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.     

The fuzzy logic of text understanding

Fuzzy graphs and fuzzy logic are used to describe the reasoning of verbal understanding of a text played to students of Unicamp. The results demonstrate that each phrase in the text is associated to two different truth values. One of them, here called the confedence degree, measures the truth of the information as related to the individual sets of beliefs; the other, called correlation degree, measures the truth of each phrase as regards the structure of the text itself. Also, empirical relations were established for each of the logical of perators used by the volunteers.     

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     

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)     

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.     

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)     

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.     

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.     

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     

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     

On witnessed models in fuzzy logic II

First the expansion of the ?ukasiewicz (propositional and predicate) logic by the unary connectives of dividing by any natural number (Rational ?ukasiewicz logic) is studied; it is shown that in the predicate case the expansion is conservative w.r.t. witnessed standard 1‐tautologies. This result is used to prove that the set of witnessed standard 1‐tautologies of the predicate product logic is Π₂‐hard. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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).