基于Coq的第三代微积分机器证明系统

本文基于证明辅助工具Coq,完整实现林群院士和张景中院士等倡导的第三代微积分|没有极限的微积分|理论构架的形式化验证,包括对张景中等发表的题为\微积分基础的新视角"论文中全部定义和定理的Coq描述.进而,对定理无例外地给出Coq的机器证明代码,所有形式化过程已被Coq验证,并在计算机上运行通过,体现了基于Coq的数学定理机器证明具有可读性和交互性的特点,其证明过程规范、严谨、可靠.本文是实践研究人员利用计算机学习、理解、构建乃至教育现代数学理论的一个尝试.     

An application of the algorithm of the cyclic coordinate descent in multidimensional optimization problems with constrained speed

In this paper we present an application of the algorithm of the cyclic coordinate descent in multidimensional variational problems with constrained speed, the physical motivation of the problem being the optimization of hydrothermal systems. The proof of the convergence of the succession generated by the algorithm was based on the use of an appropriate adaptation of Zangwill’s global theorem of convergence. We have also included an algorithm for the formal construction of the descending succession (the solution of an optimum control problem), the approximation of which we carried out using an adaptation of the Euler method in conjunction with a procedure inspired by the shooting method.     

Formal specification and proofs for the topology and classification of combinatorial surfaces

We describe one of the first attempts at using modern specification techniques in the field of geometric modeling and computational geometry. Using the Coq system, we developed a formal multi-level specification of combinatorial maps, used to represent subdivisions of geometric manifolds, and then exploited it to formally prove fundamental theorems. In particular, we outline here an original and constructive proof of a combinatorial part of the famous Surface Classification Theorem, based on a set of so-called “conservative” elementary operations on subdivisions.     

A case study in formalizing projective geometry in Coq: Desargues theorem

Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the non-degeneracy conditions lead to long technical proofs. In addition, when considering higher-dimensions, the amount of incidence relations (e.g. point–line, point–plane, line–plane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a two-dimensional setting. Regarding at least three-dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and non-incidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.     

Formal Verification of Numerical Programs: From C Annotated Programs to Mechanical Proofs

Numerical programs may require a high level of guarantee. This can be achieved by applying formal methods, such as machine-checked proofs. But these tools handle mathematical theorems while we are interested in C code, in which numerical computations are performed using floating-point arithmetic, whereas proof tools typically handle exact real arithmetic. To achieve this high level of confidence on C programs, we use a chain of tools: Frama-C, its Jessie plugin, Why and provers among Coq, Gappa, Alt-Ergo, CVC3 and Z3. This approach requires the C program to be annotated: each function must be precisely specified, and we prove the correctness of the program by proving both that it meets its specifications and that no runtime error may occur. The purpose of this paper is to illustrate, on various examples, the features of this approach.     

Robustly complete synthesis of sampled-data control for continuous-time nonlinear systems with reach-and-stay objectives

Formal methods are becoming favorable for control and verification of safety-critical systems because of the rigorous model-based computation. Relying on an over-approximated model of the original system behaviors, formal control synthesis algorithms are not often complete, which means that a controller cannot necessarily be synthesized even if there exists one. The main result of this paper shows that, for continuous-time nonlinear systems, a sample-and-hold control strategy for a reach-and-stay specification can be synthesized whenever such a strategy exists for the same system with its dynamics perturbed by small disturbances. Control synthesis is carried out by a fixed-point algorithm that adaptively partitions the system state space into a finite number of cells. In each iteration, the reachable set from each cell after one sampling time is over-approximated within a precision determined by the bound of the disturbances. To meet such a requirement, we integrate validated high-order Taylor expansion of the system solution over one sampling period into every fixed-point iteration and provide a criterion for choosing the Taylor order and the partition precision. Two nonlinear system examples are given to illustrate the effectiveness of the proposed method.     

Cyclic scheduling for F.M.S.: Modelling and evolutionary solving approach

This paper concerns the domain of flexible manufacturing systems (FMS) and focuses on the scheduling problems encountered in these systems. We have chosen the cyclic behaviour to study this problem, to reduce its complexity. This cyclic scheduling problem, whose complexity is NP-hard in the general case, aims to minimise the work in process (WIP) to satisfy economic constraints. We first recall and discuss the best known cyclic scheduling heuristics. Then, we present a two-step resolution approach. In the first step, a performance analysis is carried out; it is based on the Petri net modelling of the production process. This analysis resolves some indeterminism due to the system’s flexibility and allows a lower bound of the WIP to be obtained. In the second step, after a formal model of the scheduling problem has been given, we describe a genetic algorithm approach to find a schedule which can reach the optimal production speed while minimizing the WIP. Finally, our genetic approach is validated and compared with known heuristics on a set of test problems.     

Formal neighbourhoods, combinatory Böhm trees, and untyped normalization by evaluation

We prove the correctness of an algorithm for normalizing untyped combinator terms by evaluation. The algorithm is written in the functional programming language Haskell, and we prove that it lazily computes the combinatory Böhm tree of the term. The notion of combinatory Böhm tree is analogous to the usual notion of Böhm tree for the untyped lambda calculus. It is defined operationally by repeated head reduction of terms to head normal forms. We use formal neighbourhoods to characterize finite, partial information about data, and define a Böhm tree as a filter of such formal neighbourhoods. We also define formal topology style denotational semantics of a fragment of Haskell following Martin-Löf, and let each closed program denote a filter of formal neighbourhoods. We prove that the denotation of the output of our algorithm is the Böhm tree of the input term. The key construction in the proof is a “glueing” relation between terms and semantic neighbourhoods which is defined by induction on the latter. This relation is related to the glueing relation which was earlier used for proving the correctness of normalization by evaluation algorithm for typed combinatory logic.     

偏微分方程组的Lie群与高阶对称群的Taylor多项式逐步精化算法

本文基于生成函数的Ｔａｙｌｏｒ展开式及逐步简化步骤，提出了计算偏微分方程组的Ｌｉｅ群与高阶对称群的Ｔａｙｌｏｒ多项式算法，把标准算法中的求解超定偏微分方程组的问题转化为求解代数方程组的问题，降低了求解的难度，提高了计算效率，并且易用计算机代数系统在计算机上全过程实现，并得到重要的对称群     

A new method for constructing a minimal PERT network

A project is an enterprise consisting of several activities which are to be carried out in some specific order. The activities and the order in which they need to be carried out can be represented by a PERT network. The PERT technique is a traditional, well-known approach to the expert of project management. When networks are used, it often becomes necessary to draw dummy activities. Since the computation of project completion time is proportional to the number of arcs, including dummy arcs, it is desirable to draw a network with as few dummy activities as possible.In this paper, we propose a new method for constructing, for a given project scheduling problem, a PERT network having as small as possible the number of dummy arcs by using some results on line graphs. This algorithm deals with the existence of transitive arcs. The paper contains illustrative examples, proofs of some theoretical results as well as a comparative study with a similar algorithm known in the literature. Computational results showed the superiority of our algorithm.     

applying a reduced gradient in quadratic programming

This paper considers specific aspects of implementing an algorithm for solving problems of quadratic programming, which is based on a reduced gradient method. In the subspace of superbasis variables, minimization is carried out by a conjugate gradient method. Some examples of solving test problems are given.     

A new parallel algorithm for rooting a tree

When an undirected tree,T, and a vertex,r, in the tree are given, the problem to transformT into a rooted tree withr as its root is considered. Using Euler tour and prefix sum, an optimal algorithm has been developed [2, 3]. We will present another parallel algorithm which is optimal also on EREW PRAM. Our approach reduces the given tree step by step by pruning and pointer jumping. That is, the tree structure is retained during algorithm processing such that another tree computations can be carried out in parallel.     

Multiple lot-sizing decisions in a two-stage production with an interrupted geometric yield and non-rigid demand

In a production system with random yield, it may be more cost effective to release lots multiple times towards fulfilling a customer order. Such a decision, called the multiple lot-sizing problem, has been investigated in various contexts. This paper proposes an efficient algorithm for solving a new multiple lot-sizing problem defined in the context of a two-stage production system with non-rigid demand when its process yields are governed by interrupted geometric distributions. We formulate this problem as a dynamic program (DP) and develop lemmas to solve it. However, solving such a DP may be computationally extensive, particularly for large-scale cases with a high yield. Therefore, this study proposes an efficient algorithm for resolving computational issues. This algorithm is designed to reduce the DP network into a much simpler algorithm by combining a group of DP branches into a single one. Extensive experiments were carried out. Results indicate that the proposed reduction algorithm is quite helpful for practitioners dealing with large-scale cases characterized by high-yield.     

A derivative-free algorithm for systems of nonlinear inequalities

Recently a new derivative-free algorithm has been proposed for the solution of linearly constrained finite minimax problems. This derivative-free algorithm is based on a smoothing technique that allows one to take into account the non-smoothness of the max function. In this paper, we investigate, both from a theoretical and computational point of view, the behavior of the minmax algorithm when used to solve systems of nonlinear inequalities when derivatives are unavailable. In particular, we show an interesting property of the algorithm, namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locates a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. Moreover, we carried out numerical experimentation and comparison of the method against a standard pattern search minimization method. The obtained results confirm that the good theoretical properties of the method correspond to interesting numerical performance. Moreover, the algorithm compares favorably with a standard derivative-free method, and this seems to indicate that extending the smoothing technique to pattern search algorithms can be beneficial.     

An interacting multiple model algorithm with a switching Markov chain

A Markov chain plays an important role in an interacting multiple model (IMM) algorithm which has been shown to be effective for target tracking systems. Such systems are described by a mixing of continuous states and discrete modes. The switching between system modes is governed by a Markov chain. In real world applications, this Markov chain may change or needs to be changed. Therefore, one may be concerned about a target tracking algorithm with the switching of a Markov chain. This paper concentrates on fault-tolerant algorithm design and algorithm analysis of IMM estimation with the switching of a Markov chain. Monte Carlo simulations are carried out and several conclusions are given.     

基于粗集与概念格的属性简约讨论

概念格的属性简约是在形式背景下解决复杂问题的重要途径,通过对概念格、粗糙集的讨论,将两者有效结合,并借助粗糙集上(下)近似的方法,得出了一个对概念格属性简约的方法,方法将二维的概念格属性简约转化为一维的一种对象格的简约,避免了形式背景下的概念的计算和进一步的可辨识矩阵的计算,方法简便,算法简单易实现,是概念格属性简约有效的算法.     

Levels of arithmetic reasoning in solving an open-ended problem

This paper presents the results of an experimental teaching carried out on 12-year-old students. An open-ended task was given to them and they had not been taught the algorithmic process leading to the solution. The formal solution to the problem refers to a system of two linear equations with two unknown quantities. In this mathematical activity, students worked cooperatively. They discussed their discoveries in groups of four and then presented their answers to the whole class developing a rich communication. This study describes the characteristic arguments that represent certain different forms of reasoning that emerged during the process of justifying the solutions of the problem. The findings of this research show that within an environment conducive to creativity, which encourages collaboration, exploration and sharing ideas, students can be engaged in developing multiple mathematical strategies, posing new questions, creating informal proofs, showing beauty and elegance and bringing out that problem solving is a powerful way of learning mathematics.     

An informational approach to the global optimization of expensive-to-evaluate functions

In many global optimization problems motivated by engineering applications, the number of function evaluations is severely limited by time or cost. To ensure that each evaluation contributes to the localization of good candidates for the role of global minimizer, a sequential choice of evaluation points is usually carried out. In particular, when Kriging is used to interpolate past evaluations, the uncertainty associated with the lack of information on the function can be expressed and used to compute a number of criteria accounting for the interest of an additional evaluation at any given point. This paper introduces minimizers entropy as a new Kriging-based criterion for the sequential choice of points at which the function should be evaluated. Based on stepwise uncertainty reduction, it accounts for the informational gain on the minimizer expected from a new evaluation. The criterion is approximated using conditional simulations of the Gaussian process model behind Kriging, and then inserted into an algorithm similar in spirit to the Efficient Global Optimization (EGO) algorithm. An empirical comparison is carried out between our criterion and expected improvement, one of the reference criteria in the literature. Experimental results indicate major evaluation savings over EGO. Finally, the method, which we call IAGO (for Informational Approach to Global Optimization), is extended to robust optimization problems, where both the factors to be tuned and the function evaluations are corrupted by noise.     

汽车滚装码头泊位调度优化模型与算法研究

本文研究滚装码头混合泊位分配和劳动力分配的联合调度优化问题。首先,考虑潮汐时间窗约束、装卸劳动力约束、泊位缆桩分布约束以及泊位不规则布局因素,建立以最小化船舶总服务时间为目标的混合整数规划模型。其次,采用内外嵌套算法设计策略,提出求解该类问题的组合算法。其中,外层是多种群并行进化的遗传算法,生成多种船舶计划顺序,内层为基于规则的启发式算法,用于计算给定计划顺序的目标函数值。然后,基于实际运营数据,生成多组不同规模的算例进行全面数值实验,结果表明所提出的算法可在10分钟内求解包含50艘船、100个泊段的算例。最后,开展基于真实滚装码头运营实例的案例分析,对所提模型和算法在实际码头调度问题中的适用性与高效性进行验证。