Fuzzy reasoning based on the extension principle*

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.     

Robust reasoning with rules that have exceptions: From second-order probability to argumentation via upper envelopes of probability and possibility plus directed graphs

Rules having rare exceptions may be interpreted as assertions of high conditional probability. In other words, a rule If X then Y may be interpreted as meaning that Pr(Y∣X) 1. A general approach to reasoning with such rules, based on second-order probability, is advocated. Within this general approach, different reasoning methods are needed, with the selection of a specific method being dependent upon what knowledge is available about the relative sizes, across rules, of upper bounds on each rule's exception probabilities Pr(?Y∣X). A method of reasoning, entailment with universal near surety, is formulated for the case when no information is available concerning the relative sizes of upper bounds on exception probabilities. Any conclusion attained under these conditions is robust in the sense that it will still be attained if information about the relative sizes of exception probability bounds becomes available. It is shown that reasoning via entailment with universal near surety is equivalent to reasoning in a particular type of argumentation system having the property that when two subsets of the rule base conflict with each other, the effectively more specific subset overrides the other. As stepping stones toward attaining this argumentation result, theorems are proved characterizing entailment with universal near surety in terms of upper envelopes of probability measures, upper envelopes of possibility measures, and directed graphs. In addition, various attributes of entailment with universal near surety, including property inheritance, are examined.     

Variable precision logic

Variable precision logic is concerned with problems of reasoning with incomplete information and resource constraints. It offers mechanisms for handling trade-offs between the precision of inferences and the computational efficiency of deriving them. Two aspects of precision are the specificity of conclusions and the certainty of belief in them; we address primarily certainty and employ censored production rules as an underlying representational and computational mechanism. These censored production rules are created by augmenting ordinary production rules with an exception condition and are written in the form “if A then B unless C”, where C is the exception condition.From a control viewpoint, censored production rules are intended for situations in which the implication A ⇒ B holds frequently and the assertion C holds rarely. Systems using censored production rules are free to ignore the exception conditions when resources are tight. Given more time, the exception conditions are examined, lending credibility to high-speed answers or changing them. Such logical systems, therefore, exhibit variable certainty of conclusions, reflecting variable investment of computational resources in conducting reasoning. From a logical viewpoint, the unless operator between B and C acts as the exclusive-or operator. From an expository viewpoint, the “if A then B” part of censored production rule expresses important information (e.g., a causal relationship), while the “unless C” part acts only as a switch that changes the polarity of B to ¬B when C holds.Expositive properties are captured quantitatively by augmenting censored rules with two parameters that indicate the certainty of the implication “if A then B”. Parameter δ is the certainty when the truth value of C is unknown, and γ is the certainty when C is known to be false.     

Solutions of fuzzy relation equations based on continuous t-norms

This study is concerned with fuzzy relation equations with continuous t-norms in the form ATR = B, where A and B are the fuzzy subsets of X and Y, respectively; R ⊂ X × Y is a fuzzy relation, and T is a continuous t-norm. The problem is how to determine A from R and B. First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when X and Y are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.     

Fuzzy conditional inference under max-⊙ composition

This paper shows that the majority of fuzzy inference methods for a fuzzy conditional proposition “If x is A then y is B,” with A and B fuzzy concepts, can infer very reasonable consequences which fit our intuition with respect to several criteria such as modus ponens and modus tollens, if a new composition called “max-⊙ composition” is used in the compositional rule of inference, though reasonable consequences cannot always be obtained when using the max-min composition, which is used usually in the compositional rule of inference. Furthermore, it is shown that a syllogism holds for the majority of the methods under the max-⊙ composition, though they do not always satisfy the syllogism under the max-min composition.     

How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.     

On the possibility of using complex values in fuzzy logic for representing inconsistencies

In science and engineering, there are “paradoxical” cases in which we have some arguments in favor of some statement A (so the degree to which A is known to be true is positive (nonzero)), and we have some arguments in favor of its negation ¬A, and we do not have enough information to tell which of these two statements is correct. Traditional fuzzy logic, in which “truth values” are described by numbers from the interval [0, 1], easily describes such “paradoxical” situations: the degree a to which the statement A is true and the degree 1−a to which its negation ¬A is true can both be positive. In this case, if we use traditional fuzzy &-operations (min or product), the “truth value” a&(1−a) of the statement A&¬A is positive, indicating that there is some degree of inconsistency in the initial beliefs. When we try to use fuzzy logic to formalize expert reasoning in the humanities, we encounter the problem that is humanities, in addition to the above-described paradoxical situations caused by the incompleteness of our knowledge, there are also true paradoxes, i.e., statements that are perceived as true and false at the same time. For such statements, A&¬A=“true.” The corresponding equality a&(1−a)=1 is impossible in traditional fuzzy logic (where a&(1−a) is always≤0.5), so, to formalize such true paradoxes, we must extend the set of truth values from the interval [0, 1]. In this paper we show that such an extension can be achieved if we allow truth values to be complex numbers. © 1998 John Wiley & Sons, Inc.     

Solutions to the matrix equation

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BY, with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B), generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.     

The kernel strategy and its use for the study of combinatory logic

Barendregt defines combinatory logic as an equational system satisfying the combinatorsS andK with ((Sx)y)z=(xz)(yz) and (Kx)y=x; the set consisting ofS andK provides abasis for all of combinatory logic. Rather than studying all of the logic, logicians often focus onfragments of the logic, subsets whose basis is obtained by replacingS orK or both by other combinators. In this article, we present a powerful new strategy, called thekernel strategy, for studying fragments in the context of questions concerned with fixed point properties. Interest in such properties rests in part with their relation to normal forms and paradoxes. We show how the kernel strategy was used to answer a number of open questions, offering abundant evidence that the availability of the kernel strategy marks a singular advance for automated reasoning. In all of our experiments with this strategy applied by an automated reasoning program, the rate of success has been impressively high and the CPU time to obtain the desired information startlingly small. For each fragment we study, we use the kernel strategy to attempt to determine whether the strong or the weak fixed point property holds. WhereA is a given fragment with basisB, the strong fixed point property holds forA if and only if there exists a combinatory such that, for all combinatorsx,yx=x(yx), wherey is expressed purely in terms of elements ofB. The weak fixed point property holds forA if and only if for all combinatorsx there exists a combinatory such thaty=xy, wherey is expressed purely in terms of the elements ofB and the combinatorx. Because the use of the kernel strategy is so effective in addressing questions focusing on either fixed point property, its formulation marks an important advance for combinatory logic. Perhaps of especial interest to logicians is an infinite class of infinite sets of tightly coupled fixed point combinators (presented here), whose unexpected discovery resulted directly from the application of the strategy by an automated reasoning program. We also offer various open questions for possible research and focus on an automated reasoning program and input files that may prove useful for such research.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Linguistic hedges and the generalized modus ponens

In this paper a modification of the generalized modus ponens is presented, namely, rule: if X is bB then Y is cC; fact: X is aB, conclusion: Y is dC where a, b, c, e, and d are linguistic hedges, and B, C are fuzzy sets. The procedure that allows one to evaluate the modifier d is very simple and gives results given in Refs. 15, 18, 26, and 27. Our approach is algebraic-based and realizes Zadeh's calculus on words by means of Chang's MV algebra. ©1999 John Wiley & Sons, Inc.     

New Finite Algorithm for Solving the Generalized Nonhomogeneous Yakubovich‐Transpose Matrix Equation

In this paper, the development of the conjugate direction (CD) method is constructed to solve the generalized nonhomogeneous Yakubovich‐transpose matrix equation AXB + CX^TD + EYF = R. We prove that the constructed method can obtain the (least Frobenius norm) solution pair (X,Y) of the generalized nonhomogeneous Yakubovich‐transpose matrix equation for any (special) initial matrix pair within a finite number of iterations in the absence of round‐off errors. Finally, two numerical examples show that the constructed method is more efficient than other similar iterative methods in practical computation.     

A Note on Z-numbers

Decisions are based on information. To be useful, information must be reliable. Basically, the concept of a Z-number relates to the issue of reliability of information. A Z-number, Z, has two components, Z = (A, B). The first component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of reliability (certainty) of the first component. Typically, A and B are described in a natural language. Example: (about 45 min, very sure). An important issue relates to computation with Z-numbers. Examples: What is the sum of (about 45 min, very sure) and (about 30 min, sure)? What is the square root of (approximately 100, likely)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW). In this note, the concept of a Z-number is introduced and methods of computation with Z-numbers are outlined. The concept of a Z-number has a potential for many applications, especially in the realms of economics, decision analysis, risk assessment, prediction, anticipation and rule-based characterization of imprecise functions and relations.     

Actions,finite state sets and applications to trees

Let M be a monoid acting on a set X; for any x?X and A?X we put x ^?1 A = {m?M/xm?A}. Call A?X finite state if card {x ^?1 A/x?X} <∞.

The finite state subsets of T _Σvia the action T _Σ × P _Σ→T _Σare the recognizable forests (P _Σis the monoid of all Σ-trees with just one leaf labeled by a variable x).

Next we prove that the recognizability of forests is equivalent to the finiteness of a certain “syntactic” monoid A Mezei's-like theorem for trees is established: the finite state subsets of T _Σ × T_г are exactly the finite unions of sets of sets of the form B × C B?Rec(T _Σ) and C?Rec(T _г) Another characterization of such relations is given using bimorphisms.     

Averaging linguistic truth values in fuzzy approximate reasoning

The properties of a new rule for fuzzy conditional inference are presented and discussed. The rule is based on the extended mean operator defined on fuzzy numbers. The related propositions have the form “X is A is τ,” where τ is an element of the term set of the linguistic variable truth. The results obtained via the rule match with Fukami's and with the critical analysis carried out by Mizumoto and Zimmermann about the generalized modus ponens. © 1998 John Wiley & Sons, Inc.     

Probability and possibility models of matrix games of two subjects

Probability models and their possibility counterparts of one-matrix and bimatrix games of two subjects (A and B) were defined and analyzed. For the one-matrix game possibility model, a theorem was proven saying that maximin and minimax fuzzy strategies exist and that possibilities of A winning or losing (B) in relation to these strategies are equal. The concepts of fuzzy and randomized game strategies were defined and analyzed. The problem of statistic modeling of A and B fuzzy strategies was resolved. For possible models of bimatrix games, the existence of equilibrium points was examined. For the problem of maximization of the winning A and B possibility, it was proven that equilibrium points exist. For the problem of minimizing the possibility of losses, it was shown that if equilibrium points exist, some of them are related to clear strategies, A and B.     

Vibration perception and excitatory direction for haptic devices

Vibration feedback is one of the most popular ways to communicate between human and haptic interfaces nowadays. In order to deliver a wider variety of information accurately and efficiently, significant design factors of the vibration need to be investigated and applied to haptic devices. In this study, the excitatory direction was examined as a design factor of the vibration in terms of sensitivity and emotion. We conducted two experiments. In the first experiment, the sensitivities of three excitatory directions—X (lateral), Y (fore-and-aft) and Z (vertical) axes were estimated by the absolute thresholds of the vibration perception with two frequency levels (150 and 280 Hz). Based on ten participants’ estimated absolute thresholds, we conclude that the vibration with X axis is less sensitive than Z axis at the frequency of 150 Hz, while the vibration with Y axis is less sensitive than Z axis at the frequency of 280 Hz. In the second experiment, the agreeability of 29 emotional expressions to the vibrations was measured by a 7-point scale with a total of 12 conditions (2 frequencies × 2 amplitudes (i.e., 50 × 10⁻³ and 500 × 10⁻³ g) × 3 excitatory directions). Based on 20 participants’ responses, it is concluded that at the frequency of 150 Hz and the amplitude of 50 × 10⁻³ g, the vibration is perceived as ‘light’, and as even ‘lighter’ if the vibration is with Y axis rather than with Z axis. Likewise, at the frequency of 150 Hz and the amplitude of 500 × 10⁻³ g, the vibration is perceived as ‘repulsive’, and as even ‘more repulsive’ if the vibration is with Y or Z axis rather than with X axis. Therefore, three excitatory directions can be selectively utilized to design the distinguishable vibration by its sensitivity and emotion.     

A uniform approach to inductive posets and inductive closure

The definition scheme, “A poset P is Z-inductive if it has a subposet B of Z-compact lements such that for every element of p of P there is a Z-set S in B such that $\text{p = ?S}$, becomes meaningful when we replace the symbol of Z by such adjectives as “sirected”, “chain”, “pairwise compatible”, “singleton”, etc. Furthermore, several theorems have been proved that seem to differ only in their instantiations of Z. A simialr phenomena occurs when we comsider concepts such as Z-completeness of Z-comtinuity. This suggests that in all these different cases we are really talking about Z same thing. In this paper we show that this is indeed the case by abstracting out the essential common properties of the different instantiations of Z and proving common theorems within the resulting abstract framework.     

Aspects of neural modeling of multidimensional chaotic attractors

This article discusses various aspects of neural modeling of multidimensional chaotic attractors. The Lorenz and Rosler attractors are considered as representative cases and are thoroughly examined. These two dynamical systems are expressed, within acceptable accuracy limits, by the corresponding systems of difference equations. Initially, the complete neural models of the attractors are examined. In this case, the neural networks are supplied with the values X_n , Y_n , Z_n of the systems to predict all the next components (X_{n +1}, Y_{n +1}, and Z_{n +1}) of the attractors. In the second case, named ‘component simulation’, the neural models are trained to predict only one of the values X_n , Y_n , Z_n , when they are fed with the complete input vector as in the first case. In the third case, the proposed neural networks are trained to predict only one component (X_{n +1}, Y_{n +1}, or Z_{n +1}) of the attractors, given a number of past values of the same component. Finally, the ability of the networks to predict the Y and Z components of an X time series of the dynamical systems is examined. Since the response of some networks is not satisfactory, the distribution of absolute error is considered in order to form a realistic picture of the networks’ performance.     

Var: A program to calculate the semivariance function in three orthogonal planes

The geostatistical concept of semivariance is used in VAR to calculate and plot variograms in three orthogonal planes. Distances between all points of pairs are calculated and the vector-angles between them are resolved into components in the X - Y, Y - Z, and Z - X planes respectively. Semivariance values for each distance calculation are assigned to the appropriate angular sector in each plane and variograms for all sectors are plotted in all three orthogonal planes on a CDC CYBER 73 computer.     

An improved algorithm on the content of realizable fuzzy matrices

An n×nn{\times}n fuzzy matrix A is called realizable if there exists an n×tn{\times}t fuzzy matrix B such that A=B\odot B^T,A=B\odot B^{T}, where \odot\odot is the max–min composition. Let r(A)=min{p:A=B\odot B^T, B ? L^n×p}.r(A)={min}\{p:A=B\odot B^{T}, B\in L^{n\times p}\}. Then r(A)r(A) is called the content of A. Since 1982, how to calculate r(A) for a given n×nn{\times}n realizable fuzzy matrix A was a focus problem, many researchers have made a lot of research work. X. P. Wang in 1999 gave an algorithm to find the fuzzy matrix B and calculate r(A) within [r(A)]ⁿ²[r(A)]^{n^{2}} steps. Therefore, to find a simpler algorithm is a problem what we have to consider. This paper makes use of the symmetry of the realizable fuzzy matrix A to simplify the algorithm of content r(A)r(A) based on the work of Wang (Chin Ann Math A 6: 701–706, 1999).