Arithmetical complexity of fuzzy predicate logics – a survey

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

Copyright c 2013 Omar Salazar, Jairo Soriano, and Humberto Serrano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract It is known that any binary logic function (in binary logic) can be rep- resented by at least one logic formula, however this is not always true for a ternary logic function (in ternary logic). A ternary logic function can be represented by at least one logic formula if and only if it satises a mathematical condition which is called regularity. Also, it is known that a fuzzy logic formula (in standard fuzzy logic) can be obtained by means of truth tables in ternary logic. This paper presents a method destined for extending a binary logic function into one (or more) ter- nary logic function(s) representable by at least one logic formula, and therefore, to obtain fuzzy logic formulas from a binary logic function. Mathematics Subject Classication: 03B52, 06D72, 08A72, 94D05

A systematic review of complex fuzzy sets and logic

This study aimed to analyze the business impact of the catering sector using the fuzzy logic method. The research was conducted at a catering company in Istanbul, utilizing document review and participant observation methods to evaluate the business impact. The nominal prioritization method was used to identify critical business processes, and a model along with a mathematical formula was developed for calculating the business impact. The Fuzzy Logic Designer Toolbox in MATLAB was utilized for this calculation. The study identified eight critical business processes: (1) material supply, (2) material storage, (3) pre-preparation process, (4) cooking process, (5) portioning, (6) shipping, (7) hygiene and food safety, and (8) customer relationship management. The business impact was assessed using classical and fuzzy logic methods, and the results were compared. The fuzzy logic method provided a more flexible and comprehensive assessment, managing uncertainty and variability more effectively than classical logic. Overall, it proved to be more effective in optimizing business processes, offering a more dynamic and holistic approach to improving and prioritizing these processes.

An often raised objection to fuzzy logic can be articulated as follows. Fuzzy sets are intended to represent vague collections of objects. On the one hand, boundaries of such collections are inherently imprecise. On the other hand, fuzzy sets are defined by membership degrees which are precise. There is, therefore, an obvious discrepancy between the imprecision of a collection and the precision of a fuzzy set representing the collection. The objection leads to a question posed in the title of this paper: do exact shapes of fuzzy sets matter? That is, do the exact values of membership degrees of fuzzy sets matter? This is a fundamental question, particularly when fuzzy sets are supplied as an input to further processing. This paper presents a particular answer to this question. We focus on the case when the fuzzy sets in question are part of the input I from which the output O is produced in a way which can be described by logical formulas. This case covers several widely used fuzzy logical models including extension principle, products of fuzzy relations, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. We present formulas which say how similar is the original output O to a new output O′ when the original input I is replaced by a new input I′. The presented formulas provide us with an analytical tool which enables us to answer the question of whether (and to what extent) the exact shapes of fuzzy sets in a particular fuzzy logical model matter. We present application of our formulas to selected topics including the above-mentioned extension principle, fuzzy relational products, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. Based on the analysis, we argue that in several applications of fuzzy logic, the exact shapes of fuzzy sets do not really matter.

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

They give the soles of technological design process using fuzzy logic for metal cutting, referring to the determination of all the elements of production process: the dimensions and quality of the workpiece material, the sequence and scope of operations, the order and content of the procedures, the size of the type of machine types and tool types and gauges, regime and time of processing. It further explains manufacturability machine parts for robust design of a new product. He also offers manufacturability for cylindrical, prismatic workpieces and boxes. It explains the mathematical expressions of fuzzy logic which described above manufacturability. In fuzzy logic are used mathematical operations minimization and maximization. They are used to determine the critical solutions and choice of cost effective solutions. Provides an example of using the model to determine of the manufacturability.

<abstract><p>The application of a complex fuzzy logic system based on a linear conjunctive operator represents a significant advancement in the field of data analysis and modeling, particularly for studying physical scenarios with multiple options. This approach is highly effective in situations where the data involved is complex, imprecise and uncertain. The linear conjunctive operator is a key component of the fuzzy logic system used in this method. This operator allows for the combination of multiple input variables in a systematic way, generating a rule base that captures the behavior of the system being studied. The effectiveness of this method is particularly notable in the study of phenomena in the actual world that exhibit periodic behavior. The foremost aim of this paper is to contribute to the field of fuzzy algebra by introducing and exploring new concepts and their properties in the context of conjunctive complex fuzzy environment. In this paper, the conjunctive complex fuzzy order of an element belonging to a conjunctive complex fuzzy subgroup of a finite group is introduced. Several algebraic properties of this concept are established and a formula is developed to calculate the conjunctive complex fuzzy order of any of its powers in this study. Moreover, an important condition is investigated that determines the relationship between the membership values of any two elements and the membership value of the identity element in the conjunctive complex fuzzy subgroup of a group. In addition, the concepts of the conjunctive complex fuzzy order and index of a conjunctive complex fuzzy subgroup of a group are also presented in this article and their various fundamental algebraic attributes are explored structural. Finally, the conjunctive complex fuzzification of Lagrange's theorem for conjunctive complex fuzzy subgroups of a group is demonstrated.</p></abstract>

There are two main fuzzy system methodologies for translating expert rules into a logical formula: In Mamdani's methodology, we get a DNF formula (disjunction of conjunctions), and in a methodology which uses logical implications, we get, in effect, a CNF formula (conjunction of disjunctions). For both methodologies, universal approximation results have been proven which produce, for each approximated function f(x), two different approximating relations RDNF(x, y) and RCNF(x, y). Since, in fuzzy logic, there is a known relation FCNF(x) ≤ FDNF(x) between CNF and DNF forms of a propositional formula F, it is reasonable to expect that we would be able to prove the existence of approximations for which a similar relation RCNF(x, y) ≤ RDNF(x, y) holds. Such existence is proved in our paper. © 2002 Wiley Periodicals, Inc.

Extending possibilistic logic over Gödel logic

This paper proposes a new class of functions, i.e., fuzzy majority functions, based on the fuzzy logic which takes any value in the interval [0, 1] as the truth value of its proposition. Fuzzy majority functions are characterized by weights and thresholds, and may be represented by a logic formula involving constants. Specifically, the proposed class intersects with the well‐known fuzzy logic functions but is not a subset of them. On the other hand, the logic formula of a fuzzy majority function has a close relation with binary threshold functions. From this viewpoint, a theoretical limit in the utility of the fuzzy majority functions is discussed.

Short circuit evaluation techniques play important roles in hardware design, programming and other fields of computer science. In this paper one of the most known and used fuzzy logic system, the Godel logic is considered. Various pruning algorithms are presented to quicken the evaluations of logical formulae in Godel logic. Simulation results show the efficiency of the presented techniques.

As we sketched in [Godo and Hajek, 1996], most of fuzzy inference in Zadeh’s style is well formalized in the many-sorted many-valued Pavelka-style fuzzy predicate logic — and this can be done in several ways. The most important fact is that patterns of fuzzy inference can be presented as sound deduction rules, with all necessary assumptions expressed by some formulas of predicate logic (with the usual quantifiers ∀, ∃) in the premisse part of the rule. Here we complement the presentation of [Godo and Hajek, 1996] by a somewhat unexpected formalization of Zadeh’s Generalized Modus Ponens (GMP, cf. e.g. Zadeh [1988; 1990; 1994]). We shall be sketchy; a full detailed presentation is to be found in [Godo and Hajek, to appear]. For the reader’s convenience we repeat the definition of the many-sorted RQL (rational quantification logic, cf. [Hajek, 1995; Hajek, 1997]).

The article presents refutational resolution theorem proving system for Fuzzy Description Logic (FDL) based on general (non-clausal) resolution rule. It brings several new notions of refutation degree and proves its relation to provability degree in fuzzy predicate logic.

The Claims of Common Sense investigates the importance of ideas developed by Cambridge philosophers between the World Wars for the social sciences concerning common sense, vague concepts and ordinary language. John Coates examines the thought of Moore, Ramsey, Wittgenstein and Keynes, and traces their common drift away from early beliefs about the need for precise concepts and a canonical notation in analysis. He argues that Keynes borrowed from Wittgenstein and Ramsey their reappraisal of vague concepts, and developed the novel argument that when analysing something as complex as social reality, theory might be simplified by using concepts which lack sharp boundaries. Coates then contrasts this conclusion with the view shared by two contemporary philosophical paradigms - formal semantics and Continental post-structuralism - that the vagueness of ordinary language inevitably leads to interpretive indeterminacy. Developing a link between Cambridge philosophy and work on complexity, vague predicates and fuzzy logic, he argues that Wittgenstein's and Keynes's ideas on the economy of ordinary language present a mediating route for the social sciences between these philosophical paradigms.