Vilém Novák

Mathematical principles of fuzzy logic

Preface. 1. Fuzzy Logic: What, Why, for Which? 2. Algebraic Structures for Logical Calculi. 3. Logical Calculi and Model Theory. 4. Fuzzy Logic in Narrow Sense. 5. Functional Systems in Fuzzy Logic Theories. 6. Fuzzy Logic in Broader Sense. 7. Topoi and Categories of Fuzzy Sets. 8. Few Historical and Concluding Remarks. References. Index.

On fuzzy type theory

This paper is a generalization of classical (simple) type theory. We have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many. The structure of truth values is assumed to be an IMTL-algebra (residuated lattice with prelinearity and double negation) extended by the Baaz delta operation. Various properties of fuzzy type theory are proved including its completeness.

Fuzzy transform in the analysis of data

Fuzzy transform is a novel, mathematically well founded soft computing method with many applications. In this paper, we present this technique with applications to data analysis. First, we show how it can be used for detection and characterization of dependencies among attributes. Second, we apply it to mining associations that have a functional character. Moreover, the mined associations are characterized linguistically which means that their antecedent consists of fuzzy numbers and the consequent is characterized using pure evaluative linguistic expressions (i.e. expressions such as small, very big, more or less medium, etc).

Logical Structure of Fuzzy IF-THEN Rules

Abstract This paper provides a logical basis for manipulation with fuzzy IF-THEN rules. Our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to Lotfi A. Zadeh but also other kinds of approximate reasoning methods, e.g., perception-based deduction, provided that there exists a possibility to characterize them within a formal logical system. In contrast with other approaches employing variants of multiple-valued first-order logic, the approach presented here employs fuzzy type theory of V. Novak which has sufficient expressive power to present the essential concepts and results in a compact, elegant and justifiable form. Within the effectively formalized representation developed here, based on a complete logical system, it is possible to reconstruct numerous well-known properties of CRI-related fuzzy inference methods, albeit not from the analytic point of view as usually presented, but as formal derivations of the logical system employed. The authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy IF-THEN rule bases will be represented, formalized and backed up by proof within the well-founded logical representation presented here. An immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy IF-THEN rules are ‘first class citizens´ of the representation: there are clear, logically founded definitions for fuzzy IF-THEN rule bases to be consistent, complete, or independent.

Perception-Based Logical Deduction

In this paper, we will formalize the way, how people make inferences on the basis of the, so called, linguistic description which is a set of fuzzy IF-THEN rules understood as expressions of natural language. We will explain our idea on the following example.

First-order fuzzy logic

This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 〈0, 1〉 of reals. These are special cases of a residuated lattice 〈L, ∨, ∧, ⊗, →, 1, 0〉. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Gödels completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.

A formal theory of intermediate quantifiers

The paper provides a logical theory of a specific class of natural language expressions called intermediate quantifiers (most, a lot of, many, a few, a great deal of, a large part of, a small part of), which can be ranked among generalized quantifiers. The formal frame is the fuzzy type theory (FTT). Our main idea lays in the observation that intermediate quantifiers speak about elements taken from a class that is made “smaller” than the original universe in a specific way. Our theory is based on the formal theory of trichotomous evaluative linguistic expressions. Thus, an intermediate quantifier is obtained as a classical quantifier “for all” or “exists” but taken over a class of elements that is determined using an appropriate evaluative expression. In the paper we will characterize the behavior of intermediate quantifiers and prove many valid syllogisms that generalize classical Aristotles ones.

The relation between inference and interpolation in the framework of fuzzy systems

Abstract This papers aims at clarifying the meaning of different interpretations of the Max-Min or, more generally, the Max-t-norm rule in fuzzy systems. It turns out that basically two distinct approaches play an important role in fuzzy logic and its applications: fuzzy interpolation on the basis of an imprecisely known function and logical inference in the presence of fuzzy information.

Analysis of seasonal time series using fuzzy approach

A new methodology for the analysis and forecasting of time series is proposed. It directly employs two soft computing techniques: the fuzzy transform and the perception-based logical deduction. Thanks to the use of both these methods, and to the innovative approach, consisting of the construction of several independent models, the methodology is successfully applicable to robust long-time predictions.

Antonyms and linguistic quantifiers in fuzzy logic

Abstract The paper is a contribution to the theory of fuzzy logic in broader sense (FLb), namely the discussion of linguistic expressions fundamental for it—the evaluating linguistic predications, the pairs “nominal syntagm–antonym”, and the theory of linguistic quantifiers. The aim is to develop a theory of natural human reasoning, whose characteristic feature is the use of natural language. Formalism of FLb is based on the theory of fuzzy logic in narrow sense with evaluated syntax, which provides us means for modelling of the concepts of intension, possible world and extension. Characterization of some of the main properties of the above expressions is provided. We also propose a modified definition of the linguistic variable.