Compositional Rule Of Inference Research Articles

Type-3 fuzzy neural networks for dynamic system control

This paper presents a Type-3 Fuzzy Neural Network (T3FNN) for dynamic system control. The structure of the T3FNN controller, which is based on the compositional rule of inference of interval type-3 fuzzy systems, is proposed. TSK-type fuzzy rules, employing three-dimensional membership functions (MFs) in the antecedent part and linear functions with coefficients represented as interval sets in the consequent part, are utilized to design T3FNN. Gaussian functions with uncertain centers are employed to describe the type-3 membership functions. By employing three-dimensional MFs, type-3 fuzzy logic rules can accurately model a high degree of uncertainty and effectively handle uncertain information. The design of the T3FNN was accomplished by integrating the type-3 fuzzy logic system (T3FLS) with neural networks. The learning of the T3FNN was conducted using the genetic algorithms and gradient descent algorithm. As a result, the type-3 rule base of the system was effectively developed. The learning algorithms are presented and the proposed T3FNN structure is employed for the development of dynamic system control. The simulations of the T3FNN system for the control of the nonlinear and time-varying dynamic systems were carried out. The results obtained from the simulations indicate the suitability of using the T3FNN in controlling dynamic systems.

Solvability of systems of partial fuzzy relational equations revisited – a short note

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved.Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe.This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule.

Hierarchical fuzzy inference based on Bandler-Kohout subproduct

Fuzzy inference with the Bandler-Kohout subproduct (BKS) has been successfully applied in many fields such as fuzzy control, artificial intelligence, image processing, data mining, decision-making, prediction, classification and so on. However, one has to face with the rule explosion in these applications. To deal with this problem, hierarchical fuzzy systems with the compositional rule of inference (CRI) method have been constructed by a series of low-dimensional sub fuzzy systems. And it has been proved that hierarchical fuzzy inference method can efficiently restrain the explosion of fuzzy rules. Therefore, in order to increase the computational efficiency of the fuzzy inference based on the BKS when multi-input-single-output (MISO) fuzzy rules are involved, this paper mainly constructs two hierarchical fuzzy inference methods based on the BKS in which the if-then rules are respectively interpreted by fuzzy implications and ML-implications. Moreover, the validity of the two BKS hierarchical fuzzy inferences is studied with the GMP rules. Finally, two examples are employed to illustrate the computational efficiency of our proposed BKS hierarchical inference methods.

MP and MT properties of fuzzy inference with aggregation function

As the two basic fuzzy inference models, fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) have the important application in artificial intelligence. In order to solve FMP and FMT problems, Zadeh proposed a compositional rule of inference (CRI) method. This paper aims mainly to investigate the validity of A-compositional rule of inference (ACRI) method, as a generalized CRI method based on aggregation functions, from a logical view and an interpolative view, respectively. Specifically, the modus ponens (MP) and modus tollens (MT) properties of ACRI method are discussed in detail. It is shown that the aggregation functions to implement FMP and FMT problems provide more generality than the t-norms, uninorms and overlap functions as well-known the laws of T-conditionality, U-conditionality and O-conditionality, respectively. Moreover, two examples are also given to illustrate our theoretical results. Especially, Example 6.2 shows that the output B′ in FMP (FMT) problem is close to B (DC) with our proposed inference method when the fuzzy input and the antecedent of fuzzy rule are near (the fuzzy input near with the negation of the succedent in fuzzy rule).

Research on the problem of aggregation of multiple rules in fuzzy inference systems

In this paper, we establish the matching relation between implication operator and aggregation operator, which provides a new solution for the design and construction of multi-rule fuzzy inference system. Firstly, according to the definition and monotonicity of implication operator, a new classification method of implication operator is proposed, and then the fuzzy inference process using different implication operators is classified. Then, dynamic maximum aggregation operator and dynamic minimum aggregation operator are proposed. Based on the compositional rule of inference (CRI) method, a matching method and basis of implication operator and aggregation operator for fuzzy inference systems is given and illustrated with examples. Finally, the applicability of the proposed method in this paper is further illustrated by comparing the method with existing methods in the literature and using the nearness degree as an evaluation index.

W-Infer-polation: Approximate reasoning via integrating weighted fuzzy rule inference and interpolation

Fuzzy rule-based inference systems facilitate approximate reasoning for decision-making under circumstances where knowledge is imprecisely characterised. Compositional rule of inference (CRI) and fuzzy rule interpolation (FRI) are two typical types of technique to implement fuzzy systems. Questions of when to apply which of these two fuzzy inference techniques is often addressed by checking whether there exist certain rules that can match given observations. Both techniques have been widely investigated to improve the performance of approximate reasoning, with increasingly more attention being paid to the development of systems where rule antecedent attributes are associated with measures of their relative significance or weights. Unfortunately, they are mostly done in a separate manner within their own fields, making it difficult to achieve accurate reasoning when both techniques are required simultaneously. This paper presents an innovative fuzzy inference mechanism, W-Infer-polation, which organically integrates both enhanced versions of CRI and FRI that are equipped with rule attribute weights. In particular, the individual weights of rule antecedents and consequent are generated using the given unweighted rules only, creating an attribute weighted rule base. From this, W-Infer-polation exploits the learned weights to guide the selection of weighted rules for rule firing in CRI and that of nearest weighted rules for FRI. A systematic comparative experimentation is also presented, demonstrating the efficacy of W-Infer-polation.

PYTHAGOREAN PICTURE FUZZY SETS(PPFS), PART 2- SOME MAIN PICTURE LOGIC OPERATORS ON PPFS AND SOME PICTURE INFERENCE PROCESSES IN PPF SYSTEMS

Pythagorean picture fuzzy set (PPFS) - is a combination of Picture fuzzy set with the Yager’s Pythagorean fuzzy set [12-14]. In the first part of the paper [17] we considered basic notions on PPFS as set operators of PPFS. Unfortunately, we have not papers [18,19, 20] about spherical fuzzy sets with the same definition with some operators and applications to multi attribute group decision making problems. Now in the second part, we will present some main operators in picture fuzzy logic on PPFS: picture negation operator, picture t-norm, picture t-conorm, picture implication operators on PPFS. Last, the compositional rule of inference in PPFS setting should be presented and an numerical example was given.

A sequential method of tuning interval-type fuzzy relations for assessing the security of information systems

Today, an urgent scientific and technical problem is the creation of systems for assessing the security of information systems from threats that can process fuzzy information. These systems allow you to determine what actions are effective to minimize and prevent threats. The fuzzy model is built on the basis of the Zadeh compositional inference rule, in which the information carrier is the matrix of fuzzy "threat-damage" relations connecting the vector of measures of the significance of threats and the vector of measures of significance of damage. When designing such systems based on fuzzy relations, it is necessary to determine a set of its parameters (WІ - a set of parameters that determine a system model based on fuzzy relations of type I and WІІ - a set of parameters that define a system model based on fuzzy relations of type II). To date, for tuning interval-type fuzzy systems, an independent method is used, which assumes the determination of the set WІІ from "zero", without using the results of tuning the set of parameters WІ, which leads to an increase in the setup time. The article proposes a sequential method for adjusting fuzzy relations of interval type, which firstly provides for the determination of a set of parameters of fuzzy relations of type I using a genetic-neural algorithm, and then, on their basis, adjustment of only additional parameters, which makes it possible to reduce the average tuning time of a fuzzy model and assess the effect of uncertainty on the accuracy of the assessment.

Compositional Rule of Inference with a Complex Rule Using Lukasiewicz t-Norm

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.

Hydrotreating unit models based on statistical and fuzzy information

This paper presents the results of mathematical models’ development for hydrotreating reactor, stripping соlumn, absorbers and hydrotreating furnace which are basic units of hydrotreating block in catalytic reforming unit. Since these objects of modeling in reforming unit of Atyrau refinery operate in conditions of insufficiency and fuzziness of the initial information, their mathematical models are developed on the basis of a systematic approach, using available information of different nature (experimental-statistical data, fuzzy information from the experts), with appropriate methods of construction for mathematical models. Mathematical models, describing the dependence of the production output from the hydrotreating reactor, columns and furnace, are designed as a nonlinear regression models based on experimental-statistical data. Whereas models, evaluating the quality indicators of generated products from the hydrotreating reactor and columns, i.e., hydrogenation, hydrogen-containing and hydrocarbon-containing gases are built based on fuzzy information from specialists-experts in the form of fuzzy multiple regression equations. We have plotted the graph for the dependence of hydrogenation products output on the temperature in the hydrotreating reactor. To describe the dependence of the optimum temperature value in hydrotreating process on raw material quality, a linguistic model is designed based on compositional rules of inference and fuzzy information. Membership functions of fuzzy parameters are constructed for linguistic models.

Hydrotreating unit models based on statistical and fuzzy information

This paper presents the results of mathematical models’ development for hydrotreating reactor, stripping соlumn, absorbers and hydrotreating furnace which are basic units of hydrotreating block in catalytic reforming unit. Since these objects of modeling in reforming unit of Atyrau refinery operate in conditions of insufficiency and fuzziness of the initial information, their mathematical models are developed on the basis of a systematic approach, using available information of different nature (experimental-statistical data, fuzzy information from the experts), with appropriate methods of construction for mathematical models. Mathematical models, describing the dependence of the production output from the hydrotreating reactor, columns and furnace, are designed as a nonlinear regression models based on experimental-statistical data. Whereas models, evaluating the quality indicators of generated products from the hydrotreating reactor and columns, i.e., hydrogenation, hydrogen-containing and hydrocarbon-containing gases are built based on fuzzy information from specialists-experts in the form of fuzzy multiple regression equations. We have plotted the graph for the dependence of hydrogenation products output on the temperature in the hydrotreating reactor. To describe the dependence of the optimum temperature value in hydrotreating process on raw material quality, a linguistic model is designed based on compositional rules of inference and fuzzy information. Membership functions of fuzzy parameters are constructed for linguistic models.

Practical aspects of equivalence of Baldwin’s and Zadeh’s fuzzy inference

The article presents a thorough analysis of fuzzy inference introduced by Baldwin and compares this approach to Zaheh’s compositional rule of inference. The comparison is performed in order to analyze the equivalence of the two methods and describe practical aspects of this fact for simple and compound premises, indicating advantages and disadvantages of both approaches. The main aim of the analysis is focus on the computational complexity of the methods. The most important feature of Baldwin’s inference is transfer of the inference process into a truth space, unified for all input variables. Such environment allows to obtain one fuzzy truth value describing a compound premise in a sequence of low dimensional computations. The article proves equality of such approach with the compositional rule of inference. Therefore, this solution is much more computationally efficient in case of compound cases, for which compositional rule of inference is multidimensional.

FUZZY DIAGNOSTİCS OF SOİLS ACCORDİNG TO THE WORLD REFERENCE BASE FOR SOİL RESOURCES

Soil classification remains one of the most controversial topics in the world soil science because of differences in the principles underlying it. As of today, many countries have developed and use their own national classifications. Since scientific classification facilitates successful investigation and proper use of soils, the issue arises of representation of one classification in another. In the paper the possibility of applying fuzzy technologies to soil diagnostics based on the World Reference Base for Soil Resources is considered. A trapezoidal form of membership functions of the parameter set of diagnostic horizons is proposed. The algorithm of fuzzy diagnostics is presented on the basis of the proposed functions. Conclusion. The paper proposes a fuzzi fication mechanism for the set of indicators of diagnostic parameters and describes the algorithm of the system of fuzzy inference about the identification of the values of the corresponding parameters in accordance with the compositional inference rule. Thus, the effectiveness of the use of fuzzy technologies in soil diagnostics by the World Reference Base for Soil Resources is demonstrated.

Fractional Fuzzy Inference System: The New Generation of Fuzzy Inference Systems

This paper presents a new machinery of compositional rule of inference called fractional fuzzy inference system (FFIS). An FFIS is a fuzzy inference system (FIS) in which consequent parts of a rule base consist of a new type of membership functions called fractional membership functions. Fractional membership functions are characterized using fractional indices. There are two types of fractional indices. Each type can be either constant or dynamic. An FFIS intelligently considers not only the truth degrees of information included in membership functions, but also the volume of the information in the process of making a conclusion. In other words, the volume of information extracted from a membership function depends on the truth degree of information. Concretely, the higher the truth degree, the larger the volume of information that is involved in the process of making a conclusion. It is shown that typical FISs, e.g. Mamdani's or Larsen's FISs, are special cases of FFISs. Specifically, as the fractional indices approach one, the FFIS approaches a typical FIS. In addition, using two theorems proved in this paper, it is demonstrated that, independent of the problem in question, a typical FIS never leads to results which are more satisfactory than those obtained by the FFIS corresponding to the typical FIS provided that a particular set of fractional indices is taken into account. Put another way, it seems sound to expect that applying FFIS always leads to more satisfactory results than applying its corresponding FIS. It is also shown that FFIS grants a special dynamic to FIS which can be also customized according to a new concept called reaction trajectories map (RTM). Particularly, the RTM enables decision makers to select an FFIS more suitable for their purpose. Some more concepts such as the left and right orders of an FFIS and the fracture index are also introduced in this paper.

Proof of Theorems in Fuzzy Logic Based on Structural Resolution

An approach to proving theorems with fuzzy and not quite true argumentation is considered. Zadeh’s compositional rule of inference is used as the rule of provably correct reasoning, and its procedural implementation is enabled by a refutation mechanism. As such a mechanism, structural resolution (S-resolution) is proposed that is a generalization of the principle of resolutions to fuzzy statements. S-resolution is based on semantic indices of letters and their similarity. Semantic indices are essential in S-resolution. They contain data used as control information in the process of inference. And similarity implies finding letters to obtain an S-resolvent. Combining Zadeh’s compositional rule of inference and S-resolution allows, on the one hand, to withdraw the problem of correctness of resolvents in fuzzy logic and, on the other hand, to ensure the regularity of the process of a proof in two-valued and fuzzy logics.

Triple-I FMP algorithm for double hierarchical fuzzy system based on manifold learning

As is well known, Zadeh presented the compositional rule of inference to discuss complex inference modes. After that other researchers also investigated this problem. However, the classic fuzzy control systems in the application often encounter the curse of dimensionality. To overcome these difficulties in the classic fuzzy control systems with high-dimensional input variables, the entire triple-I algorithm for double fuzzy control systems and manifold learning of dimensionality reduction will be discussed in this paper. Specifically, triple-I FMP algorithm is presented for double hierarchical fuzzy control system based on Guojun Wang’s implication operator and its reversibility is proved. Using manifold learning, dimension reduction SNE algorithm is given for double-layer hierarchical fuzzy control systems to keep the distribution of peak possibly point, so as to minimize the control stability impact due to reducing dimension. Since any type of multi-layer fuzzy control system is regarded as multiple fusion of double-layer hierarchical fuzzy system, the proposed algorithms and their reversibility are universal.

Fuzzy inference systems preserving Moser–Navara axioms

Satisfiability of three naturally proposed axioms, which are desirable for fuzzy inference system, was discussed by Moser and Navara in [36]. The fuzzy rule base models considered in this study were mainly the Mamdani–Assilian ones and partly also logically motivated implicative fuzzy systems. The investigation leads to the fact that often none of these fuzzy systems do satisfy the given axioms simultaneously. Therefore, Moser and Navara proposed so-called conditionally firing rules and proved that the simultaneous satisfaction of the three axioms is feasible under very mild and practically non-restrictive conditions. Note that the inference mechanism, tailored to the fuzzy rules in the investigation, was the compositional rule of inference (abb. CRI) and similarly, the concept of conditionally firing rules were stemming from the CRI and the Mamdani–Assilian model of a fuzzy rule base. For the implicative interpretation of fuzzy rules, the original axioms proposed by Moser and Navara were later on re-defined by Štěpnička and Mandal with the aim to express the meaning of the original axioms initially proposed for the Mamdani–Assilian rules. Furthermore, conditionally firing implicative rules were proposed based on the so-called Bandler–Kohout subproduct (abb. BKS) inference mechanism and a study similar to the one by Moser and Navara was undertaken for them. The study encompasses the satisfaction of the modified axioms for the use of implicative rules and the BKS. Furthermore, some additional results confirming the applicability of the conditionally firing rules satisfying the Moser–Navara axioms are also provided, including the universal approximation ability. However, the solvability criterions for fuzzy relational equations give a strong preference to the usage of the other combinations, namely, to the combination of CRI with the implicative interpretation of fuzzy rules and to the combination of BKS with the Mamdani–Assilian interpretation of fuzzy rules. Therefore, also these combinations of fuzzy inference mechanisms and interpretations of fuzzy rules should be investigated with respect to the satisfaction of Moser–Navara axioms. The main purpose of this article, which extends the short conference contribution [47] by Štěpnička, is to provide such an investigation jointly with the review on conditionally firing rules and supplementary results.

A short note on fuzzy relational inference systems

This paper is a short note contribution to the topic of fuzzy relational inference systems and the preservation of their desirable properties. It addresses the two main fuzzy relational inferences – compositional rule of inference (CRI) and the Bandler–Kohout subproduct (BK-subproduct) – and their combination with two fundamental fuzzy relational models of fuzzy rule bases, namely, the Mamdani–Assilian and the implicative models.The goal of this short note article is twofold. Firstly, we show that the robustness related to the combination of BK-subproduct and implicative fuzzy rule base model was not proven correctly in [24]. However, we will show that the result itself is still valid and a valid proof will be provided. Secondly, we shortly discuss the preservation of desirable properties of fuzzy inference systems and conclude that neither the above mentioned robustness nor any other computational advantages should automatically lead to a preference of the combinations of CRI with Mamdani–Assilian models or of the BK-subproduct with the implicative models.

A New Look at Type-2 Fuzzy Sets and Type-2 Fuzzy Logic Systems

We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other, whereas in the type-2 fuzzy set framework, the secondary variable depends on the primary variable by definition. It is this dependence between the primary and secondary variables that makes the type-2 fuzzy sets a complex subject, while the conditional fuzzy sets do not have this built-in dependence and, thus, are much easier to analyze. With the fuzzy relation formulation, powerful tools in fuzzy set theory such as Zadeh's compositional rule of inference can be used to obtain the marginal fuzzy sets of the type-2 and conditional fuzzy sets, transforming the type-2 problems back to the conventional type-1 domain. With the help of the marginal fuzzy set concept, we show that a type-2 fuzzy logic system can be designed in the same way as designing a type-1 fuzzy logic system.

Fuzzy Systems Based on Universal Triple I Method and Their Response Functions

The fuzzy systems based on the universal triple I method are investigated, and then their response functions are analyzed. First, the conclusions show that 100 fuzzy systems via the universal triple I method are approximately interpolation functions, which can be used in practical systems, and that 90 ones are approximately fitted functions, which may be usable. Second, as its special cases, the Compositional Rule of Inference (CRI) method and the triple I method are discussed, with the results that 19 fuzzy systems via the CRI method and 2 ones via the triple I method are practicable. Therefore, the universal triple I method has larger effective choosing space, which can obtain more usable fuzzy systems than the others. Lastly, it is found that the first implication and second implication, respectively, embody the function of rule base and reasoning mechanism, further demonstrating the reasonability of the universal triple I method.