Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.     

Feature selection via Boolean independent component analysis

We devise a feature selection method in terms of a follow-out utility of a special classification procedure. In turn, we root the latter on binary features which we extract from the input patterns with a wrapper method. The whole contrivance results in a procedure that is progressive in two respects. As for features, first we compute a very essential representation of them in terms of Boolean independent components in order to reduce their entropy. Then we reverse the representation mapping to discover the subset of the original features supporting a successful classification. As for the classification, we split it into two less hard tasks. With the former we look for a clustering of input patterns that satisfies loose consistency constraints and benefits from the conciseness of binary representation. With the latter we attribute labels to the clusters through the combined use of basically linear separators.We implement out the method through a relatively quick numerical procedure by assembling a set of connectionist and symbolic routines. These we toss on the benchmark of feature selection of DNA microarray data in cancer diagnosis and other ancillary datasets.     

The algebraic structures of generalized rough set theory

Rough set theory is an important technique for knowledge discovery in databases, and its algebraic structure is part of the foundation of rough set theory. In this paper, we present the structures of the lower and upper approximations based on arbitrary binary relations. Some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended. Based on the concepts of definable sets in rough set theory, two important Boolean subalgebras in the generalized rough sets are investigated. An algorithm to compute atoms for these two Boolean algebras is presented.     

基于ROBDD的布尔函数同构判定算法研究

布尔函数是密码体制设计与分析中一个不可缺少的工具,在布尔函数的应用中,判定两个布尔函数的同构问题具有广泛的需求,但是,判定布尔函数同构是NP-难问题,并且采取穷举法也将随着变量的增多,因极高的时间复杂度而使其难以实现.该文基于图的思想,提出了一种基于ROBDD(简化有序二元决策图)的布尔函数同构判定算法,其算法的复杂度...     

Decomposition of systems of Boolean functions determined by binary decision diagrams

A method for the decomposition of systems of completely defined Boolean functions given a two-block partition of the set of variables and a decision algorithm for the partition of this set are proposed. The system of functions is determined by a binary decision diagram, which simplifies the search for the variable partition and the decomposition based on this partition. Results of experimental studies demonstrating the efficiency of the proposed decomposition in the synthesis of logic circuits are presented.     

Counterexamples to the long-standing conjecture on the complexity of BDD binary operations

In this article, we disprove the long-standing conjecture, proposed by R.E. Bryant in 1986, that his binary decision diagram (BDD) algorithm computes any binary operation on two Boolean functions in linear time in the input–output sizes. We present Boolean functions for which the time required by Bryant?s algorithm is a quadratic of the input–output sizes for all nontrivial binary operations, such as ∧, ∨, and ⊕. For the operations ∧ and ∨, we show an even stronger counterexample where the output BDD size is constant, but the computation time is still a quadratic of the input BDD size. In addition, we present experimental results to support our theoretical observations.     

一种基于熵的OBDD变量排序算法

有序二叉决策图（OBDD）是一种有效表示布尔函数的数据结构,其大小依赖于所采用的变量序。熵是定量描述布尔函数中变量重要性的一种方法。基于变量的熵值分析了高质量变量序的特征,给出了一种基于熵的OBDD变量排序算法。实验结果表明：该算法与模拟退火算法和遗传算法结果相当。时间仅为相应算法的80．84％和29．79％。     

The use of binary decision diagrams for the decomposition of programmable logic arrays

The decomposition method of programmable logic arrays based on the two-block partitioning of a set of variables and the algorithm of the selection of variable partitioning are suggested. The method utilizes the representation of the system of disjunctive normal functions of completely specified Boolean functions implemented by PLA via a binary decision diagram. The results of the experimental comparison of the software implementing the suggesting method with the known PLA decomposition software packages are presented.     

A training algorithm for binary feedforward neural networks

The authors present a new training algorithm to be used on a four-layer perceptron-type feedforward neural network for the generation of binary-to-binary mappings. This algorithm is called the Boolean-like training algorithm (BLTA) and is derived from original principles of Boolean algebra followed by selected extensions. The algorithm can be implemented on analog hardware, using a four-layer binary feedforward neural network (BFNN). The BLTA does not constitute a traditional circuit building technique. Indeed, the rules which govern the BLTA allow for generalization of data in the face of incompletely specified Boolean functions. When compared with techniques which employ descent methods, training times are greatly reduced in the case of the BLTA. Also, when the BFNN is used in conjunction with A/D converters, the applicability of the present algorithm can be extended to accept real-valued inputs.     

Dualization, decision lists and identification of monotone discrete functions

Many data-analysis algorithms in machine learning, datamining and a variety of other disciplines essentially operate on discrete multi-attribute data sets. By means of discretisation or binarization also numerical data sets can be successfully analysed. Therefore, in this paper we view/introduce the theory of (partially defined) discrete functions as an important theoretical tool for the analysis of multi-attribute data sets. In particular we study monotone (partially defined) discrete functions. Compared with the theory of Boolean functions relatively little is known about (partially defined) monotone discrete functions. It appears that decision lists are useful for the representation of monotone discrete functions. Since dualization is an important tool in the theory of (monotone) Boolean functions, we study the interpretation and properties of the dual of a (monotone) binary or discrete function. We also introduce the dual of a pseudo-Boolean function. The results are used to investigate extensions of partially defined monotone discrete functions and the identification of monotone discrete functions. In particular, we present a polynomial time algorithm for the identification of so-called stable discrete functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

基于自适应LBP和SVM的织物疵点检测算法

为准确提取不同种类织物纹理的特征,提出一种新的纹理特征描述方法——自适应局部二值模式(ALBP)。该方法为不同纹理结构创建相应的主要概率模式子集,避免了均匀局部二值模式(ULBP)使用同一模式集描述不同纹理而导致的描述不准确问题。在该算法基础上构建一种基于支持向量机(SVM)的织物疵点检测算法,将疵点检测问题转化为分类问题。实验结果证明,该算法不仅保持了传统局部二值模式(LBP)的旋转不变、多分辨率等特点,而且疵点检测结果在视觉上更加清晰、误检率更低、适用范围更广,SVM的优秀分类性能也有效地提高了疵点检测的准确率。     

Calculation of the paired Haar transform through shared binary decision diagrams

A concept of paired Haar transform (PHT) for representation and efficient optimization of systems of incompletely Boolean functions has recently been introduced. In this article, a method to calculate PHT for incompletely specified switching functions through shared binary decision diagrams (SBDDs) is presented. The algorithm converts switching functions in the form of SBDDs into their paired Haar spectra and can operate on functions with many variables.     

GACH: a grid-based algorithm for hierarchical clustering of high-dimensional data

This paper proposes a grid-based hierarchical clustering algorithm (GACH) as an efficient and robust method to explore clusters in high-dimensional data with no prior knowledge. It discovers the initial positions of the potential clusters automatically and then combines them hierarchically to obtain the final clusters. In this regard, GACH first projects the data patterns on a two-dimensional space (i.e., on a plane established by two features) to overcome the curse of dimensionality problem in high-dimensional data. To choose these two well-informed features, a simple and fast feature selection algorithm is proposed. Then, through meshing the plane with grid lines, GACH detects the crowded grid points. The nearest data patterns around these grid points are considered as initial members of some potential clusters. By returning the patterns back to their true dimensions, GACH refines these clusters. In the merging phase, GACH combines the closely adjacent clusters in a hierarchical bottom-up manner to construct the final clusters’ members. The main features of GACH are: (1) it automatically discovers the clusters, (2) the obtained clusters are stable, (3) it is efficient for data sets with high dimensions, and (4) its merging process involves a threshold which can be obtained in advance for well-clustered data. To assess our proposed algorithm, it is applied on some benchmark data sets and the validity of obtained clusters is compared with the results of some other clustering algorithms. This comparison shows that GACH is accurate, efficient and feasible to discover clusters in high-dimensional data.     

奇数元二阶相关免疫对称布尔函数的构造

n元m阶相关免疫对称函数的构造等价于方程[i=0n-2Cin-2xi=i=0n-2Cin-2xi+1]在二元域上的求解。通过对该方程及其等价方程解的关系讨论,给出了构造奇数元二阶相关免疫对称函数的算法。     

Case-factor diagrams for structured probabilistic modeling

We introduce a probabilistic formalism handling both Markov random fields of bounded tree width and probabilistic context-free grammars. Our models are based on case-factor diagrams (CFDs) which are similar to binary decision diagrams (BDDs) but are more concise for circuits of bounded tree width. A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give versions of the inside–outside algorithm and the Viterbi algorithm for these models.     

Mathemtical model of composite objects and its application fororganizing engineering databases

The authors introduce a clustering concept called component aggregation which considers assemblies having the same types of parts as equivalent objects. The notion of equivalent objects is used to develop a mathematical model of composite objects. It is shown that the set of equivalence classes of objects form a Boolean algebra whose minterms represent the objects that are not considered composite at the current viewing level. The algebraic structure of composite objects serves as a basis for developing a technique for organizing composite objects and supporting materialization of explosion views. The technique provides a clustering mechanism which partitions the database into meaningful and application-oriented clusters, and allows any desired explosion view to be materialized using a minimal set of stored views. A simplified relational database for design data and a set of frequent access patterns in design applications are outlined and used to demonstrate the benefits of database organizations based on the mathematical model of composite objects     

Discovering colocation patterns from spatial data sets: a general approach

Given a collection of Boolean spatial features, the colocation pattern discovery process finds the subsets of features frequently located together. For example, the analysis of an ecology data set may reveal symbiotic species. The spatial colocation rule problem is different from the association rule problem since there is no natural notion of transactions in spatial data sets which are embedded in continuous geographic space. We provide a transaction-free approach to mine colocation patterns by using the concept of proximity neighborhood. A new interest measure, a participation index, is also proposed for spatial colocation patterns. The participation index is used as the measure of prevalence of a colocation for two reasons. First, this measure is closely related to the cross-K function, which is often used as a statistical measure of interaction among pairs of spatial features. Second, it also possesses an antimonotone property which can be exploited for computational efficiency. Furthermore, we design an algorithm to discover colocation patterns. This algorithm includes a novel multiresolution pruning technique. Finally, experimental results are provided to show the strength of the algorithm and design decisions related to performance tuning.     

On multivariate binary data clustering and feature weighting

This paper presents an approach that partitions data sets of unlabeled binary vectors without a priori information about the number of clusters or the saliency of the features. The unsupervised binary feature selection problem is approached using finite mixture models of multivariate Bernoulli distributions. Using stochastic complexity, the proposed model determines simultaneously the number of clusters in a given data set composed of binary vectors and the saliency of the features used. We conduct different applications involving real data, document classification and images categorization to show the merits of the proposed approach.     

硬件组合技术在数据库查询优化中的应用

查询优化技术是关系数据库成功运作的关键技术之一。随着现代数据库规模不断扩大到以十亿字节(GB)计量,对能够处理如此巨大的数据信息的系统的需求也随之而来。找到一种高效的信息提取方法对于使研发过程更快、更容易地进行是十分必要的。文章介绍了一种将与或图和数字逻辑电路技术应用于SQL查询优化,得到数据库中有效信息的技术方法。该方法中把与或图作为一种中间数据结构,用来描述布尔值域上的查询集合的子集;数字逻辑电路则用来表示二进制数集合上的各项逻辑运算功能的一种实现方式。该文同时给出了相关实验结果,实验表明这是一个十分有效的方法。     

Binary rule generation via Hamming Clustering

The generation of a set of rules underlying a classification problem is performed by applying a new algorithm called Hamming Clustering (HC). It reconstructs the AND-OR expression associated with any Boolean function from a training set of samples. The basic kernel of the method is the generation of clusters of input patterns that belong to the same class and are close to each other according to the Hamming distance. Inputs which do not influence the final output are identified, thus automatically reducing the complexity of the final set of rules. The performance of HC has been evaluated through a variety of artificial and real-world benchmarks. In particular, its application in the diagnosis of breast cancer has led to the derivation of a reduced set of rules solving the associated classification problem.