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Beyond [0,1] to Intervals and Further: Do We Need All New Fuzzy Values?
 Proceedings of The Eighth International Fuzzy Systems Association World Congress IFSA'99
, 1999
"... In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0; 1] and to consider more general fuzzy values: e.g., intervals, or real numbers outside the interval [0; 1]. When we increase the set of possible fuzzy values, we thus increase the number of bits necess ..."
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In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0; 1] and to consider more general fuzzy values: e.g., intervals, or real numbers outside the interval [0; 1]. When we increase the set of possible fuzzy values, we thus increase the number of bits necessary to store each degree, and therefore, increase the computation time which is needed to process these degrees. Since in many applications, it is crucial to get the result on time, it is therefore desirable to make the smallest possible increase. In this paper, we describe such smallest possible increases. I. Introduction In classical (twovalued) logic, there are only two truth values: "true" (which, in the computer, is usually denoted by 1) and "false" (which is usually denoted by 0). To represent the uncertainty of human reasoning, L. Zadeh proposed, in his fuzzy logic, to use additional truth values, including truth values which are intermediate between "true" and "false", i.e., inter...
Possible New Directions in Mathematical Foundations of Fuzzy Technology: A Contribution to the Mathematics of Fuzzy Theory
 Proceedings of the VietnamJapan Bilateral Symposium on Fuzzy Systems and Applications VJFUZZY'98, HaLong Bay, Vietnam, 30th September2nd
, 1998
"... this paper, we describe new possible applicationoriented directions towards formalizing these new ideas: ..."
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Cited by 7 (7 self)
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this paper, we describe new possible applicationoriented directions towards formalizing these new ideas:
Why Unary and Binary Operations in Logic: General Result Motivated by IntervalValued Logics
 Proceedings of the Joint 9th World Congress of the International Fuzzy Systems Association and 20th International Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS 2001
, 2001
"... Traditionally, in logic, only unary and binary operations are used as basic ones  e.g., "not", "and", "or"  while the only ternary (and higher order) operations are the operations which come from a combination of unary and binary ones. For the classical logic, with ..."
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Traditionally, in logic, only unary and binary operations are used as basic ones  e.g., "not", "and", "or"  while the only ternary (and higher order) operations are the operations which come from a combination of unary and binary ones. For the classical logic, with the binary set of truth values f0; 1g, the possibility to express an arbitrary operation in terms of unary and binary ones is well known: it follows, e.g., from the well known possibility to express an arbitrary operation in DNF form. A similar representation result for [0; 1]based logic was proven in our previous paper. In this paper, we expand this result to finite logics (more general than classical logic) and to multiD analogues of the fuzzy logic  both motivated by intervalvalued fuzzy logics. 1.
Approximate Nature of Traditional Fuzzy Methodology Naturally Leads to ComplexValued Fuzzy Degrees
"... Abstract—In the traditional fuzzy logic, the experts ’ degrees of confidence in their statements is described by numbers from the interval [0; 1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical ..."
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Abstract—In the traditional fuzzy logic, the experts ’ degrees of confidence in their statements is described by numbers from the interval [0; 1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical degrees – e.g., complexvalued degrees. While these complexvalued degrees are helpful in solving practical problems, their intuitive meaning is not clear. In this paper, we provide a possible explanation for the success of complexvalued degrees which makes their use more intuitively understandable – namely, we show that these degrees naturally appear due to the approximate nature of the traditional fuzzy methodology. I. FORMULATION OF THE PROBLEM Fact: complexvalued fuzzy degrees are sometimes useful in practice. One of the main motivations for fuzzy logic
Why ComplexValued Fuzzy? Why Complex Values in General? A Computational Explanation
"... Abstract—In the traditional fuzzy logic, as truth values, we take all real numbers from the interval [0, 1]. In some situations, this set is not fully adequate for describing expert uncertainty, so a more general set is needed. From the mathematical viewpoint, a natural extension of real numbers is ..."
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Abstract—In the traditional fuzzy logic, as truth values, we take all real numbers from the interval [0, 1]. In some situations, this set is not fully adequate for describing expert uncertainty, so a more general set is needed. From the mathematical viewpoint, a natural extension of real numbers is the set of complex numbers. Complexvalued fuzzy sets have indeed been successfully used in applications of fuzzy techniques. This practical success leaves us with a puzzling question: why complexvalued degree of belief, degrees which do not seem to have a direct intuitive meaning, have been so successful? In this paper, we use latest results from theory of computation to explain this puzzle. Namely, we show that the possibility to extend to complex numbers is a necessary condition for fuzzyrelated computations to be feasible. This computational result also explains why complex numbers are so efficiently used beyond fuzzy, in physics, in control, etc.
Analysis and algorithms of bifuzzy systems
 Proceedings of the Royal Society of London Series A, Vol.460
, 2004
"... A fuzzy variable is a function from a possibility space to the set of real numbers, while a bifuzzy variable is a function from a possibility space to the set of fuzzy variables. In this paper, a concept of chance distribution is originally presented for bifuzzy variable, and the linearity of expect ..."
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A fuzzy variable is a function from a possibility space to the set of real numbers, while a bifuzzy variable is a function from a possibility space to the set of fuzzy variables. In this paper, a concept of chance distribution is originally presented for bifuzzy variable, and the linearity of expected value operator of bifuzzy variable is proved. Furthermore, bifuzzy simulations are suggested and illustrated by some numerical experiments.
PostStructuralist Game Theory
"... The manuscript offers a conceptualization and formalization of the notion of choice undecidability of an individual actor within a poststructural approach to social theory. In contrast to conventional decision theories which posit that the actors are able to delineate the set of alternative choices ..."
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The manuscript offers a conceptualization and formalization of the notion of choice undecidability of an individual actor within a poststructural approach to social theory. In contrast to conventional decision theories which posit that the actors are able to delineate the set of alternative choices, this paper explores how choice undecidability shapes the decisionmaking process in strategic situations. Drawing a conceptual analogy and borrowing terminology from quantum mechanics, I use the technologies of quantum superposition and entanglement to suggest a framework for formally studying choice undecidability in a poststructuralist game theory. Contact:
Extending TNorms Beyond [0,1]: Relevant Results of Semigroup Theory
, 1999
"... Originally, fuzzy logic was proposed to describe human reasoning. Lately, it turned out that fuzzy logic is also a convenient approximation tool, and that moreover, sometimes a better approximation can be obtained if we use real values outside the interval [0, 1]; it is therefore necessary to descri ..."
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Originally, fuzzy logic was proposed to describe human reasoning. Lately, it turned out that fuzzy logic is also a convenient approximation tool, and that moreover, sometimes a better approximation can be obtained if we use real values outside the interval [0, 1]; it is therefore necessary to describe possible extension of tnorms and tconorms to such new values. It is reasonable to require that this extension be associative, i.e., that the set of truth value with the corresponding operation form a semigroup. Semigroups have been extensively studied in mathematics. In this short paper, we describe several results from semigroup theory which we believe to be relevant for the proposed extension of tnorms and tconorms.
1. Fuzzy Techniques as an EasiertoCompute Continuous Approximation for DifficulttoCompute Discrete Objects and Processes
"... Abstract. While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes ..."
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Abstract. While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.
From 1D to 2D Fuzzy: A Proof that IntervalValued and ComplexValued Are the Only Distributive Options
"... Abstract—While the usual 1D fuzzy logic has many successful applications, in some practical cases, it is desirable to come up with a more subtle way of representing expert uncertainty. A natural idea is to add additional information, i.e., to go from 1D to 2D (and multiD) fuzzy logic. At present ..."
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Abstract—While the usual 1D fuzzy logic has many successful applications, in some practical cases, it is desirable to come up with a more subtle way of representing expert uncertainty. A natural idea is to add additional information, i.e., to go from 1D to 2D (and multiD) fuzzy logic. At present, there are two main approaches to 2D fuzzy logic: intervalvalued and complexvalued. At first glance, it may seem that many other options are potentially possible. We show, however, that, under certain reasonable conditions, intervalvalued and complexvalued are the only two possible options. I.