expert systems — Results and prospects

This paper surveys the use of fuzzy methods in commercial applications of the technology of expert systems and in commercial products which aim to support such applications. It attempts to evaluate the utility of such an approach to uncertainty management compared to other well known methods of handling uncertainty in expert systems. Starting from this base it attempts an evaluation of the prospects for fuzzy expert systems in the medium term. As a survey it attempts to list applications and commercial products as comprehensively as is practical and includes an extensive bibliography on the topic.     

Fuzzy logic for auditory evoked response monitoring and control of depth of anaesthesia

This paper describes a self-organizing fuzzy model of patients undergoing surgery which was created from 10 clinical trials with off-line analysis during maintenance of anaesthesia using the drug propofol. The effects of patient sensitivity and surgical disturbances are also represented in this patient model. Hence, this model can be considered to be a qualitative pharmacologically related model for propofol during the anaesthetic maintenance stage. Furthermore, a closed-loop simulation has been designed to validate the patient model and compare the performance of a self-organizing fuzzy logic controller algorithm against a clinically derived linguistic controller. The successful results obtained provide proof-of-concept and encouragement to perform on-line clinical trials using fuzzy logic-based monitoring and control in operating theatre in the near future.     

Fuzzy logic control of electrodynamic levitation devices coupled to dynamic finite volume method analysis

Electrodynamic levitation devices, which utilizing eddy currents induced in the levitated item to produce the repulsive force, are being involved in many engineering applications due to its fast response. This kind of repulsion is particularly used in electromagnetic launcher, electromagnetic brake and other applications. To analyze and improve the dynamic behavior and performances of such devices, the conventional way is using the finite element method (FEM), due to its ability of using adaptive mesh to handle complex geometries. Nevertheless, it has a serious limitation in efficiency for large number of variables which is reflected by the high cost in terms of computational properties. During the past few years, the finite volume method FVM formulations have gained attention inside the electromagnetic community. The method has been proved its effectiveness in the solution of different kinds of problems, such as in magnetostatic field computation and eddy current nondestructive testing. The FVM method is particularly attractive thanks to its small required storage memory and reduced CPU time. In this paper an FVM model is developed to analyze the dynamic characteristic of the motion of the electrodynamic levitation device TEAM Workshop Problem 28. The dynamic characteristic of the motion is obtained by solving the electromagnetic equation coupled to the mechanical one. The repulsive force applied to the levitated plate of TEAM Workshop Problem 28, is computed by the interaction between eddy current induced in the plate and the magnetic flux density. A comparison between experimental and numerical results is carried out to show the efficiency of the developed model. What’s more, based on the developed FVM model, a fuzzy logic controller FLC is designed and implemented to control the position of the levitated item.     

Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic

 Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. Received: 20 June 2001 / Published online: 2 September 2002 This paper was prepared while the first author was visiting the Universidad de Barcelona supported by INTERCAMPUS Program E.AL 2000. The second author was partially supported by Grants 2000SGR-0007 of D. G. R. of Generalitat de Catalunya and PB 97-0888 of D. G. I. C. Y. T. of Spain. Mathematics Subject classification (2000): 03B50, 03B52, 03G25, 06D35 Keywords or Phrases: Basic fuzzy logic – Łukasiewicz logic – BL-algebras – MV-algebras – Glivenko's theorem     

Coalgebraic logic

We present a generalization of modal logic to logics which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a fragment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called Δ. This fragment generalizes to a wide range of coalgebraic logics. Each coalgebraic logic is determined by a functor on sets satisfying a few properties, and the formulas of each logic are interpreted on coalgebras of that functor. Among the logics obtained are the fragment of infinitary modal logic mentioned above as well as versions of natural logics associated with various classes of transition systems, including probabilistic transition systems. For most of the interesting cases, there is a characterization result for the coalgebraic logic determined by a given functor. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of a final coalgebra.     

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown.