# Possibility theory: fuzzy sets and fuzzy logic

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### Advances In Infection Surveillance and Clinical Decision Support With Fuzzy Sets and Fuzzy Logic

Moni—a system for monitoring nosocomial infections in intensive care units for adult and neonatal patients—employs knowledge bases that were written with extensive use of fuzzy sets and fuzzy logic, allowing the inherent un-sharpness of clinical terms and the inherent uncertainty of clinical conclusions to be a part of Moni’s output. Thus, linguistic as well as propositional uncertainty became a part of Moni, which can now report retrospectively on HAIs according to traditional crisp HAI surveillance definitions, as well as support clinical bedside work by more complex crisp and fuzzy alerts and reminders. This improved approach can bridge the gap between classical retrospective surveillance of HAIs and ongoing prospective clinical-decision-oriented HAI support.

### Barinder Paul Singh. Keywords Fuzzy Sets, Artificial Intelligence, Fuzzy Logic, Computational Intelligence, Soft Computing.

Keywords— Fuzzy Sets, Artificial Intelligence, Fuzzy Logic, Computational Intelligence, Soft Computing. I. I NTRODUCTION Fuzzy mathematics is the study of fuzzy structures, or structures that involve fuzziness i.e., such mathematical structures that at some points replace the two classical truth values 0 and 1 with a larger structure of degrees. Most of our traditional tools for formal modelling, reasoning, and computing are crisp, deterministic, and precise in character. Crisp means dichotomous, that is, yes-or-no type rather than more-or-less type [1]. In traditional dual logic, for instance, a statement can be true or false—and nothing in between. In set theory, an element can either belong to a set or not; in optimization a solution can be feasible or not. Precision assumes that the parameters of a model represent exactly the real system that has been modelled. This, generally, also implies that the model is unequivocal, that is, that it contains no ambiguities.

### Type-2 Fuzzy Sets Applied to Multivariable Self-Organizing Fuzzy Logic Controllers for Regulating Anesthesia

Accepted Manuscript find that even the worst case of the 42-IT2-SOFLC and 42-GT2-SOFLC can still produce a lower error than the best case of the direct adaptive IT2-FLC in [16]. Nevertheless, it should be noted that the RMSE measured for direct adaptive IT2-FLC also counted the settling stage, which leads to a larger overall error. Hence, further research will provide a benchmark comparison to evaluate the performance of the two controllers more precisely. More recent research described in [17] uses genetic algorithms to design fuzzy PID controllers to regulate BIS. The parameters of the fuzzy PID controllers are optimized offline and a cost function is defined to evaluate the performance of the fuzzy PID controllers. However, this controlling structure is quite different from our study so it is difficult to directly compare the performance of this system with our approach. Future research could compare this control strategy to the SOFLCs if we change our monitoring of DoA to use BIS instead of BP as is the case in this study. In our simulations, we have used a fixed amount of initial bolus which is administered to patients to reach rapid anesthesia. The initial bolus is effective in muscle relaxation in the first 15 minutes and settles down towards the set point. The simulation results show that the bolus can let the system stabilize close to the desired set point which can then be regulated through the adaptive SOFLC. In a real clinical setting it is still however difficult for the anesthetists to decide the amount of the bolus to give. Different patients have different physiological response such as height and weight, and the dosage of initial bolus can therefore also vary in terms of its effectiveness on the patient. Anesthetists generally tend to guess the initial bolus and adjust the amount according to the patient’s physiological response. Future work will simulate different amount of initial bolus and see how the type-2 system can handle the related uncertainties and adjustment of the bolus amount to be initially administered.

### Type-2 fuzzy sets applied to multivariable self-organizing fuzzy logic controllers for regulating anesthesia

find that even the worst case of the 42-IT2-SOFLC and 42-GT2-SOFLC can still produce a lower error than the best case of the direct adaptive IT2-FLC in [16]. Nevertheless, it should be noted that the RMSE measured for direct adaptive IT2-FLC also counted the settling stage, which leads to a larger overall error. Hence, further research will provide a benchmark comparison to evaluate the performance of the two controllers more precisely. More recent research described in [17] uses genetic algorithms to design fuzzy PID controllers to regulate BIS. The parameters of the fuzzy PID controllers are optimized offline and a cost function is defined to evaluate the performance of the fuzzy PID controllers. However, this controlling structure is quite different from our study so it is difficult to directly compare the performance of this system with our approach. Future research could compare this control strategy to the SOFLCs if we change our monitoring of DoA to use BIS instead of BP as is the case in this study. In our simulations, we have used a fixed amount of initial bolus which is administered to patients to reach rapid anesthesia. The initial bolus is effective in muscle relaxation in the first 15 minutes and settles down towards the set point. The simulation results show that the bolus can let the system stabilize close to the desired set point which can then be regulated through the adaptive SOFLC. In a real clinical setting it is still however difficult for the anesthetists to decide the amount of the bolus to give. Different patients have different physiological response such as height and weight, and the dosage of initial bolus can therefore also vary in terms of its effectiveness on the patient. Anesthetists generally tend to guess the initial bolus and adjust the amount according to the patient’s physiological response. Future work will simulate different amount of initial bolus and see how the type-2 system can handle the related uncertainties and adjustment of the bolus amount to be initially administered.

### Kabbalah Logic and Semantic Foundations for a Postmodern Fuzzy Set and Fuzzy Logic Theory

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foun- dations of a postmodern fuzzy set and fuzzy logic theory addressing these issues by deconstructing fuzzy truth values and fuzzy set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal logic of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal logic. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal logic aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom sets. We constructively define a fractal multi-modal Kabbalistic L-fuzzy set as the central concept of the postmodern fuzzy set theory based on Kabbalah logic and semantics.

### Fuzzy in 3-D: Contrasting Complex Fuzzy Sets with Type-2 Fuzzy Sets

A. Fuzzy Inferencing Systems It is via the Fuzzy Inferencing System (FIS) 1 that fuzzy sets are put to use. An FIS is a decision making program which works by applying fuzzy logic operators to common- sense linguistic rules. In this paper we are concerned with the Mamdani FIS, in which a crisp numerical input passes through three stages: fuzzification, inferencing, and finally defuzzification. The output of inferencing is a fuzzy set known as the aggregated set. During the defuzzification stage the aggregated set is converted into a crisp number, which is the output of the FIS. Figure 1 provides a representation of a Mamdani-style type-2 FIS. A Mamdani-style complex FIS differs from the type-2 version only in that defuzzification is a one stage procedure. Defuzzification is beyond the scope of this paper; the focus will be on the inferencing stages of the FIS.

### Application Of Fuzzy Sets Theory In Students’ Performance Appraisal

Step 2: Definition of Fuzzy rules for the fuzzy logic system and inference method Fuzzy sets and fuzzy operators are subjects and verves of fuzzy logic. These if-then rule statements were used to formulate the conditional statements that comprise fuzzy logic. CATs and EXAM were taken as parts of antecedent. All parts of antecedent were calculated simultaneously and resolved to a single number using the logical operator AND. The performance of the student was considered as the consequent affected by the antecedent. Next is sample of 10 rules taken from 25 if-then rules generated for the system :

### A. WHY FUZZY LOGIC?

B. FUZZY INFERENCE SYSTEM Steps in fuzzy inference system: 1: STEP 1: FUZZIFICATION The first step in a fuzzy inference system is the fuzzification of crisp inputs. It transforms the exact logic problem into a fuzzy logic problem. Unlike crisp logic, fuzzy logic deals with linguistic variables instead of numerical variables. The process of converting numerical variables of the problem into grades of membership for linguistic terms of fuzzy sets is called fuzzification. Thus it is a mapping from a certain input space to fuzzy sets in certain input universes of discourse.

### A Theory of Lattice Valued Fuzzy Sets and Fuzzy Maps Between Different Lattice Valued Fuzzy Sets – Revisited

India Abstract: F-Set Theory is a natural generalization of Goguen's L-Fuzzy Set Theory which itself is a generalization of Zadeh's, both Fuzzy and Interval Valued Fuzzy Set Theories. It naturally and neatly extends several of the crisp (Sub)Set-Map-Properties to: L-valued f-(sub) sets, f-maps between L-valued f-sets and M-valued f-sets, where the complete lattice L-may possibly different from the complete lattice M, M-valued f- image of an L-valued f-subset of the domain L-valued f-set and L-valued f-inverse image of an M-valued f-subset of the co-domain M-valued f- set. However, for several of the results in this theory, the complete homomorphisms are assumed to be one or a combination of: 0-preserving, 0- reflecting, 1-preserving and 1-reflecting. Further, some of the results use the infinite meet distributivity of the underlying complete lattice of the domain and/or range f-set.

### An application of fuzzy logic to prospect theory

error) than FDM for out particular dataset, but there advantages of FDM are clear. It has less parameters and a more descriptive model if we use fuzzy concepts. Also, consider the following: ”Usually, fuzzy controllers are implemented as software running on standard microprocessors. A few special-purpose microprocessors have been built that do fuzzy operations directly in hardware, but even these use digital binary signals at the lowest hardware level. There are some research prototypes of computer chips that use analog signals at the lowest level, but these chips simulate the operation of neurons rather than fuzzy logic.” So, if fuzzy com- puters become more widely available in the future it is conceivable that this kind of computations will become more efficient both for programming and running. 1

### ABSTRACT : Fuzzy logic controller (FLC) is the most widely used applications of fuzzy set theory. A Fuzzy Logic

PG Scholar, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 1 Professor, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 2,3 ABSTRACT: Fuzzy logic controller (FLC) is the most widely used applications of fuzzy set theory. A Fuzzy Logic Controlled (FLC) buck-boost converter for solar energy-battery systems is analyzed. Solar Photovoltaic system has been increasingly playing a significant role in new energy systems. PV system has some disadvantages including low conversion efficiency and inconsistent values of output voltage due to irregular solar power which is caused by weather changes and shading effects. To address these disadvantages, DC-DC converters have been used to control PV output voltage and output power. Fuzzy logic controllers are non linear controllers which do not require exact information of mathematical model. Hybridization of the two controllers can be done to use the positive sides of both controllers to obtain better performance. The comparison of result obtained from fuzzy logic controller, fuzzy - PI hybrid controller, fuzzy - PID hybrid controller for DC-DC converter shows the benefit of the hybrid algorithm in terms of transient response. General design of a fuzzy logic controller based on MATLAB/Simulink is performed. The control system has been developed, analyzed, and validated by simulation study.

### A Theory of Lattice Interval Valued Fuzzy Sets and Fuzzy Maps Between Different Lattice Interval Valued Fuzzy Sets

Now coming back to the developments in this side of this paper, Goguen further generalized the two types of fuzzy subsets of Zadeh, namely the fuzzy subset and the interval valued f-subset, to those that take the truth values in a complete lattice. However, even though Goguen unified both of them mathematically, one must observe here that, as mentioned earlier, when it comes to practical applications, the fuzzy subsets and the interval valued f-subsets are quite different because fuzzy sets require a specific real number between 0 and 1 to be associated with each of its elements while interval valued f-sets require a reasonable interval to be associated with each of its elements, offering a representation of even more uncertainty in belonging of certain elements to a set than the fuzzy sets themselves.

### R fuzzy sets and grey system theory

Nanjing, China Abstract—This paper investigates the use of grey theory to en- hance the concept of an R-fuzzy set, with regards to the precision of the encapsulating set of returned significance values. The use of lower and upper approximations from rough set theory, allow for an R-fuzzy approach to encapsulate uncertain fuzzy membership values; both collectively generic and individually specific. The authors have previously created a significance measure, which when combined with an R-fuzzy set provides one with a refined approach for expressing complex uncertainty. This pairing of an R-fuzzy set and the significance measure, replicates in part, the high detail of uncertainty representation from a type-2 fuzzy approach, with the relative ease and objectiveness of a type-1 fuzzy approach. As a result, this new research method allows for a practical means for domains where ideally a generalised type-2 fuzzy set is more favourable, but ultimately unfeasible due to the subjectiveness of type-2 fuzzy membership values.

### Fuzzy Logic

This approach to set theory was not applied to control systems until the 70's due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that people do not require precise, numerical information input, and yet they are capable of highly adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement. Unfortunately, U.S. manufacturers have not been so quick to embrace this technology while the Europeans and Japanese have been aggressively building real products around it.

### Possibility of Applying Fuzzy Logic in Determining Phase Order for Signalized Intersections

SUMMARY POSSIBILITY OF APPLYING FUZZY LOGIC IN DETERMINING PHASE ORDER FOR SIGNALIZED INTERSECTIONS In this bachelor's thesis an adaptive control system for signalized intersections has been presented. The system uses fuzzy logic to fully optimize traffic flow, minimize wait times, queue length and controlling arrival and exit flow in order to prevent traffic congestion throughout the observed signalized intersection. Given that road traffic's use has been largely increased over years, it is necessary to optimize traffic flow in an already existing traffic infrastructure using technologies within intelligent transport systems. The designed system makes an exclusive use of changing the signal phase order to optimize functionality for all traffic demand scenarios covered in this work. Fuzzy logic enables the use of predefined rules which serve as a base for predicting possible cases of traffic flow, thus giving a better insight over further optimizing phase order in the intersection.

### Soft rough fuzzy sets and soft fuzzy rough sets

© 2011 Elsevier Ltd. All rights reserved. 1. Introduction To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems. While probability theory, fuzzy set theory [1], rough set theory [2,3], and other mathematical tools are well known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [4,5]. In 1999, Molodtsov [4] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set theory is free from the difficulties affecting existing methods.

### A Generalized Interval Valued Intuitionistic Fuzzy Sets theory

In this paper, a novel generalized interval-valued intuitionistic fuzzy sets (GIVIFS) is presented, which is the generalization of conventional intuitionistic fuzzy sets (IFS) and interval-valued intuitionistic fuzzy sets (IVIFS). By analyzing the degree of hesitancy, this paper introduces generalized interval-valued intuitionistic fuzzy sets with parameters (GIVIFSP). And then, it is proved that GIVIFS is a closed algebraic system as IFS and IVIFS.

### Fundamentals. Chapter Fuzzy programming Fuzzy sets

Chapter 2 Fundamentals This chapter is devoted to reviewing optimization concepts and the related compu- tational methods that will be used in the remaining chapters. In particular, we deal with three optimization concepts incorporating fuzziness and ambiguity of human judgments, uncertainty of events characterizing decision making problems, and mul- tiplicity of evaluation criteria. Fuzzy programming which is developed in order to take into account fuzziness and ambiguity of human judgments, and this method is ﬁrst presented together with the basic concepts in fuzzy set theory. Fundamentals of stochastic programming are also provided for decision problems under probabilistic uncertainty. Speciﬁcally, two-stage programming and chance constraint program- ming are covered. To meet expectations of diversiﬁed evaluations, multiobjective programming has been investigated. After presenting interactive methods for multi- objective linear programming problems, we give some techniques to deal with not only multiple objectives but also fuzzy goals for the objectives. In an organization with a hierarchical structure, we often ﬁnd decision making with two or more deci- sion makers (DMs) attempting to optimize their objective functions. To model such a decision making problem mathematically, a two-level programming problem is formulated, and cooperative and noncooperative solution methods are presented for two-level programming. Genetic algorithms are considered to be one of the most practical and proven meta heuristics for difﬁcult classes of optimization problems, and ﬁnally basic concepts of genetic algorithms are provided.