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This study introduces an innovative interval-valued fuzzy inference system (IFIS) integrated with federated learning (FL) to enhance posture detection, with a particular emphasis on fall detection for the elderly. Our methodology significantly advances the accuracy of fall detection systems by addressing key challenges in existing technologies, such as false alarms and data privacy concerns. Through the implementation of FL, our model evolves collaboratively over time while maintaining the confidentiality of individual data, thereby safeguarding user privacy. The application of interval-valued fuzzy sets to manage uncertainty effectively captures the subtle variations in human behavior, leading to a reduction in false positives and an overall increase in system reliability. Furthermore, the rule-based system is thoroughly explained, highlighting its correlation with system performance and the management of data uncertainty, which is crucial in many medical contexts. This research offers a scalable, more accurate, and privacy-preserving solution that holds significant potential for widespread adoption in healthcare and assisted living settings. The impact of our system is substantial, promising to reduce the incidence of fall-related injuries among the elderly, thereby enhancing the standard of care and quality of life. Additionally, our findings pave the way for future advancements in the application of federated learning and fuzzy inference in various fields where privacy and precision in uncertain environments are of paramount importance. [ABSTRACT FROM AUTHOR]

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In fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication. That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms. In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning. Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement. Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises. We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm. Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way. Lastly, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

Fuzzy sets theory (cf. Zadeh [71]) and its related fuzzy logic has been proposed for dealing with and solving various problems in which variables, parameters, relations, etc. are only imprecisely known and hence approximate reasoning schemes should be used. Such a situation is common in the case of virtually all non-trivial, in particular human centered, phenomena, processes and systems that prevail in reality and it is difficult to use conventional mathematics, based on binary logic, for their adequate characterization. The classic fuzzy set with the degrees of membership and non-membership of elements summing up to 1, has then been extended to many other concepts, and in our context these can be notably the intuitionistic (ordinary and interval) fuzzy sets (cf. Atanassov [1–5], hesitant fuzzy sets (cf. Torra [65] and Torra and Narukawa [66]) or vague sets (cf. Gau and Buehrer [22], Xu, Ma, Wang and Hao [70] (2010). Here we use the concept and tools and technique of vague sets in which basically for each element of a universe of discourse a true membership degree and a false membership degree, both from [0, 1], are assigned, and their sum is up to 1. We are concerned with a general decision making problem in medical diagnosis under imprecise information in which imprecision is represented by vague sets, In particular, a new definition of the basic concept of a distance between the vague sets is proposed as a crucial element of the approach. A graph theoretic perspective is assumed. An example is shown. [ABSTRACT FROM AUTHOR]

Inference systems are intelligent software performed generally to help people take appropriate decisions and solve problems in specific domains. Fuzzy inference systems are a kind of these systems that are based on fuzzy knowledge. To handle the fuzziness in the inference, the compositional rule of inference is used, which has two parameters: a t-norm and an implication operator. However, most of the combinations of t-norm/implication do not give an adequate inference result that coincides with human intuitions. This was the motivation for several works to study these combinations and to identify those that are compatible, in order to guarantee a performance close to that of humans. We are interested in this paper to a more general form of rules, which is complex rules, whose premise is a conjunction of propositions. To obtain the consequence in a fuzzy inference system using the compositional rule of inference with a complex rule, we study, in this work, Lukasiewicz t-norm which was not investigated before in this context. We combine it with known implications, and we verify the satisfaction of some criteria that model human intuitions. [ABSTRACT FROM AUTHOR]

Motivated by several problems on constructions, algebraic properties, in particular the law of importation (LI) and the flexible ordering property (FOP), and applications in approximate reasoning, the present article defines a new class of fuzzy implications, called $(\theta,t)$ -generated implications, by a pair of multiplicative and additive generators of t-norms and the usual addition instead of the multiplication or division usually used in the literature. The relations to other generator generated implications are clarified, the intersections with $(S,N)$ -implications are discussed in detail and the intersections with $R$ -implications are proved to be conjugates of the Łukasiewicz implication. The t-norm solutions to the (LI) equation for $(\theta,t)$ -generated implications in several cases are characterized in terms of generated t-norms, and two special subclasses of $(\theta,t)$ -generated implications are characterized through (LI) and (FOP) on the other hand. A particularly interesting corollary shows that a binary operation on [0,1] enjoying (OP) satisfies (LI) with the Łukasiewicz t-norm if and only if it is the Łukasiewicz implication. These results will partially solve or enrich the related open problems pending for fuzzy implications. As applications of $(\theta,t)$ -generated implications, two fuzzy reasoning methods, called triple I method and parallel hierarchical triple I method, based on (LI) and (FOP) are proposed in order to set a logic foundation for Zadeh’s compositional rule of inference (CRI) and to remedy the dependency of Jayaram’s hierarchical CRI on the order of inputs, respectively. [ABSTRACT FROM AUTHOR]

The fuzzy logic reasoning based on the "If... then..." rule is not the inaccurate reasoning of AI against ambiguity because fuzzy reasoning is antilogical. In order to solve this problem, a redundancy theory for discriminative weight filtering containing six theorems and one M(1,2,3) model was proposed and the approximate reasoning process was shown, the system logic of AI handling ambiguity as an extension of the classical logic system was proposed. The system is a generalized dynamic logic system characterized by machine learning, which is the practical-application logic system of AI, and can effectively deal with practical problems including conflict, noise, emergencies and various unknown uncertainties. It is characterized by combining approximate reasoning and computing for specific data conversion through machine learning. Its core is data and calculations and the condition is "sufficient" high-quality training data. The innovation is that we proposed a discriminative weight filtering redundancy theory and designed a computable approximate reasoning logic system that combines approximate reasoning and calculation through machine learning to convert specific data. It is a general logic system for AI to deal with uncertainty. The study has significance in theory and practice for AI and logical reasoning research. [ABSTRACT FROM AUTHOR]

The law of importation, given by the equality $(x\wedge y) \longrightarrow z\equiv (x\longrightarrow (y\longrightarrow z))$ , is a tautology in classical logic and has been proved to be widely used in approximate reasoning and image processing. Some open problems of fuzzy implication dealing with the law of importation were suggested on 8th International Conference on Fuzzy Set Theory and Applications (FSTA 2006). In this article, we partially solve one open problem associated with this property. Specifically, we mainly devote ourselves to solving the general form of the law of importation $I(U(x,y),z)=I(x,I(y,z))$ , where $I$ is a fuzzy implication and $U$ is a conjunctive uninorm with a continuous underlying t-norm and a continuous underlying t-conorm. Along this study, given a fixed uninorm with continuous underlying operators, all fuzzy implications that satisfy the law of importation with respect to this uninorm, and having an $\alpha$ -section that is a continuous negation, are characterized. [ABSTRACT FROM AUTHOR]

Its goals can be characterized as follows: To express all concepts capable of being expressed in natural language, To characterize all the valid inferences that can be made in natural language, To mesh with adequate linguistic descriptions of all natural languages. In this contribution, the Fuzzy Property Grammar permits to describe linguistic complexity of a natural language and linguistic universality (presence of a grammatical characteristics in all or most natural languages) as vague concepts. In [[1]], we gave reasons to focus on one of the possible future directions of fuzzy logic towards the concept of fuzzy natural logic (FNL). [Extracted from the article]

Approximate reasoning systems facilitate fuzzy inference through activating fuzzy if–then rules in which attribute values are imprecisely described. Fuzzy rule interpolation (FRI) supports such reasoning with sparse rule bases where certain observations may not match any existing fuzzy rules, through manipulation of rules that bear similarity with an unmatched observation. This differs from classical rule-based inference that requires direct pattern matching between observations and the given rules. FRI techniques have been continuously investigated for decades, resulting in various types of approach. Traditionally, it is typically assumed that all antecedent attributes in the rules are of equal significance in deriving the consequents. Recent studies have shown significant interest in developing enhanced FRI mechanisms where the rule antecedent attributes are associated with relative weights, signifying their different importance levels in influencing the generation of the conclusion, thereby improving the interpolation performance. This survey presents a systematic review of both traditional and recently developed FRI methodologies, categorised accordingly into two major groups: FRI with non-weighted rules and FRI with weighted rules. It introduces, and analyses, a range of commonly used representatives chosen from each of the two categories, offering a comprehensive tutorial for this important soft computing approach to rule-based inference. A comparative analysis of different FRI techniques is provided both within each category and between the two, highlighting the main strengths and limitations while applying such FRI mechanisms to different problems. Furthermore, commonly adopted criteria for FRI algorithm evaluation are outlined, and recent developments on weighted FRI methods are presented in a unified pseudo-code form, easing their understanding and facilitating their comparisons. [ABSTRACT FROM AUTHOR]

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8]. [ABSTRACT FROM AUTHOR]

Fuzzy logic is one of the precisely constructed and wellmotivated logical systems that find successful applications in various fields. They are especially suitable in cases when it is necessary to use approximate reasoning. In this paper, I present the main views on the application of fuzzy logic and fuzzy sets in the field of medicine and I express my attitude to them. [ABSTRACT FROM AUTHOR]

Unification is a central concept in logic systems based on the resolution principle. As well, in knowledge representation, proximity relations (i.e., reflexive, symmetric, fuzzy binary relations) are useful for introducing semantics into a syntactic level by modeling the semantic closeness of different syntactic objects and managing vague or imprecise information. Proximity relations, in combination with the unification algorithm, make possible expressing certain forms of approximate reasoning in a logic programming framework. In this article, we use proximity relations in the context of a (fuzzy) logic programming system, called Bousi ~ Prolog, as a way of solving the limitations introduced by similarity relations (i.e., transitive proximity relations) to correctly represent fuzzy information. Recently, we introduced an accurate definition of proximity between expressions (terms or atomic formulas) and a new unification algorithm able to manage proximity relations properly. However, the so-called weak unification algorithm, which is an extension of Martelli and Montanari's unification algorithm supported by the new notion of proximity, does not have an efficient implementation. In this article, we present a method that facilitates such an efficient implementation, including an adaptation of the weak SLD resolution rule based on the new unification algorithm, and its integration and implementation into the fuzzy logic programming system Bousi ~ Prolog. A performance analysis to show its efficiency is also presented. [ABSTRACT FROM AUTHOR]

In this research, we introduce a classification procedure based on rule induction and fuzzy reasoning. The classifier generalizes attribute information to handle uncertainty, which often occurs in real data. To induce fuzzy rules, we define the corresponding fuzzy information system. A transformation of the derived rules into interval type-2 fuzzy rules is provided as well. The fuzzification applied is optimized with respect to the footprint of uncertainty of the corresponding type-2 fuzzy sets. The classification process is related to a Mamdani type fuzzy inference. The method proposed was evaluated by the F-score measure on benchmark data. [ABSTRACT FROM AUTHOR]

Implications are one of the key operations in fuzzy logic and approximate reasoning. We prove that the Dishkant arrow in a lattice effect algebra is a partial triangular implication if and only if the lattice effect algebra is an MV-effect algebra. Then we show that a certain implication based on the Sasaki arrow is a partial triangular implication. Furthermore, we propose a new partial triangular implication, which is also a fuzzy implication in lattice effect algebras. [ABSTRACT FROM AUTHOR]

This paper presents multi-adjoint algebras as a natural and formal extension of the original algebraic structure taken into account by Zadeh in the theories of fuzzy sets and fuzzy logic. [ABSTRACT FROM AUTHOR]

Introduction: Fuzzy logic is gaining popularity with predictive healthcare applications in the field of medicine. It is consistently being used in recognising various chronic diseases with optimum accuracy. Fuzzy logic is also supposed to be a beneficial tool to diagnose celiac disease with approximate reasoning. Aim: To design a fuzzy logic inference system to predict celiac disease based on individual symptoms in North-Indian patients. Materials and Methods: The study was conducted in Punjab, Northern India, on 700 individuals using a questionnaire stratagem. SPSS software was used for the analysis of celiac data. The fuzzy logic system was implemented in Python software with 6 inputs and 1 fuzzy output parameter to determine the probability level of celiac disease. Results: The fuzzy logic system produced 98.8% accuracy, 98.5% sensitivity, and 98.39% specificity with minimal error rate. The prevalence of celiac disease obtained a 1:19 ratio from the survey in Punjab with 134 celiac patients. Conclusion: Fuzzy logic yields better results in terms of sensitivity and specificity in comparison with other existing disease probabilistic systems. Hence, the fuzzy system is a fruitful tool for celiac disease disclosure without any painful testing strategy. [ABSTRACT FROM AUTHOR]

In recent years, descriptive evaluation has been introduced as a new model for educational evaluation of Iranian students. The current descriptive evaluation method is based on four-valued logic. Assessing all students with only four values is led to a lack of relative justice and creation of unrealistic equality. Also, the complexity of the evaluation process in the current method increases teacher error's likelihood. As a suitable solution, in this paper, a fuzzy descriptive evaluation system has been proposed. The proposed method is based on fuzzy logic, which is an infinite-valued logic, and it can perform approximate reasoning on natural language propositions. By the proposed fuzzy system, student assessment is performed over the school year with infinite values instead of four values. In order to eliminate the diversity of assigned values to students, at the end of the school year, the calculated values for each student will be rounded to the nearest value of the four standard values of the current descriptive evaluation method. It can be implemented in an appropriate smartphone application, which makes it much easier for teachers to assess the educational process of students. In this paper, the evaluation process of the elementary third-grade mathematics course in Iran during the period from the beginning of the MEHR (the seventh month of Iran) to the end of BAHMAN (the eleventh month of Iran) is examined by the proposed system. To evaluate the validity of this system, the proposed method has been simulated in MATLAB software. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present FTCLogic, a formal first-order logic that can manage fuzzy temporal constraints between variables efficiently. In this logic, the use of explicit temporal axioms is unnecessary, and therefore the deduction mechanism doesn't slow down for this reason. FTCLogic has an immediate precedent: the Extended Fuzzy Temporal Constraint Logic or EFTCL. However, while EFTCL is based on Timed Possibilistic Logic, FTCLogic uses the Possibilistic Logic to formulate an original semantics according to its syntax. In fact, FTCLogic defines both syntax and semantics from a powerful combination of two formalisms: the Possibilistic Logic and the Fuzzy Temporal Constraints Networks. FTCLogic has provided the basis for the creation of a fuzzy temporal PROLOG: PROLogic, which is implemented through Haskell, and which is currently undergoing evaluation. [ABSTRACT FROM AUTHOR]

Nowadays, knowledge-based system has to be able to model and treat imperfect knowledge. Among the knowledge imperfection, we cite imprecision. Imprecise information are generally represented in a quantitative way using fuzzy logic or in a qualitative way using symbolic multi-valued logic. As far as we knew, no work has considered both fuzzy and symbolic multi-valued knowledge simultaneously in the same knowledge-based system. However, the user is often in need of both data types to insure a relevant decision-making. In order to improve the decision-making process performance, we propose in this paper an approach, that is able to standardize input knowledge. In fact, we propose a fuzzy-to-symbolic conversion of inputs by projecting them over the abscissa axis. We apply the proposed conversion module in symbolic inference systems. Thus, a symbolic approximate reasoning can be executed. The conversion process involves the expert by asking him to express its tolerance threshold toward handled fuzzy knowledge. Thus, a minimum of fuzzy information loss will be insured according to the expert preferences and the reasoning context. Our proposal is also useful even when the rule conclusion is originally fuzzy. In that case, a symbolic-to fuzzy conversion of the inference result is required to make the inference result more intelligible for the user and to maintain the transparency of the fuzzy-to-symbolic conversion. A numerical study is provided to illustrate the potential applications of the proposed methodology. [ABSTRACT FROM AUTHOR]

Distributivity between two operations is a property posed many years ago — that is especially interesting in the framework of logical connectives because of its applications to fuzzy logic and approximate reasoning as their applications. Since semi-uninorms have been used in these topics, the study of the distributivity between two semi-uninorms becomes of particular interest that calls for thorough studies. The distributivity between two semi-uninorms, which are non-commutative and non-associative uninorms, has been developed only in the cases when both semi-uninorms are examples of very special classes of semi-uninorms. On the other hand, in general, the distributivity does not rely on the commutativity and associativity. The objective of this work is twofold. The first one is to show new solutions to distributivity equations for semi-uninorms. The second one is to check whether the results concerning the distributivity between two uninorms are valid for semi-uninorms. We investigate the distributivity involving two semi-uninorms when only one semi-uninrom lies in the most studied classes of semi-uninorms, achieving the above two objectives simultaneously. [ABSTRACT FROM AUTHOR]

Fuzzy possibilistic logic is an important formalism for approximate reasoning. It extends the well-known basic propositional logic BL, introduced by Hájek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Pseudomonadic BL-algebras. These algebras turn to be a generalization of both Pseudomonadic algebras introduced by Bezhanishvili (Math Log Q 48:624-636, 2002) and serial, Euclidean and transitive Bimodal Gödel algebras proposed by Caicedo and Rodriguez (J Log Comput 25:37-55, 2015). We present the connection between this class of algebras and possibilistic BL-frames, as a first step to solve an open problem proposed by Hájek (Metamathematics of fuzzy logic. Trends in logic, Kluwer, Dordrecht, 1998, Chap. 8, Sect. 3). [ABSTRACT FROM AUTHOR]