The algebra of fuzzy logic

In the following, human thinking based on premises with no complete truth value is reviewed for controlling the algebra of fuzzy sets operations. Assuming a system may be developed in this sphere, it should be considered as the algebra of fuzzy sets, as the same algebra is satisfied by classical logic and sets. As will be proved, this algebra is not a lattice and consequently the Zadeh definitions do not constitute an adequate representation. The binary operations of my algebra are “interactive” types. An axiom system is given that, in my opinion, is the foundation of the conception, adequately and without redundancy. The agreement of the theorems deduced from the axiom system with the intuitive expectations is shown. A special arithmetical structure satisfying this algebra is given, and the relation between this structure and the theory of probability is analyzed.Adapting a process of classical logics, fuzzy quantifiers are defined on the basis of the operations of propositional algebra. A “qualifier” is also defined. The qualifier is functional; applying it to Ax we get the statement “usually Ax” s a middle cource between the statements “at least once Ax” and “always Ax”. The concept of entailment of fuzzy logics is introduced. This concept is an innovative generalization of the classical deduction theory, opposite to the concept of entailment of classical multi-valued logics. An important error of the abbreviated system of notation of the fuzzy theory [e.g. m(x, AvB)] appears: the functional type operations (e.g. quantifiers) cannot be interpreted in propositional calculus. Therefore a new system of symbols is proposed in this paper.     

Distributions in fuzzy logic

Probability distributions associated with several ‘ply’-operators are discussed. These exact distributions are compared with relevant Gaussian approximations.     

Interpolation in fuzzy logic

We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences. Received: 10 December 1997     

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.     

Array approach to fuzzy logic

Promising results from applying an array-based approach to two-valued logic suggests its application to fuzzy logic. The idea is to limit the domain of truth-values to a discrete, finite domain, such that a logical relationship can be evaluated by an exhaustive test of all possible combinations of truth-values. The paper presents a study of the topic from an engineer's viewpoint. As an example 31 logical sentences valid in two-valued logic were tested in three-valued logic using the nested interactive array language, Nial. Out of these, 24 turned out to be valid in a three-valued extension based on the well-known S^* implication operator, also called “Gödel's implication operator”. Applications to automated approximate reasoning and fuzzy control are also illustrated.     

Analysis of a fuzzy logic controller

An analysis is performed of the fuzzy logic controller which results in the identity between this controller and a multilevel relay. This tool is used in stability analysis.     

Very true on CBA fuzzy logic

CBA logic was introduced as a non-associative generalization of the Łukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the “very true” connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.     

Birkhoff variety theorem and fuzzy logic

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity , i.e. a fuzzy equivalence relation, such that each operation f is compatible with . Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic. On the other hand, they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones. In this paper, we present a generalization of the well-known Birkhoffs variety theorem: a class of algebras with fuzzy equality is the class of all models of a fuzzy set of identities iff it is closed under suitably defined morphisms, substructures, and direct products. and Institute for Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic Mathematics Subject Classification (2000):03B52, 08B05     

Chaos synchronization using fuzzy logic controller

The design of a rule-based controller for a class of master-slave chaos synchronization is presented in this paper. In traditional fuzzy logic control (FLC) design, it takes a long time to obtain the membership functions and rule base by trial-and-error tuning. To cope with this problem, we directly construct the fuzzy rules subject to a common Lyapunov function such that the master–slave chaos systems satisfy stability in the Lyapunov sense. Unlike conventional approaches, the resulting control law has less maximum magnitude of the instantaneous control command and it can reduce the actuator saturation phenomenon in real physic system. Two examples of Duffing–Holmes system and Lorenz system are presented to illustrate the effectiveness of the proposed controller.     

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.     

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     

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).     

Selection of parameters for a fuzzy logic controller

The fuzzy logic controller is reviewed and its parameter are explicitly identified. The problem of initial selection and subsequent adjustment of the parameters are discussed in detail by example.     

The interpretation of fuzzy integrals and their application to fuzzy systems

Fuzzy integrals, in general, and Sugeno integrals, in particular, are well known aggregation operators. They can be used in a great variety of decision making applications. Nevertheless, their use is not easy as their interpretation is not straightforward. In this paper we study the interpretation of fuzzy integrals, focusing on Sugeno ones, and we show their application to fuzzy inference systems when the rules are not independent.     

The merging of neural networks,fuzzy logic,and genetic algorithms

During the last decade, there has been increased use of neural networks (NNs), fuzzy logic (FL) and genetic algorithms (GAs) in insurance-related applications. However, the focus often has been on a single technology heuristically adapted to a problem. While this approach has been productive, it may have been sub-optimal, in the sense that studies may have been constrained by the limitations of the technology and opportunities may have been missed to take advantage of the synergies between the technologies. For example, while NNs have the positive attributes of adaptation and learning, they have the negative attribute of a “black box” syndrome. By the same token, FL has the advantage of approximate reasoning but the disadvantage that it lacks an effective learning capability. Merging these technologies provides an opportunity to capitalize on their strengths and compensate for their shortcomings.This article presents an overview of the merging of NNs, FL and GAs. The topics addressed include the advantages and disadvantages of each technology, the potential merging options, and the explicit nature of the merging.     

Ten questions and one problem on fuzzy logic

Recently, I had a very interesting friendly e-mail discussion with Professor Parikh on vagueness and fuzzy logic. Parikh published several papers concerning the notion of vagueness. They contain critical remarks on fuzzy logic and its ability to formalize reasoning under vagueness [10,11]. On the other hand, for some years I have tried to advocate fuzzy logic (in the narrow sense, as Zadeh says, i.e. as formal logical systems formalizing reasoning under vagueness) and in particular, to show that such systems (of many-valued logic of a certain kind) offer a fully fledged and extremely interesting logic [4, 5]. But this leaves open the question of intuitive adequacy of many-valued logic as a logic of vagueness. Below I shall try to isolate eight questions Parikh asks, add two more and to comment on all of them. Finally, I formulate a problem on truth (in)definability in Łukasiewicz logic which shows, in my opinion, that fuzzy logic is not just “applied logic” but rather belongs to systems commonly called “philosophical logic” like modal logics, etc.