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One of main topic of fuzzy logic is approximate reasoning, that refers to theory and methodologies for reasoning with imprecise antecedents to obtain meaning full consequents.
Z-extinction of fuzzy sets porridges a richer knowledge representation and reliable approximate reasoning. In literature on approximate reasoning exist huge number works based on use of Type-1, Type-2 and other extensions of fuzzy sets. Unfortunately up today there is no studes on Z-fuzzy set-based approximate reasoning. In this study for first time we develop theoretical basis and computational methods of Z-fuzzy information-based approximate reasoning. We suggest Z-fuzzy valued implication and investigate its properties. Then using Z-fuzzy implication approximate reasoning is studied. A numerical example and application in business problem are discussed.
SHORT BIO ABOUT THE AUTHOR :
We assume as the point of departure the classic Bellman and Zadeh’s [2] problem of multistage decision making and control under fuzzy constraints on inputs (decisions or controls) and fuzzy goals on the outputs (states, for simplicity), in a more comprehensive form presented in Kacprzyk [10]. As opposed to the basic case of a fixed and specified termination time (planning horizon), for instance „10 years” in the case of economic planning, we assume a fuzzy termination time exemplified, for instance, by „more or less 10 years”, „much more than 5 years”, „much less than 15 years”, etc. Such a fuzzy termination time is quite often what is really perceived by the human stakeholders, for instance, decision makers, in real life even if, for formal reasons, a fixed and specified termination time, here „10 years” is set. The fuzzy termination time, introduced by Kacprzyk [3,4,5] is assumed to be a fuzzy set in the set of control (decision making) stages which is assumed to be finite. We assume, first, that the system under control is deterministic and its dynamics (stage transitions) is given as state transition matrix, both the same at all control stategs and also time varying, that is, different for different control stages.
To extend the expressive power of the formulation, and to acccount for a partial reliability, the fuzzy constraints and fuzzy goals are represented by the Z-numbers defined in finite sets of states and controls, and then, we also mention an extended formulation with the system under control represented by a Z-relation. We assume in the first shot that the fuzzy termination time is defined just as a fuzzy number defined in the set of control stages.
First, we recall the source Z-number based basic approach to the problem of multistage fuzzy control with a fixed and specified termination time by Aliev, Pedrycz, Guirimov, Huseynov and Aliyev (2024) in which the termination time is assumed to be fixed and specified, exemplified by „10 years” as in our example.
Then, we consider the 3 basic approaches to the formulation and solution of the multusgae fuzzy control (decision making) problem with a fuzzy termination time proposed by Kacprzyk [3, 4, 5, 10]:
· dynamic programming,
· branch-and-bound,
· evolutionary computation (mostly genetic algorithms).
We start with the presentation of a Z-number extension of Kacprzyk’s [3, 4, 5] basic fuzzy dynamic programming model of the multistage fuzzy control (decision making) problem with a fuzzy termination time combined with some Z-number related solutions employed for the fuzzy dynamic programming by Aliev, Pedrycz, Guirimov, Huseynov and Aliyev [1]. This new approach proposed involves an additional degree of membership of a particular control stage in the fuzzy termination time.
Next, we show first a Z-numbr based extension of the branch-and-bound approach (cf. Kacprzyk [6]) to the formuation and solution of the problem of multistage fuzzy control (decision making) with a fuzzy termination time which can be a good alternative to dynamic programming for larger problems because it makes it possible to alleviate the infamous course of dimensionality of dynamic
programming, including clearly fuzzy dynamic programming and the Z-number based dynamic programming alike. We show mainly how to implement the partitioning of a problem into smaller problems, and to devise and implememt a bounding function in the case of the Z-numbers.
Then, we show a Z –number based extension of a conceptually simple genetic algorithm approach by Kacprzyk [11].
Finally, we briefly indicate possible further extensions related to the use of the Z-numbers for the formulation and solution of multistage fuzzy control (decision making) with a fuzzy termination time in the case of the stochastic and fuzzy systems under control, and also a very interesting case of a fuzzy termination time specified as a Z-number. We mention a possible Z-number based extension of a neural network approach to the solution of the multistage fuzzy control (decision making) problem, an extension of Francelin, Kacprzyk and Gomide [12, 13], as well as the use of belief qualification introduced by Kacprzyk and [14]. We also mention some applications, notably in inventory control.
References
1. R.A. Aliev, W. Pedrycz, B.G. Guirimov, O.H. Huseynov, R.R. Aliyev, Z-relation-based multistage decision making, Information Sciences, vol. 653, p. 119799, 2024.
2. R.E. Bellman and L.A. Zadeh (1970) Decision-Making in a Fuzzy Environment,” Management Science., vol. 17, no. 4, p. B-141-B-164.
3. J. Kacprzyk (1978) , Control of a stochastic system in a fuzzy environment with a fuzzy termination time, Systems Science, Vol. 4, 291-300, 1978.
4. J. Kacprzyk, Control of a nonfuzzy system in a fuzzy environment with a fuzzy termination time, Systems Science, Vol. 3, 320-334, 1977.
5. J. Kacprzyk, Decision-making in a fuzzy environment with fuzzy termination time, Fuzzy Sets and Systems, Vol. 1, 169-179, 1978.
6. J. Kacprzyk, A branch-and-bound algorithm for the multistage control of a nonfuzzy system in a fuzzy environment, Control and Cybernetics, Vol. 7, 51-64, 1978.
7. J. Kacprzyk and P. Staniewski, Long-term inventory policy-making through fuzzy decision- making models, Fuzzy Sets and Systems, Vol. 8, 117-132, 1982.
8. J. Kacprzyk, A generalization of fuzzy multistage decision making and control via linguistic quantifiers, International Journal of Control, Vol. 38, 1249-1270, 1983.
9. J. Kacprzyk and R.R. Yager, Linguistic quantifiers and belief qualification in fuzzy multicriteria and multistage decision making, Control and Cybernetics, Vol.13, 155-173, 1984.
10. J. Kacprzyk, Multistage Fuzzy Control: A Model-Based Approach to Control and Decision-Making, Wiley, Chichester, 1997.
11. J. Kacprzyk, Multistage Control of a Stochastic System in a Fuzzy Environment Using a Genetic Algorithm, International Journal of Intelligent Systems, vol. 13, 1011-1023, 1998.
12. R.A.F. Francelin, J. Kacprzyk and F.A. Gomide, Neural network based algorithm for dynamic system optimization. Asian Journal of Control, vol. 3, No. 2, 131-142, 2001.
13. R.A. Francelin, F.A.C. Gomide and J. Kacprzyk, A biologically inspired neural network for dynamic programming. International Journal of Neural Systems, vol. 11, no. 6, 561—572, 2001.
SHORT BIO ABOUT THE AUTHOR :
Classical Artificial Intelligence models primarily rely on probabilistic frameworks to represent uncertainty. However, many real-world systems involve not only randomness but also imprecision, linguistic ambiguity, and varying levels of information reliability. Traditional AI methodologies do not adequately capture these structured forms of uncertainty.
This talk introduces Z-AI, a mathematical framework for Artificial Intelligence based on Z-information theory, where each piece of information is modeled as a pair consisting of a restriction component and a reliability component. By embedding credibility directly into the learning and inference process, Z-AI enables intelligent systems to distinguish between uncertain data and unreliable data.
The presentation outlines the theoretical foundations of Z-numbers, their linguistic extensions, and their integration into machine learning and optimization models. Applications in decision support and healthcare systems are discussed to illustrate how reliability-aware uncertainty modeling can enhance robustness, interpretability, and trust in AI systems.
Z-AI represents a step toward mathematically rigorous, reliability-conscious intelligent systems capable of operating in complex, uncertainty-rich environments.
SHORT BIO ABOUT THE AUTHOR :
This keynote, titled “Advancement in Type-3 Fuzzy Systems and Control,” surveys the rapid evolution of type-3 (T3) fuzzy systems and their growing adoption across a broad range of scientific and technological domains. In recent years, T3 fuzzy frameworks have attracted significant attention due to their enhanced ability to model deep uncertainty, complex nonlinearities, and highly variable operating environments, capabilities increasingly demanded in modern intelligent systems. The talk first reviews the principal methodological advancements in T3 fuzzy systems, highlighting key developments in representation, learning, inference, and computational implementation that have enabled practical deployment. It then introduces Type-3 Adaptive Neuro-Fuzzy Inference Systems (T3-ANFIS) as a unifying learning-and-inference architecture that extends classical ANFIS to higher-order uncertainty modeling. Finally, the keynote presents and critically discusses a recently developed intelligent control scheme based on T3-ANFIS, emphasizing design rationale, validation results, and a real-world implementation to demonstrate feasibility, robustness, and performance in practical control applications.
SHORT BIO ABOUT THE AUTHOR :
Traditional fuzzy reasoning is based on compositional inference. In high-order fuzzy systems, this inference mechanism requires both type-reduction and defuzzification operations. In type-3 fuzzy systems, a three-dimensional membership function is first reduced to a two-dimensional representation, then to a one-dimensional form, and finally converted into a crisp output. As the number of variables and rules increases, compositional inference becomes computationally expensive. This research investigates the use of similarity in approximate reasoning. Similarity-based fuzzy reasoning is an alternative inference approach in which conclusions are drawn by measuring how closely a new input resembles previously defined fuzzy rule antecedents, rather than relying solely on strict rule matching. This approach is particularly effective when rule structures are complex (high-order) and inputs do not clearly match existing rules. It compares observed input patterns with stored fuzzy rule antecedents and determines outputs according to their degree of similarity. The proposed mechanism more closely reflects human reasoning, is robust to uncertainty and incomplete knowledge, demonstrates strong generalization capability, and effectively handles nonlinear, high-order interactions. Compared with compositional inference, similarity-based reasoning offers stronger interpolation capability and is more robust to sparse rule bases. In this research, common similarity measures—including set-theoretic and distance-based measures—are investigated to design an inference mechanism for type-3 fuzzy systems, and real-time applications of similarity-based reasoning are examined.
SHORT BIO ABOUT THE AUTHOR :
For some time now, we have been focusing on the intersection between linguistics and fuzzy logic in these studies. We began with “Some Contributions of Fuzzy Logic and Artificial Intelligence Studies to Linguistics” and continued with “Systems, Language, and Fuzzy Logic.” Perhaps one of the most interesting topics within this line of research is Paradox. This topic has been an area of interest in linguistics for a very long time. Our esteemed professor Rafiq Aliyev emphasized the numerical dimension of this topic and, in fact, the context of this paper, particularly in the additions section of his published book Fuzzy Logic and Language-Speech. In my preface, I stated: “At this point, let us simply say that paradox is perhaps the most important issue being addressed in human systems and motivation science, or in other words, contemporary linguistics.” However, paradox is vital in terms of the human mind, mental health, communication, and interaction. Beyond that, it also gives rise to our beliefs and doubts about the universe we live in. Today, paradox is related in some way to the intersection of numerical and linguistic sciences, meta-mathematics, proof theory, logical type theory, and issues of consistency, computability, and decision. As a starting point for these studies, I would like to focus on this very important topic in linguistics. I will focus on its importance in practical life and communication, beyond its place in history, mythology, and writing.
SHORT BIO ABOUT THE AUTHOR :
Main books:
- Economic Sketches, 1996
- Azerbaijan's Ranking (in International Comparative Studies), 2006
- Rehabilitation of Post-Conflict Territories of Azerbaijan, 2010
- Social Orientation of the Economy in Right-Wing and Left-Wing Systems, 2014
- Post-Conflict Territories: Economic Potential and Comparative Advantages, 2023
- Government Regulation of Pricing: Cross-Country Analysis and Outcomes for Azerbaijan, 2024