Linear Quantifier Elimination

This paper presents verified quantifier elimination procedures for dense linear orders (two of them novel), for real and for integer linear arithmetic. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection). The formalization of the different theories is highly modular.     

Quantifier Elimination and Provers Integration

We exploit quantifier elimination in the global design of combined decision and semi-decision procedures for theories over non-disjoint signatures, thus providing in particular extensions of Nelson-Oppen results.     

Reduction and Quantifier Elimination Techniques for Program Validation

We study the automatic verification of programs with infinite or parameterized state space. This paper presents methods allowing the transformation of some second-order formulas expressing Hoare triples into equivalent formulas expressed in a weaker but decidable logic. Two techniques are considered: quantifier elimination and reduction to a finite domain. We illustrate these techniques on the validation of memory coherency protocols expressed in Unity.     

Simulation and Optimization by Quantifier Elimination

We present a highly optimized method for the elimination of linear variables from a Boolean combination of polynomial equations and inequalities. In contrast to the basic method described earlier, the practical applicability of the present method goes far beyond academic examples. The optimization is achieved by various strategies to prune superfluous branches in the elimination tree constructed by the method.The main application concerns the simulation of large technical networks of (electric, mechanical or hydraulic) components, whose characteristic curves are piecewise linear (or quadratic) in the variables to be eliminated. Typical goals are the computation of admissible ranges for certain variables and the detection of a malfunction of a network component. The algorithms are currently used in a commercial software system for industrial applications.Moreover, we extend the author's elimination method for parametric linear programming to the non-convex case by allowing arbitraryand–orcombinations of parametric linear inequalities as constraints. We present a new strategy for finding smaller elimination sets and thus smaller elimination trees for parametric linear programming. Some benchmark examples from thenetliblibrary oflpproblems show the significance of this strategy even for convex linear programming problems.     

Applying Quantifier Elimination to the Birkhoff Interpolation Problem

This paper is devoted to show how to use Computer Algebra and Quantifier Elimination to solve some particular instances of the Birkhoff Interpolation Problem. In particular, this problem is completely solved for degrees less than or equal to 3 and any number of nodes and several instances of degree 4 and 5 by computing all the incidence normal poised matrices with such characteristics. The used Computer Algebra and Quantifier Elimination includes manipulation of multivariate polynomials, computation of determinants of matrices with polynomial entries and the formal manipulation of univariate polynomial inequalities by using Sturm—Habicht sequences and the Sign Determination Scheme.     

带n-元存在量词的描述逻辑MSC推理

分析了描述逻辑非标准推理的重要性,特别分析了描述逻辑MSC(Most Specific Concept)推理的研究现状和存在的问题.针对目前描述逻辑MSC推理不能处理n-元存在量词的不足,研究了带n-元存在量词的描述逻辑εL(n)的MSC推理问题.提出了一种新的εL(n)一描述图,利用描述树和描述图给出了描述逻辑εL(n)的MSC近似推理算法,并利用εL(n)-描述树嵌套和εL(n)-描述树描述图同态证明了MSC近似推理算法的正确性.作为一个附带的结果,利用εL(n)-描述树描述图同态给出了εL(n)-的实例推理算法,也证明了实例推理算法的正确性.     

碰撞问题中的量词消去算法的并行化研究

机器人技术中的碰撞问题可以被表示成量词消去问题，但由于有些碰撞问题的复杂性使得这些问题在单个微机上求解需要花费的时间很长或者根本就解不出来。本文提出了基于分布Maple系统下量词消去算法的并行化．并针对分布Maple系统的特点以及算法的特点，通过实例分析，给出了两种并行策略，以达到在Maple软件环境下提高处理器利用率，提高量词消去算法的效率的目的。     

Improved Algorithms for Partial Curve Matching

We revisit the problem of deciding whether a given curve resembles some part of a larger curve under a fixed Fréchet distance, achieving a running time of O(nm), for n and m being the number of segments in the two curves. This improves the long-standing result of Alt and Godau by an O(log(nm)) factor. Our solution is based on constructing a simple data structure which we call free-space map. Using this data structure, we obtain improved algorithms for several variants of the Fréchet distance problem, including the Fréchet distance between two closed curves, and the so-called minimum/maximum walk problems. We also improve the map matching algorithm of Alt et al. for the particular case in which the map is a directed acyclic graph.     

Improved Approximation Algorithms for Data Migration

Our work is motivated by the need to manage data items on a collection of storage devices to handle dynamically changing demand. As demand for data items changes, for performance reasons, the system needs to automatically respond to changes in demand for different data items. The problem of computing a migration plan among the storage devices is called the data migration problem. This problem was shown to be NP-hard, and an approximation algorithm achieving an approximation factor of 9.5 was presented for the half-duplex communication model in Khuller, Kim and Wan (Algorithms for data migration with cloning. SIAM J. Comput. 33(2):448–461, 2004). In this paper we develop an improved approximation algorithm that gives a bound of 6.5+o(1) using new ideas. In addition, we develop better algorithms using external disks and get an approximation factor of 4.5 using external disks. We also consider the full duplex communication model and develop an improved bound of 4+o(1) for this model, with no external disks.     

遗传算法的改进

简要介绍了一般遗传算法的基本原理，由此提出了一个新的改进算法，它导向以适应度比较高的模式为祖先的染色体“家族”方向。文中给出了两个典型求最大值的例子，从结果中看，改进后的遗传算法大大提高了算法的速度，精度也有所提高。     

Improved Algorithms for Constructing Fault-Tolerant Spanners

Abstract. Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

Improved Algorithms for Constructing Fault-Tolerant Spanners

Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short'' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

Improved Distributed Algorithms for SCC Decomposition

We study and improve the OBF technique [Barnat, J. and P.Moravec, Parallel algorithms for finding SCCs in implicitly given graphs, in: Proceedings of the 5th International Workshop on Parallel and Distributed Methods in Verification (PDMC 2006), LNCS (2007)], which was used in distributed algorithms for the decomposition of a partitioned graph into its strongly connected components. In particular, we introduce a recursive variant of OBF and experimentally evaluate several different implementations of it that vary in the degree of parallelism. For the evaluation we used synthetic graphs with a few large components and graphs with many small components. We also experimented with graphs that arise as state spaces in real model checking applications. The experimental results are compared with that of other successful SCC decomposition techniques [Orzan, S., “On Distributed Verification and Verified Distribution,” Ph.D. thesis, Free University of Amsterdam (2004); Fleischer, L.K., B. Hendrickson and A. Pinar, On identifying strongly connected components in parallel, in: Parallel and Distributed Processing, IPDPS Workshops, Lecture Notes in Computer Science 1800, 2000, pp. 505–511].