A general purpose fuzzy controller for monotone functions

In this work, a general purpose fuzzy controller is proposed to handle the class of monotonic functions. A set of rules on the selection of fuzzy subsets and decision tables based on the mean-of-inversion (MOI) defuzzification method for guaranteed convergence and accuracy is given and proved. Unlike the mean-of-maximum (MOM) and the center-of-area (COA) methods, the MOI method defuzzifies each fired rule separately instead of superimposing fired rules before defuzzification     

An explicit construction of short monotone formulae for the monotone symmetric functions

We construct formulae that assume the value 1 when and only when at least k of their n variables assume the value 1, using only conjunction and disconjunction, and having (for any fixed k) only

$\text{O}\text{(n}\text{log}\text{n)}\text{k}\text{2}{}^{\mathrm{<mtext>log</mtext><mtext>1n</mtext>}}$

occurences of variables.     

Monotone circuits for monotone weighted threshold functions

Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. In this work we by-pass this addition step and construct a polynomial size logarithmic depth unbounded fan-in monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC¹. (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.)Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) information-theoretic secret sharing schemes where the size of each share is quasi-polynomial in the number of users, and (2) computational secret sharing schemes where the size of each share is polynomial in the number of users.     

A mode-based meta-model for the frequency response functions of uncertain structural systems

Frequency response functions (FRFs) play an important role in the assessment of the structural response of linear systems subjected to dynamic forces. In this work a functional relation (meta-model) between the uncertain structural parameters and the model properties is presented. The meta-model provides a computationally fast solution to approximate the eigenfrequencies and mode shapes needed thereafter for evaluating the FRFs. The meta-model circumvents the repeated solution of the eigenvalue problem for each set of uncertain structural input parameters. The provided relations can be helpful in the design stage to control the dynamic response within certain frequency bands. Moreover, the meta-model can be used for optimization and reliability assessments based on Monte Carlo sampling procedures. Numerical examples show the application of the method focusing mainly on the variability of the FRFs. The efficiency and accuracy of the meta-model to compute approximate FRFs is assessed by a comparison with the solution by Finite Element (FE) analyses.     

Replaceability and computational equivalence for monotone boolean functions

Summary Replacement rules have played an important role in the study of monotone boolean function complexity. In this paper, notions of replaceability and computational equivalence are formulated in an abstract algebraic setting, and examined in detail for finite distributive lattices — the appropriate algebraic context for monotone boolean functions. It is shown that when computing an element f of a finite distributive lattice D, the elements of D partition into classes of computationally equivalent elements, and define a quotient of D in which all intervals of the form [t f, t f] are boolean. This quotient is an abstract simplicial complex with respect to ordering by replaceability. Other results include generalisations and extensions of known theorems concerning replacement rules for monotone boolean networks. Possible applications of computational equivalence in developing upper and lower bounds on monotone boolean function complexity are indicated, and new directions of research both abstract mathematical and computational, are suggested.     

Combined subgradient monotone methods for minimization of nonsmooth functions

Translated from Kibernetika, No. 3, pp. 59–63, May–June, 1988.     

基于粗糙分类的不确定可拓群决策数据挖掘及应用

针对可拓分类的不足及分类中存在的不确定性,将关联函数与决策函数相结合,研究不确定条件下可拓群决策的可拓变换与粗糙分类、属性约简与规则提取、数据分析与方案识别,以实现决策对象在不确定条件下的比较与选择;解决了多方案可拓分类和动态识别的系统分类问题,提高了可拓群决策数据挖掘的准确性和可信度.     

The complexity of monotone boolean functions

We study the realization of monotone Boolean functions by networks. Our main result is a precise version of the following statement: the complexity of realizing a monotone Boolean function ofn arguments is less by the factor (2/n)^1/2, where is the circular ratio, than the complexity of realizing an arbitrary Boolean function ofn arguments. The proof combines known results concerning monotone Boolean functions with new methods relating the computing abilities of networks and machines.     

Proof of two conjectures on correlation inequalities for one class of monotone functions

We prove two correlation inequalities conjectured in [1] for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions.     

Quality of cascade operations control based on uncertain variables

This article concerns a problem of control quality evaluation for a control plant composed of cascade operations. The execution time of each operation is described by an expert in the form of inequality containing an unknown parameter. All unknown parameters are assumed to be values of uncertain variables characterized by certainty distributions. This article gives a formulation and solution of the knowledge-based control problem, a proposition for a control quality index, a comprehensive example illustrating the approach presented, and the results of simulations.This work was presented, in part, at the 9th International Symposium on Artificial Life and Robotics, Oita, Japan, January 28–30, 2004     

On learning monotone Boolean functions under the uniform distribution

In this paper, we prove two general theorems on monotone Boolean functions which are useful for constructing a learning algorithm for monotone Boolean functions under the uniform distribution.     

Credal classification rule for uncertain data based on belief functions

In this paper we present a new credal classification rule (CCR) based on belief functions to deal with the uncertain data. CCR allows the objects to belong (with different masses of belief) not only to the specific classes, but also to the sets of classes called meta-classes which correspond to the disjunction of several specific classes. Each specific class is characterized by a class center (i.e. prototype), and consists of all the objects that are sufficiently close to the center. The belief of the assignment of a given object to classify with a specific class is determined from the Mahalanobis distance between the object and the center of the corresponding class. The meta-classes are used to capture the imprecision in the classification of the objects when they are difficult to correctly classify because of the poor quality of available attributes. The selection of meta-classes depends on the application and the context, and a measure of the degree of indistinguishability between classes is introduced. In this new CCR approach, the objects assigned to a meta-class should be close to the center of this meta-class having similar distances to all the involved specific classes? centers, and the objects too far from the others will be considered as outliers (noise). CCR provides robust credal classification results with a relatively low computational burden. Several experiments using both artificial and real data sets are presented at the end of this paper to evaluate and compare the performances of this CCR method with respect to other classification methods.     

On the operational matrices of block pulse functions

It is pointed out that the mathematical bases for block pulse operational methods to various dynamic system problems are to represent the delay operator exp(— hs)approximately by and to represent exp ( — αhs) approximately by (1 — α) + α exp ( — hs), where 0 ≤ α ≤ 1. On these bases, all the existing block pulse operational matrices—the integration matrix, the convolution matrix, the correlation matrix, the delay matrix, and the stretch matrix—can be derived in a mathematically rigorous way without having to use graphical demonstrations.     

Reset law design based on robust model predictive strategy for uncertain systems

In a continuous time control system, if some of the controller states are reset, certain limitations on the system response can be removed. Moreover, the stability and performance of such a reset control system may be improved. Resetting action has two main characteristics which should be determined: (a) when the controller states are reset and (b) how the after reset values of the controller states are determined. By defining a reset set, when the system states enter this set, the instants for the controller states reset, can be determined. This paper addresses the second question on how to determine the after reset values. In order to design a reset law for real time applications, a model predictive strategy is proposed that specifies the after reset values by minimizing a quadratic performance index. The quadratic minimization problem is converted to a LMI formulation and the reset law is determined by solving this LMI optimization problem at certain reset times. This approach is applied to a typical CSTR system to demonstrate the effectiveness of the proposed method for industrial process control application.     

A monotone framework for CCS

The calculus of communicating systems, CCS, was introduced by Robin Milner as a calculus for modelling concurrent systems. Subsequently several techniques have been developed for analysing such models in order to get further insight into their dynamic behaviour.In this paper we present a static analysis for approximating the control structure embedded within the models. We formulate the analysis as an instance of a monotone framework and thus draw on techniques that often are associated with the efficient implementation of classical imperative programming languages.We show how to construct a finite automaton that faithfully captures the control structure of a CCS model. Each state in the automaton records a multiset of the enabled actions and appropriate transfer functions are developed for transforming one state into another. A classical worklist algorithm governs the overall construction of the automaton and its termination is ensured using techniques from abstract interpretation.     

Odd extension for fourier approximation of nonperiodic functions

Approximation of functions by Fourier series plays an important role in applied digital signal processing. The method of odd extension of a nonperiodic function, which increases smoothness compared to existing methods, is proposed. The method is shown to result in a considerable acceleration of convergence of the Fourier series for this function. The method is generalized to the function of two variables. For the two-dimensional Fourier approximation, the optimal technique of truncating the coefficient matrix is found. The advantage of the method is illustrated by test calculations.