퍼지함의를 일반화한 대수구조

Truth values in fuzzy logic are extended into interval truth values {[a, b]: 0≤a≤b≤1} from the numerical truth values of [0, 1]. Next we extend this interval truth values into two directions; one is to n-dimensional case such as [a 1,...,a n ] where a 1≤...≤a n , and the other is to dropping the condition a≤b in [a, b]. We examine the algebraic structures of these truth values on which three kinds of logic operations AND(·), OR(∨), NOT(⊃ are defined, where we interpret (x·y), (x∨y), 0 and (⊃x) as min(x, y), max(x, y), and (1−x), respectively. It is shown that we can find out structures of Kleene algebra and De Morgan algebra in these truth values.

A kind of intuitionistic linguistic truth-valued reasoning approach for decision making with both comparable and incomparable truth values is proposed in this paper. By using the lattice implication algebra, an six-element linguistic truth-valued intuitionistic propositional logic system is established which can express both truth degree and falsity degree. The implication operation of linguistic truth-valued intuitionistic propositional logic can be deduced from four times implication of their truth values. Therefore, we can use more information in the process of reasoning and eventually improve the precision of reasoning. As reasoning and operation are directly acted by linguistic truth values in the decision process, the issue on how to obtain the weight for rational decision making results is discussed. An illustration example shows the proposed approach seems more effective for decision making under a linguistic information environment with both truth degree and falsity degree.

It is well known that an important task of the artificial intelligence is to make a computer simulate a human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. For investigation of several properties, such logics have been represented as algebras, that is, sets with one, two, or more algebraic operations satisfying some conditions inspired by these logics. This inspiration is illustrated by the similarities between the names. We have BCK-algebras and a BCK positive logic, BCI-algebras and a BCI positive logic, and BL-algebras and a basic logic. In many cases, the connection between such algebras and their corresponding logics is much stronger. In this case, one can give a translation procedure which translates all well-formed formulas and all theorems of a given logic L into terms and theorems of the corresponding algebras. In some cases, one can give also an inverse translation. In this case, we say that the given logic and class of such algebras are isomorphic. Nevertheless, the study of algebras motivated by logics is interesting and very useful also in the case when these structures are not isomorphic. An important class of algebras inspired by logic is BL-algebras introduced by Hajek in order to provide an algebraic proof of the completeness theorem of Basic Logic. A classical example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. Other important examples of BL-algebras are MV-algebras introduced earlier as a model of some infinitely valued Łukasiewicz logic. All these algebras are also strongly connected with residuated lattices. Hence, the filter theory plays an important role in the study of these algebras. From logic point of view, various filters correspond to various sets of provable formulae in the respective logic. Below we present several papers on filters in various algebras. In one of these papers, it is proved that there is a one-to-one correspondence between the set of all vague filters and all vague congruence of a lattice implication algebra. In another, the relations among fuzzy t-filters on residuated lattices are described. Another paper is devoted to the filter topology on lattice implication algebras. In further articles is characterized the relationship between the main types of fuzzy filters in BE-algebras and EQ-algebras. The role of ideals and soft sets in BL-algebras is described in another two papers. In papers on pseudo-weak-R0 algebras, the most simplified axioms system is presented and it is proved that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras. Pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Several presented papers are devoted to the possible applications of algebras inspired by logic and fuzzy sets. For example, a robust fuzzy logic controller is proposed for stabilization and disturbance rejection in nonlinear control systems of a particular type. The dynamic feedback controller is designed as a combination of a control law that compensates for nonlinear terms in a control system and a dynamic fuzzy logic controller that addresses unknown model uncertainties and an unmeasured disturbance. A mathematical derivation is used to prove that the controller is able to achieve asymptotic stability by processing state measurements. Properties of some hesitant triangular fuzzy aggregation operators based on Bonferroni means also are discussed. In one paper, the method of a construction of a ranking of the physical matches is proposed. This method is based on the fuzzy clustering analysis. Decisions are often made by many decision experts having preference of risks. In view of the grey situation group decision-making problems, a grey situation group decision-making method on the basis of prospect theory is shown. The method takes the positive and negative ideal situation distance as reference points, defines positive and negative prospect value function, and introduces decision experts' risk preference into grey situation decision-making to make the final decision more in line with decision experts' psychological behavior. One possibility of determination of the weight of each decision expert sets up comprehensive prospect value matrix for decision experts' evaluation, and finally determination of the optimal situation is presented. Jianming Zhan Bijan Davvaz Wieslaw A. Dudek Young Bae Jun Hee Sik Kim

In 1973 Lotfi Zadeh introduced the theory of fuzzy logic [17]. Fuzzy logic was an extension of Boolean logic so that it allowed using not only Boolean values to express reality. One kind of basic logical operations in fuzzy logic are so-called fuzzy implications. From over eight decades a number of different fuzzy implications have been described [3] - [16]. In the family of all fuzzy implications the partial order induced from [0,1] interval exists. Pairs of incomparable fuzzy implications can generate new fuzzy implications by using min(inf) and max(sup) operations. As a result the structure of lattice is created ([1], page 186). This leads to the following question: how to choose the correct functions among basic fuzzy implications and other generated as described above. In our paper, we propose a new method for choosing implications. Our method allows to compare two fuzzy implications. If the truth value of the antecedent and the truth value of the implication are given, by means of inverse fuzzy implications we can easily optimize the truth value of the implication consequent. In other words, we can choose the fuzzy implication, which has the greatest or the smallest truth value of the implication consequent or which has greater or smaller truth value than another implication. Primary results regarding this problem are included in the paper [14].

Boolean algebra (BA) was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847) and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term “Boolean algebra” was first suggested by Sheffer in 1913. Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1, considered to be “fuzzy.” By contrast, in Boolean logic, the truth values of variables may only be the “crisp” values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific (membership) functions. The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Lukasiewicz and Tarski. In this chapter we look at both of these logics holistically; any extensive details are beyond the scope of this book, and we encourage our readers to refer to so many books and articles that can be found on the Internet or Amazon.

In fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc.

In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative.Keywords: Implicative algebra, lattice implication algebras, Autometrized Algebras, regular Autometrized Algebras

In this paper, several equivalent conditions for fuzzy filters are given in lattice implication algebra. The relation between fuzzy filters and fuzzy implicative filters is discussed. It is also proved that fuzzy implicative filters are fuzzy filters in lattice implication algebra.

A many‐valued sentential logic with truth values in an injective MV‐algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV‐algebra. In the well‐defined fuzzy sentential logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good‐behaviour, MV‐valued logic can be regarded as a mathematical basis of fuzzy reasoning.

We introduce two variants of first-order fuzzy logic that can deal with non-denoting terms, or terms that lack existing referents, e.g., Pegasus, the current king of France, the largest number, or 0/0. Logics designed for this purpose in the classical setting are known as free logics. In this paper we discuss the features of free logics and select the options best suited for fuzzification, deciding on the so-called dual-domain semantics for positive free logic with truth-value gaps and outer quantifiers. We fuzzify the latter semantics in two levels of generality, first with a crisp and subsequently with a fuzzy predicate of existence. To accommodate truth-valueless statements about nonexistent objects, we employ a recently proposed first-order partial fuzzy logic with a single undefined truth value. Combining the dual-domain semantics with partial fuzzy logic, we define several kinds of ‘inner-domain’ quantifiers, relativized by the predicate of existence. Finally, we make a few observations on some of the resulting rules of free fuzzy quantification that illustrate the differences between the two proposed systems of free fuzzy logic and their well known non-free or non-fuzzy variants.

We present some model theoretic results for Lukasiewicz predicate logic by using the methods of continuous model theory developed by Chang and Keisler. We prove compactness theorem with respect to the class of all structures taking values in the Lukasiewicz BL-algebra. We also prove some appropriate preservation theorems concerning universal and inductive theories. Finally, Skolemization and Morleyization in this framework are discussed and some natural examples of fuzzy theories are presented. ajek and Cintula wrote: \In the last few decades many formal systems of fuzzy logics have been developed. Since the main dierences between fuzzy and classical logics lie at propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this paper we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate In that paper the authors proved a generalized completeness result for a wide class of fuzzy predicate logics, namely core fuzzy logics. To do this they used a generalized Henkin method for a through survey of the present status of fuzzy predicate logic, see (3). In the present paper we continue the work on fuzzy predicate logics by developing further model theory of these logics. We choose Lukasiewicz predicate logic. The reason is that the Lukasiewicz BL-algebra is equipped with continuous operations and this is essential in proving the compactness theorem which is a fundamental result in model theory. Here, the compactness theorem says that if each nite subset of a set of sentences has a Lukasiewicz model, then the set itself has a Lukasiewicz model. As consequences of compactness, we prove the most basic model theoretic results including the upward and downward Lowenheim-Skolem theorems. We also prove suitable versions of the most well known preservation theorems, including preservation theorems for universal and inductive theories. At the end, we introduce some fuzzy theories and study their model theoretic properties. To prove the compactness theorem, we use ultraproduct and its basic properties which we rst investigate. We follow the old methods developed by Chang and

The development of fuzzy logic in the companies mentioned by the authors is meant to automate the plant adaptively. The goal is to reduce downtimes in the plant, and hence, the framework done by the researchers will solve the problems by making the plant intelligent. Fuzzy logic is human concept, potentially applicable to a wide range of processes and tasks that require human intuition and experience. In computer, truth values are either 1 or 0, which correspond to true/false duality. In fuzzy logic, truth is the matter of degree, thus truth-values range between 1 and 0 in a continuous manner. Fuzzy logic is a method for representing information in a way that resembles natural human communication.

This paper proved that when we have a binary operation rightarrow and a nullary operations rarr on a non-empty set L, if these operations satisfy the four axioms of lattice implication algebra, then (L, rarr, 0) will be a lattice implication algebra. So, when we define lattice implication algebra, we needn't start on a complemented lattice with universal bounds, but we can begin with an algebra of type (2,0).

The concept of quasi-coincidence of a fuzzy point with a fuzzy set is extended to the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, the notions of interval valued (∈, ∈ ⋁q)-fuzzy filters in lattice implication algebras which is a generalization of ordinary fuzzy filters of lattice implication algebras, is defined, and their related properties and equivalent description are given. In particular, the concept of a fuzzy subgroup with thresholds is generalized to the concept of an interval valued fuzzy filter with thresholds in lattice implication algebras.

We investigate some related properties of fuzzy filters and fuzzy implicative filters in lattice implication algebras. We find a characterization of fuzzy filters and fuzzy implicative filters, and we discuss a relation between fuzzy filters and fuzzy implicative filters in lattice implication algebras. Also we give an extension theorem of fuzzy implicative filters.