In this short communication, we introduce a class of functions defined via quasi-arithmetic means on the unit interval. We demonstrate that when these means are induced by strictly convex and differentiable additive generators of strict triangular norms, the resulting functions are distance functions that satisfy the properties of continuous and symmetric monometrics as well.

Approximate reasoning Threshold-based Modus PonensIn 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.

BACKGROUND: Accurate prediction of copy number variations (CNVs) from targeted capture next-generation sequencing (NGS) data relies on effective normalization of read coverage profiles. The normalization process is particularly challenging due to hidden systemic biases such as GC bias, which can significantly affect the sensitivity and specificity of CNV detection. In many cases, the kit manifests provide only the genome coordinates of the targeted regions, and the exact bait design of the oligo capture baits is not available. Although the on-target regions significantly overlap with the bait design, a lack of adequate information allows less accurate normalization of the coverage data. In this study, we propose a novel approach that utilizes a 1D convolution neural network (CNN) model to predict the positions of capture baits in complex whole-exome sequencing (WES) kits. By accurately identifying the exact positions of bait coordinates, our model enables precise normalization of GC bias across target regions, thereby allowing better CNV data normalization. RESULTS: We evaluated the optimal hyperparameters, model architecture, and complexity to predict the likely positions of the oligo capture baits. Our analysis shows that the CNN models outperform the Dense NN for bait predictions. Batch normalization is the most important parameter for the stable training of CNN models. Our results indicate that the spatiality of the data plays an important role in the prediction performance. We have shown that combined input data, including experimental coverage, on-target information, and sequence data, are critical for bait prediction. Furthermore, comparison with the on-target information indicated that the CNN models performed better in predicting bait positions that exhibited a high degree of overlap (>90%) with the true bait positions. RESULTS: This study highlights the potential of utilizing CNN-based approaches to optimize coverage data analysis and improve copy number data normalization. Subsequent CNV detection based on these predicted coordinates facilitates more accurate measurement of coverage profiles and better normalization for GC bias. As a result, this approach could reduce systemic bias and improve the sensitivity and specificity of CNV detection in genomic studies.

In this short communication, we will show that a quasi-arithmetic mean induced by an additive generator of a strict t-norm (strict t-conorm, respectively) can be represented by the composition of a unary operator (called the tau function) and the strict t-norm (strict t-conorm, respectively), both induced by the same generator function as the quasi arithmetic mean. Here, we will also state a connection between the idempotency of a transformed strict t-norm (strict t-conorm, respectively) and the tau function.

We present a novel idea of the membership function using logical expressions over inequalities. This is achieved by introducing the multidimensional membership function (distending function) using inequalities by including more variables instead of just one. In this paper, we concentrate on the application of this new approach to different kinds of feasibility problems. This new concept serves as a good tool for describing various kinds of feasibility regions. Another result here is that we present algorithms that can handle linear, nonlinear, convex and non-convex feasibility problems. Usually, in practice only the conjunction operator is used in the case of feasibility problems. A novelty of our result is that we drop this restriction, and based on our construction we will describe regions where other types of operators can also be used. We shall introduce the fundamentals of this new concept and we will show how it can be used in practical applications. Later, this approach may be applied to neural networks as well.

Abstract Following tfhe seminal paper of Offerman et al. (2009), in this study, adaptations of constructions of continuous-valued logic to prospect theory are presented. Here, we demonstrate that the so-called kappa function and its special cases are viable alternatives to some elements of quadratic scoring rule prospects theory. After, we present the tau-eta scoring rule prospect and show that it may be treated as a generalization of the quadratic scoring rule prospect. Furthermore, we prove that if this new prospect for an uncertain event is evaluated using specific kappa functions as utility functions, then (1) the weighting measure of the event is a function of the optimal value of its reported probability, (2) the inverse of the latter function, and (3) the (risk-) corrected reported probability of the event, also as a function of the optimal value of its reported probability, all have a common formula. The parameters of the common formula are unambiguously determined by four tuning parameters. Lastly, we show that with our approach, by fitting one of the abovementioned functions to corresponding empirical data, we can immediately obtain the other two functions as well.

Fuzzy type-2 modeling techniques are increasingly being used to model uncertain dynamical systems. However, some challenges arise when applying the existing techniques. These are: 1) A large number of rules are required to complete cover the whole input space; 2) A large of parameters associated with type-2 membership functions have to be determined; 3) The identified fuzzy model is usually difficult to interpret due to the large number of rules; 4) Designing a fuzzy type-2 controller using these models is a computationally expensive task. To overcome these limitations, a procedure is proposed here to identify the fuzzy type-2 model directly from the data. This model is called the Distending Function-based Fuzzy Inference System (DFIS). This model consists of rules and interval Type-2 Distending Functions (T2DFs). First, a few key rules are identified from the data and later more rules are added until the error is less than the threshold. The proposed procedure is used to model the altitude controller of a quadcopter. The DFIS model performance is compared with various fuzzy models. Furthermore a simplified procedure based on the rules is presented to design a computationally low-cost type-2 controller. The effectiveness of the controller is shown by regulating the height of a quadcopter in the presence of noisy sensory data. The performance of this controller is compared with various other controllers. Lastly, the proposed type-2 controller was implemented on a Parrot Mambo quadcopter to demonstrate its real-time performance.

In the application of fuzzy sets, the greatest uncertainty appears when the membership grade in a fuzzy set is equal to the membership grade in the complement of this fuzzy set. That is, the membership grade is equal to the value of the fixed point ($\nu$) of the underlying complement (negation) operator. The fixed point of the standard Zadeh negation is 0.5. In fuzzy set theory, the complement of a fuzzy set can be defined using a strong complement (negation) operator that differs from the standard Zadeh negation. In this case, the fixed point of the strong complement (negation) operator is not necessarily 0.5. In this short paper, we present the concept of a parametric fuzziness measure called the$\nu$-maximal vagueness measure. This new measure may be regarded as a generalized fuzziness measure. Here, we present an operator system-dependent kernel, which we call the$\nu$-maximal vagueness entropy, and using this, we construct a$\nu$-maximal vagueness measure. The proposed vagueness entropy is based on a common generator function of a strict triangular norm and a strong negation operator. The$\nu$-maximal vagueness entropy is flexible and operator-dependent, and it can be readily adapted to a continuous-valued logical system. Furthermore, the parameter$\nu$has a clear semantic meaning.

In this short communication, we will present a connection between two well-known representations of the strong negations (involutive negations). Namely, we will provide a necessary and sufficient condition for the equality of Trillas and Dombi forms of negations. We will also show a connection between the additive generators of nilpotent and strict triangular norms.

In this study, we present a continuous-valued logical approach to generalize the sigmoid function. Our starting point is the kappa function, which is known as a unary operator in continuous-valued logic. First we extend the kappa function to the (a,b) interval, and then we interpret the generalized sigmoid function as the limit of the extended kappa function when a and b tend to the negative and positive infinity, respectively. Since the extended kappa function is induced by an additive generator of a strict t-norm or strict t-conorm, the generalized sigmoid function is operator dependent. Based on the properties of this new function, we show that it can be viewed as the generalization of the classical sigmoid function. Also, we demonstrate that the classical sigmoid function is a special case of the generalized sigmoid function. Next, we provide a sufficient condition for the equality of two generalized sigmoid functions. It is well known that the classical sigmoid function can be utilized in logistic regression and in preference modeling. Here, we demonstrate how the logistic regression can be generalized using the generator function-based sigmoid function. Also, we show that the generalized sigmoid function can be viewed as a preference measure.

Here, we present an operator-dependent, analytic membership function family that is derived from two soft inequalities by using the so-called middle hedge operator. This operator is defined over a pair of a strictly increasing and a strictly decreasing parametric membership functions that represent two soft inequalities. The middle hedge operator can also be treated as a membership function of an interval on a bounded or unbounded domain. Owing to its construction, the values of this membership function at the end points of the interval, which it represents, are equal to a given parameter value. This parameter may be interpreted as an intersection parameter. That is, if a fuzzy partition of a bounded or unbounded interval is created by using middle hedge operators with a fixed value of the intersection parameter, then each two successive membership functions in the fuzzy partition intersect at the same level. Here, we demonstrate that this new membership function family is very flexible and it can be readily used for constructing a membership function system in a fuzzy control system.

In this paper, we study monotone set functions defined as the composition of an additive measure with a strictly increasing function. This function is a unary operator in continuous-valued logic, called the tau function, and it is a generator function-based parametric mapping. We provide a necessary and sufficient condition for the equality of two tau functions that are induced by different generator functions. Using the tau function and its properties, we introduce a new monotone measure that we call the tau-additive measure. This measure is computationally simple and it can be viewed as an upper or lower probability depending on the parameter settings of the tau function. We present the parameter-dependent submodularity and supermodularity of the tau-additive measure and show how this measure can be constructed from a set function on a finite set. This procedure is analogous to how the well-known lambda-additive measure can be constructed, but our method is computationally simpler. We demonstrate that the tau-additive measure can be used to approximate the lambda-additive measure. Lastly, exploiting these theoretical results, we present an application in the area of human resource management.

In this study, we present the operator-dependent sigmoid function, which is derived from a universal unary operator called the kappa function. Here, we describe how the generalized sigmoid function is related to representable uninorms (i.e., Dombi's aggregative operator). Namely, we show that the inverse of a generalized sigmoid function is an additive generator of the aggregative operator. We provide the necessary and sufficient conditions for the form of the function that transforms the aggregative operator into a conjunctive or disjunctive logical operator. This transformation is also based on the generalized sigmoid function. Here, we show how conjunctive and disjunctive operators, which form a De Morgan system with a negation, can be derived from the aggregative operator. We point out that, under certain conditions, a set of generalized sigmoid functions is closed under the negation and modifier operators. Lastly, we demonstrate an important connection between the weighted aggregative operator and the generalized sigmoid function. Based on this connection, we provide a new interpretation of the feed-forward neural networks. We show that a perceptron-based neural network can be modeled using the aggregative operator and the generalized sigmoid function.

In this study, the notion of a dual pair of modal operators is interpreted according to the algebraic criteria for necessity and possibility operators on De Morgan lattices presented by Cattaneo, Ciucci and Dubois, 2011. Here, a representation theorem is introduced which demonstrates that, in this algebraic model, a dual pair of modal operators can be represented by compositions of two strong negations, where one of them is stricter than the other. Then, the Pliant negation operator is utilized to derive dual modal operators. It is demonstrated that using the generator function of Dombi operators, the composition of two Pliant negations results in modal operators that have simple forms and easy-to-use characteristics. Next, we examine how the proposed modal operators are connected with the drastic necessity and possibility operators. Also, the necessary and sufficient condition for the distributivity of modal operators induced by compositions of strong negations over strict t-norms and strict t-conorms is presented. Lastly, a connection between the modal operators and hedges is highlighted.

In this paper, we study operator dependent modifiers and we interpret the dual pair of modal operators based on an algebraic definition. It is a known fact that the substantiating and weakening modifier operators can be induced by repeating the arguments of conjunctive and disjunctive operators. We provide the conditions for which these modifier operators satisfy the requirements for a dual pair of necessity and possibility operators. Next, the necessary and sufficient condition for the distributivity of unary operators over conjunctive and disjunctive operators is presented. This also means that setting the distributivity as a requirement results in a unary operator that is identical to the modal operators mentioned above. Using this property, we establish an important connection between modal operators and linguistic hedges. Previously, we demonstrated that the unary operators induced by compositions of two strong negations satisfy the requirements for a dual pair of modal operators. Here, we view the negation operator as a modifier operator. Then, it is shown that (1) the strong negations, (2) the substantiating and weakening modifier operators, modal operators and linguistic hedges mentioned above, and (3) the unary operators, which are distributive over conjunctive and disjunctive operators, may be viewed as special cases of a unified unary operator class.

Most multi-layer neural networks used in deep learning utilize rectified linear neurons. In our previous papers, we showed that if we want to use the exact same activation function for all the neurons, then the rectified linear function is indeed a reasonable choice. However, preliminary analysis shows that for some applications, it is more advantageous to use different activation functions for different neurons - i.e., select a family of activation functions instead, and select the parameters of activation functions of different neurons during training. Specifically, this was shown for a special family of squashing functions that contain rectified linear neurons as a particular case. In this paper, we explain the empirical success of squashing functions by showing that the formulas describing this family follow from natural symmetry requirements.

Following our previous paper Dombi and Jónás (2019) [16], we will now present new inequalities using the general Poincaré formula for λ-additive measures. These inequalities represent bounds for the well-known Poincaré formula of probability theory.

Huynh et al. introduced a probability-based fuzzy relation for comparing fuzzy numbers (see V. Huynh, Y. Nakamori and J. Lawry (2008) [40]), but they did not detail how to compute it. Here, we will consider this fuzzy relation as a probability-based preference intensity index and present closed formulas for the integrals needed to compute this index for fuzzy sets that have trapezoidal membership functions. Also, we will propose an algorithm to compute this index and a numerical method to approximate it. The comparison of two fuzzy numbers should also be able to capture the situation where the order of the fuzzy numbers cannot be judged; and so, their order may be considered as being indifferent. Here, using the probability-based preference intensity index, we will introduce two crisp relations, which have a common parameter, over a collection of fuzzy sets that have trapezoidal membership functions. Next, we will show that - depending on the parameter value - one of them is a strict order relation and the other one may be interpreted as a relation that expresses the order indifference of fuzzy numbers. We will call this latter one the order indifference relation. Lastly, we will demonstrate how these two relations can be utilized to rank a collection of fuzzy sets that have trapezoidal membership functions.

In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.

Combining neural networks with continuous logic and multicriteria decision-making tools can reduce the black-box nature of neural models. In this study, we show that nilpotent logical systems offer an appropriate mathematical framework for hybridization of continuous nilpotent logic and neural models, helping to improve the interpretability and safety of machine learning. In our concept, perceptrons model soft inequalities; namely membership functions and continuous logical operators. We design the network architecture before training, using continuous logical operators and multicriteria decision tools with given weights working in the hidden layers. Designing the structure appropriately leads to a drastic reduction in the number of parameters to be learned. The theoretical basis offers a straightforward choice of activation functions (the cutting function or its differentiable approximation, the squashing function), and also suggests an explanation to the great success of the rectified linear unit (ReLU). In this study, we focus on the architecture of a hybrid model and introduce the building blocks for future applications in deep neural networks.

The theories of multi-criteria decision-making (MCDM) and fuzzy logic both aim to model human thinking. In MCDM, aggregation processes and preference modeling play the central role. This paper suggests a consistent framework for modeling human thinking by using the tools of both fields: fuzzy logical operators as well as aggregation and preference operators. In this framework, aggregation, preference, and the logical operators are described by the same unary generator function. Similarly to the implication being defined as a composition of the disjunction and the negation operator, preference operators were introduced as a composition of the aggregative operator and the negation operator. After a profound examination of the main properties of the preference operator, our main goal is the implementation into neural networks. We show how preference can be modeled by a perceptron, and illustrate the results in practical neural applications.

The aim of this paper is twofold. On the one hand, the \(\lambda \) -additive measure (Sugeno \(\lambda \) -measure) is revisited, and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called \(\nu \) -additive measure as an alternatively parameterized \(\lambda \) -additive measure is introduced. Here, the advantages of the \(\nu \) -additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the \(\nu \) -additive measure lies in the fact that its parameter \(\nu \in (0,1)\) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the \(\nu \) -additive measure, some well-known results concerning the \(\lambda \) -additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the \(\nu \) -additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two \(\nu \) -additive measures, with the parameters \(\nu _1\) and \(\nu _2\) , are a dual pair of belief- and plausibility measures if and only if \(\nu _1+\nu _2 = 1\) . Furthermore, it is demonstrated how a \(\nu \) -additive measure (or a \(\lambda \) -additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the \(\nu \) -additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic.

Abstract In this study, we present a novel methodology that can be used to generate parametric probability weighting functions, which play an important role in behavioral economics, by making use of the Dombi modifier operator of continuous-valued logic. Namely, we will show that the modifier operator satisfies the requirements for a probability weighting function. Next, we will demonstrate that the application of the modifier operator can be treated as a general approach to create parametric probability weighting functions including the most important ones such as the Prelec and the Ostaszewski, Green and Myerson (Lattimore, Baker and Witte) probability weighting function families. Also, we will show that the asymptotic probability weighting function induced by the inverse of the so-called epsilon function is none other than the Prelec probability weighting function. Furthermore, we will prove that, by using the modifier operator, other probability weighting functions can be generated from the dual generator functions and from transformed generator functions. Finally, we will show how the modifier operator can be used to generate strictly convex (or concave) probability weighting functions and introduce a method for fitting a generated probability weighting function to empirical data.

It is widely accepted that distributivity properties play a key role in fuzzy research, especially in fuzzy control. Making use of the solution of the autodistributivity functional equations, we give a characterization of all the four types of distributivity of fuzzy implication. The necessary and sufficient condition for all the four distributive equation is that the operator belongs to the pliant operator class. This theorem leads to a new implication called preference implication. It is well known that there is no implication in (continuous-valued) fuzzy logic that satisfies all the properties that are valid in (classical) two-valued logic. We show that preference implication fulfills: 1) the law of contraposition, 2) the T-conditionality, 3) the ordering property, 4) the exchange principle, 5) the law of importation, 6) the identity principle, and 7) the general hypothetical reasoning. Preference implication has a ν parameter, i.e., the fixed point of the negation. This ν value serves as a threshold and with this value we can go back by projection to the two-valued logic case. At the end of the article, we indicate that the preference implication is closely related to the preference relation used in multicriteria decision making. We point out that if the preference implication is multiplicative, transitive, and reciprocal, then the pliant system is reduced to some particular generator function of Dombi.

The purpose of the current study is to consider the main unary operators of a nilpotent logical system in an integral framework and to reveal the underlying general structure of all the previously examined operators in nilpotent logical systems.The unary operators are obtained by repeating the argument in multivariable operators.This enables us to provide a widely applicable system, where all the operators are connected to each other, and where the modalities and hedges are operator-dependent.It becomes possible to describe all the operators by using a generator function and a few parameters.The possibility, necessity and sharpness operators are thoroughly examined and it is also shown how the multivariable operators can be derived from the unary ones.

Intelligent decision support systems may be required to adapt to changes in their operating environment. FPGA-based hardware accelerators can be used to design embedded systems that are reprogrammable and reconfigurable. A new algorithm was proposed to implement a dynamic, adaptive inference mechanism. A hardware accelerator was developed for this fuzzy automaton model. A Zynq-7000 FPGA platform was used for the hardware implementation. An application example was given in the context of eye-hand coordination problems with handicapped children. Simulations results were presented to verify the hardware implementation.

In our study we give a general form for modifiers that includes negation, different types of hedge and the sharpness operators. We will show that the four operators have a common form in the Pliant system and they will be called modifier operators. By changing the parameter value of a modifier we get the modalities, negation and the sharpness operators.

Our starting point is the multiplicative utility function which is extensively used in the theory of multicriteria decision making. Its associativity is shown and as its generalization a class of operators is introduced with fine and useful properties. As in special cases, it reduces to well-known operators of fuzzy set theory: min/max, product, Einstein, Hamacher, Dombi, and drastic. As a consequence, we generalize the addition of velocities in Einstein's special relativity theory to multiple moving objects. Also, a new form of the Hamacher operator is given, which makes multiargument calculations easier. We examine the De Morgan identity which connects the conjunctive and disjunctive operators by a negation. It is shown that in some special cases (min/max, drastic, and Dombi) the operator class forms a De Morgan triple with any involutive negation.