一个在Horn子句中求解极大缩减的算法

在信念修正理论中,一个核心问题是求解一个公式集合关于事实集合的所有极大协调子集,即极大缩减.本文尝试从算法的角度来解决这一问题,研究在Horn子句中求解所有极大缩减的算法.首先,本文指出并证明了公式集合和事实集合并集的极小不协调子集与公式集合关于事实集合的极大缩减之间的转化关系.其次,给出并证明了Horn子句集合极小不协调的一个必要条件.然后,基于上述两个结论,本文提出了一个在Horn子句中枚举公式集合和事实集合并集的极小不协调子集的交互式算法和一个通过这些极小不协调子集计算所有极大缩减的算法.最后,综合这两个算法,提出了一个在Horn子句中求解所有极大缩减的交互式算法.     

Maximal contractions in Boolean algebras

In the present paper,the concepts of deductive element and maximal contraction are introduced in Boolean algebras,and corresponding theories of consistency and maximal contractions are studied.An algorithm principle is proposed to compute all maximal contractions for a consistent set with respect to its refutation in Boolean algebras.It is pointed out that the quotient algebra of the first-order language with respect to its provable equivalence relation constitutes a Boolean algebra,and hence the computation of R-contractions for closed formulas in first-order languages can be converted into the one in Boolean algebras proposed in this paper.Furthermore,the concept of basic element is introduced in Boolean algebras,which contributes to the definitions of clause and Horn clause transplanted from logic to a special type of Boolean algebras generated by basic elements.It is also pointed out that the computation of R-contractions for clauses in the classical propositional logic can be converted into the one in Boolean algebras generated by basic elements proposed in this paper.     

An efficient algorithm for Horn description

We give a new algorithm for computing a prepositional Horn CNF formula given the set of its models. Its running time is O(|R|n(|R|+n)), where |R| is the number of models and n that of variables, and the computed CNF contains at most |R|n clauses. This algorithm also uses the well-known closure property of Horn relations in a new manner.     

An algorithm to compute state feedback matrices for multi-input deadbeat control

An algorithm for the computation of a state feedback controller shifting all eigenvalues of the closed loop system to the origin is presented. The available freedom in multi-input systems is used to reduce the norm of the feedback matrix. The algorithm places one or more eigenvalues in each step using partial feedback laws of minimal norm. This results in an overall feedback matrix whose norm is close to its minimum value in most cases.     

一个基于模型的故障诊断算法

基于模型的诊断方法是重要的故障诊断方法之一,本文首先分析了现有的基于模型的故障诊断的优缺点,指出了这些方法的不足之处,然后再对这些方法进行了改进。因为求诊断问题是针对受限语言的,所以我们只求受限语言的质蕴含项,并且在算法中增加了极小性检查,以确保得到的结论都是质蕴含项,在此基础上给出了一个基于模型的诊断算法。     

利用结构信息的故障诊断方法

基于模型的故障诊断方法是重要的故障诊断方法之一,该方法主要的问题就是如何求得所有的诊断．该文利用系统的结构信息,给出了求极小冲突集的一个算法,证明了算法的正确性,分析了算法的复杂性;然后介绍了如何由极小冲突求得诊断．其次,还给出了利用结构信息直接求诊断的一个算法,证明了其正确性．最后与相关工作进行了比较．该文给出的算法,对于一些特殊结构的系统可在多项式时间内结束．     

An algorithm for Exact Satisfiability analysed with the number of clauses as parameter

We give an algorithm for Exact Satisfiability with polynomial space usage and a time bound of poly(L)⋅m!, where m is the number of clauses and L is the length of the formula. Skjernaa has given an algorithm for Exact Satisfiability with time bound poly(L)⋅m² but using exponential space. We leave the following problem open: Is there an algorithm for Exact Satisfiability using only polynomial space with a time bound of cm, where c is a constant and m is the number of clauses?     

An optimal parallel algorithm for c-vertex-ranking of trees

For a positive integer c, a c-vertex-ranking of a graph G=(V,E) is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. The c-vertex-ranking problem is to find a c-vertex-ranking of a given graph using the minimum number of ranks. In this paper we give an optimal parallel algorithm for solving the c-vertex-ranking problem on trees in O(log₂n) time using linear number of operations on the EREW PRAM model.     

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a -approximation algorithm, where c is an arbitrary constant.In this paper, we present a -approximation algorithm based on an LP relaxation, where χ^′(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also -approximable. From edge-coloring theory, the approximation ratio of our algorithm is , where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least . Moreover, χ^′(G) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.     

An effective heuristic algorithm for sum coloring of graphs

Given an undirected graph G=(V,E), the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of G, using colors represented by natural numbers (1,2,…) such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present EXSCOL, a heuristic algorithm based on independent set extraction for this NP-hard problem. EXSCOL identifies iteratively collections of disjoint independent sets of equal size and assign to each independent set the smallest available color. For the purpose of computing large independent sets, EXSCOL employs a tabu search based heuristic. Experimental evaluations on a collection of 52 DIMACS and COLOR2 benchmark graphs show that the proposed approach achieves highly competitive results. For more than half of the graphs used in the literature, our approach improves the current best known upper bounds.     

An efficient approach to compute the threshold for multi-dimensional linear classifiers

In this paper, we theoretically analyze some properties that relate Fisher's classifier and the optimal quadratic classifier, when the latter is derived utilizing a particular covariance matrix for the classes. We propose an efficient approach which is used to select the threshold after a linear transformation onto the one-dimensional space is performed. We achieve this by selecting the decision boundary that minimizes the classification error in the transformed space, assuming that the univariate random variables are normally distributed. Our empirical results on synthetic and real-life data sets show that our approach lead to smaller classification error than the traditional Fisher's classifier. The results also demonstrate that minimizing the classification error in the transformed space leads to smaller classification error in the original multi-dimensional space.     

An enhancement for heuristic attribute reduction algorithm in rough set

Attribute reduction is one of the most important issues in the research of rough set theory. Numerous significance measure based heuristic attribute reduction algorithms have been presented to achieve the optimal reduct. However, how to handle the situation that multiple attributes have equally largest significances is still largely unknown. In this regard, an enhancement for heuristic attribute reduction (EHAR) in rough set is proposed. In some rounds of the process of adding attributes, those that have the same largest significance are not randomly selected, but build attribute combinations and compare their significances. Then the most significant combination rather than a randomly selected single attribute is added into the reduct. With the application of EHAR, two representative heuristic attribute reduction algorithms are improved. Several experiments are used to illustrate the proposed EHAR. The experimental results show that the enhanced algorithms with EHAR have a superior performance in achieving the optimal reduct.     

An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states

An algorithm to find a polyhedral approximant to the maximal state constraint set, given state and input bounds, is suggested for linear discrete-time dynamical systems.     

An output sensitive algorithm for computing a maximum independent set of a circle graph

We present an output sensitive algorithm for computing a maximum independent set of an unweighted circle graph. Our algorithm requires O(nmin{d,α}) time at worst, for an n vertex circle graph where α is the independence number of the circle graph and d is its density. Previous algorithms for this problem required Θ(nd) time at worst.