On witnessed models in fuzzy logic

Witnessed models of fuzzy predicate logic are models in which each quantified formula is witnessed, i.e. the truth value of a universally quantified formula is the minimum of the values of its instances and similarly for existential quantification (maximum). Systematic theory of known fuzzy logics endowed with this semantics is developed with special attention paid to problems of arithmetical complexity of sets of tautologies and of satisfiable formulas. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On arithmetic in the Cantor- Łukasiewicz fuzzy set theory

Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic. But we cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.Long before them, Klaua and Gottwald studied various forms of iterated fuzzy power set constructions inside classical set theory, see the references.     

Joint Consistency of Fuzzy Theories

This paper is a contribution to the development of fuzzy logic in narrow sense with evaluated syntax and connectives interpreted in Łukasiewicz algebra. The main results concern model theory of fuzzy logic (various kinds of submodels, chains of models) and generalization of the Craig‐Robinson's theorem on joint consistency of fuzzy theories as well as Craig's interpolation theorem.     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

A discrete representation of free MV‐algebras

We prove that the m ‐generated free MV‐algebra is isomorphic to a quotient of the disjoint union of all the m ‐generated free MV⁽ⁿ⁾‐algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV⁽ⁿ⁾‐algebras and special maps between them as morphisms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

In this paper we study extensions of the arithmetic operators +, -, ·, ÷ to the lattice of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on which can be generated using these operators; these classes provide natural extensions of the Łukasiewicz, product, Frank, Schweizer–Sklar and Yager t-norms to .     

Probability distributions in fuzzy logic II

Probability distributions associated with several ‘iff’ ply operators are discussed. These exact distributions are compared with relevant normal approximants. Two possible applications are noted.     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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.     

Formal systems of fuzzy logic and their fragments

Formal systems of fuzzy logic (including the well-known Łukasiewicz and Gödel–Dummett infinite-valued logics) are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction (an essential connective in the usual axiomatic system of fuzzy predicate logics).     

Fuzzy topology representation for MV‐algebras

Let M be an MV‐algebra and Ω_M be the set of all σ ‐valuations from M into the MV‐unit interval. This paper focuses on the characterization of MV‐algebras using σ ‐valuations of MV‐algebras and proves that a σ ‐complete MV‐algebra is σ ‐regular, which means that a ≤ b if and only if v (a) ≤ v (b) for any v ∈ Ω_M. Then one can introduce in a natural way a fuzzy topology δ on Ω_M. The representation theorem forMV‐algebras is established by means of fuzzy topology. Some properties of fuzzy topology δ and its cut topology U are investigated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the use of fuzzy inference techniques in assessment models: part II: industrial applications

In this paper, we study the applicability of the monotone output property and the output resolution property in fuzzy assessment models to two industrial Failure Mode and Effect Analysis (FMEA) problems. First, the effectiveness of the monotone output property in a single-input fuzzy assessment model is demonstrated with a proposed fuzzy occurrence model. Then, the usefulness of the two properties to a multi-input fuzzy assessment model, i.e., the Bowles fuzzy Risk Priority Number (RPN) model, is assessed. The experimental results indicate that both the fuzzy occurrence model and Bowles fuzzy RPN model are able to fulfill the monotone output property, with the derived conditions (in Part I) satisfied. In addition, the proposed rule refinement technique is able to improve the output resolution property of the Bowles fuzzy RPN model.     

Nonparametric regression estimation in models with weak error's structure

In this paper we propose nonparametric estimates of the regression function and its derivative when it is only assumed a weak error's structure. We study their local and global asymptotic behaviour when we observe dependent trajectories.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).     

Archimedean classes in integral commutative residuated chains

This paper investigates a quasi‐variety of representable integral commutative residuated lattices axiomatized by the quasi‐identity resulting from the well‐known Wajsberg identity (p → q) → q ≤ (q → p) → p if it is written as a quasi‐identity, i. e., (p → q) → q ≈ 1 ? (q → p) → p ≈ 1 . We prove that this quasi‐identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi‐variety is in fact a variety and provide an axiomatization. The obtained results shed some light on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL‐algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL‐algebras is generated by its Archimedean members (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)