Unified Approach for Developing EfficientAlgorithmic Programs

A unified approach called partition-and-recur for developing efficient and correct algorithmic programs is presented.An algorithm(represented by recurrence and initiation)is separated from program,and special attention is paid to algorithm manipulation rather than proram calculus.An algorithm is exactly a set of mathematical formulae.It is easier for formal erivation and proof.After getting efficient and correct algorithm,a trivial transformation is used to get a final rogram,The approach covers several known algorithm design techniques,e.g.dynamic programming,greedy,divide-and-conquer and enumeration,etc.The techniques of partition and recurrence are not new.Partition is a general approach for dealing with complicated objects and is typically used in divide-and-conquer approach.Recurrence is used in algorithm analysis,in developing loop invariants and dynamic programming approach.The main contribution is combining two techniques used in typical algorithm development into a unified and systematic approach to develop general efficient algorithmic programs and presenting a new representation of algorithm that is easier for understanding and demonstrating the correctness and ingenuity of algorithmicprograms.     

Formal Derivation of Graph AlgorithmicPrograms Using Partition-and-Recur

In this paper,we derive,by presenting some suitable notations,three typical graph algorithms and corresponding programs using a unified approach,partition-and-recur.We putemphasis on the derivation rather than the algorithms themselves.The main ideas and ingenuity of these algorithms are revealed by formula deduction.Success in these examples gives us more evidence that partition-and-recur is a simple and practical approach and developing enough suitable notations is the key in designing and deriving efficient and correct algorithmic programs.     

Garbage Collector Verification for Proof-Carrying Code

We present the verification of the machine-level implementation of a conservative variant of the standard mark- sweep garbage collector in a Hoare-style program logic.The specification of the collector is given on a machine-level memory model using separation logic,and is strong enough to preserve the safety property of any common mutator program.Our verification is fully implemented in the Coq proof assistant and can be packed immediately as foundational proof-carrying code package.Our work makes important attempt toward building fully certified production-quality garbage collectors.     

Optimal decentralized control of large scale systems

This paper presents a new optimized decentralized controller design method for solving the tracking and disturbance rejection problems for large-scale linear time-invariant systems, using only low-order decentralized controllers. To illustrate the type of results which can be obtained using the new optimized decentralized control design method, the control of a large flexible space structure is studied and compared with the standard centralized LQR-observer controller. The order of the resultant decentralized controller is much smaller than that of the standard centralized LQR-observer controller. The proposed controller also has certain fail-safe properties and, in addition, it can be five orders of magnitude more robust than the standard LQR-observer controller based on their real stability radii. The new decentralized controller design method is applied to a large flexible space structure system with 5 inputs and 5 outputs and of order 24.     

Exact Computation of the Topology and Geometric Invariants of the Voronoi Diagram of Spheres in 3D

In this paper, we are addressing the exact computation of the Delaunay graph (or quasi-triangulation) and the Voronoi diagram of spheres using Wu’s algorithm. Our main contributions are first a methodology for automated derivation of invariants of the Delaunay empty circumsphere predicate for spheres and the Voronoi vertex of four spheres, then the application of this methodology to get all geometrical invariants that intervene in this problem and the exact computation of the Delaunay graph and the Voronoi diagram of spheres. To the best of our knowledge, there does not exist a comprehensive treatment of the exact computation with geometrical invariants of the Delaunay graph and the Voronoi diagram of spheres. Starting from the system of equations defining the zero-dimensional algebraic set of the problem, we are applying Wu’s algorithm to transform the initial system into an equivalent Wu characteristic (triangular) set. In the corresponding system of algebraic equations, in each polynomial (except the first one), the variable with higher order from the preceding polynomial has been eliminated (by pseudo-remainder computations) and the last polynomial we obtain is a polynomial of a single variable. By regrouping all the formal coefficients for each monomial in each polynomial, we get polynomials that are invariants for the given problem. We rewrite the original system by replacing the invariant polynomials by new formal coefficients. We repeat the process until all the algebraic relationships (syzygies) between the invariants have been found by applying Wu’s algorithm on the invariants. Finally, we present an incremental algorithm for the construction of Voronoi diagrams and Delaunay graphs of spheres in 3D and its application to Geodesy.     

S-and T-Invariants in Cyber Net Systems

Cyber nets are also known as self-modifying nets.Though proposed and defined some 20 years ago.They have never been under thorough study ever since.The reason for this is simple:the nonlinear nature of such nets keeps them away from applications of well developed methods known to the whole Petri Net Society in the world.This paper attempts to make a start of studying cyber nets in depth by proposing a way to defing and to verify S-invariants and T-invariants in such nets.These invariants reflect important dynamic properties of cyber nets.Invariants in cyber nets play a role similar to loop invariants proposed and studied by E.W.Dijkstra and D.Gries when cyber nets are used for program specification.     

Symbolic algorithmic verification of intransitive generalized noninterference

Generalized noninterference can be used to formulate transitive security policies,but is unsuitable for intransitive security policies.We propose a new information flow security property,which we call intransitive generalized noninterference,that enables intransitive security policies to be specified formally.Next,we propose an algorithmic verification technique to check intransitive generalized noninterference.Our technique is based on the search for counterexamples and on the window induction proof,and can be used to verify generalized noninterference.We further demonstrate that the search of counterexamples and induction proof can be reduced to quantified Boolean satisfiability.This reduction enables us to use efficient quantified Boolean decision procedures to perform the check of intransitive generalized noninterference.It also reduces spatial requirement by representing the space compactly,and improves the efficiency of the verification procedure.     

Program Constructionby Verifying Specification

A program construction method based on Gamma language is proposed.The problem to be solved is specified by first-order predicate logic and a semantic verification program is constructed directly from the specification.Ways for improving efficiency of the program are also studied.The method differs from the one proposed by Manna and Waldinger,where a program is extracted from the proof of the existence of an object meeting the given specification.On the other hand,it also differs from the classical one used for deriving Gamma programs of Banatre and Le metayes,which consists in decomposing the specification into an invariant and a termination conditon.     

The application of Extensible Forces to the controlling in complex economic intelligence system

Economic system is a complex large-scale system with the complexity derived from the emergence of economic entity. In this paper, Extensible Forces, All set theory and Set Pair Analysis （SPA） are introduced to build the problem equation for solving the transformation strategies by Extensible Forces analysis of complex economic entity. At last, this paper shows the program flow chart for solving the problem equation and the application of Extensible Forces in economic entity.     

循环不变式开发新策略及其应用

循环不变式体现了循环程序的本质特征,在算法程序的开发、证明和推导中具有十分重要的作用。而传统的循环不变式开发策略并没有很好地解决循环不变式开发难的问题。文章在阐述现有策略局限性的基础上,详细阐述了刻画循环不变式本质特征的新定义及基于此定义的开发循环不变式的新策略,并通过三个典型的实例,对开发新策略的具体应用作了比较深入的探索。     

若干算法程序的形式化推导与生成技术研究

PAR方法基于分划与递推、量词变换规则、循环不变式开发新策略和软件转换工具,实现了复杂算法问题的形式化开发.采用PAR方法形式化推导几个典型的算法问题.通过量词变换规则对程序规约进行形式化推导,可以得到具有数学引用透明性、易于形式化证明的求解算法问题的递推关系;并在此基础上,自然地导出循环不变式.在得到简短、易于理解、高可靠性的Apla算法程序之后,通过转换工具自动生成Java,C 等可执行程序.     

PAR方法和循环不变式的范畴语义

范畴论对理解程序规约及程序设计和正确性证明十分有用。PAR方法则是建立在严格的数学基础之上的一种统一的算法程序设计方法。循环不变式在循环算法程序的设计中至关重要。使用格理论和范畴论作为工具对PAR方法建立一个理论框架,并对其用范畴论的概念加以解释,从而使得PAR有更强的理论基础。在此基础上引入不动点原理深入刻划循环不变式的含义,循环不变式可以表示为谓词泛函的最小不动点,并从范畴论的角度解释该过程。     

后序遍历二叉树非递归算法的推导及形式化证明

开发涉及非线性数据结构算法程序的循环不变式一直是形式化方法的难点。本文使用PAR方法开发循环不变式的新策略,对后序遍历二叉树问题循环不变式的开发使用递归定义技术,得到了该问题循环不变式的简单精确的表达形式,简化了算法程序的推导和证明过程;利用PAR平台提供的抽象程序设计语言Ap1a中的数据抽象机制,使所得的算法程序结构简洁清晰且易于证明;最后,使用Dijkstra-Gries标准程序证明法形式证明了该问题的核心算法程序(只有4行代码),并使用PAR平台将Apla程序转换成正确的C++代码。实例的成功进一步说明PAR方法提供的循环不变式的开发技术对推导和证明非线性数据结构算法程序的有效性。     

基于算法框架的可重用部件设计与实现

对算法程序的功能规约进行等价变换,可以自然而且方便地得到求解问题设计思想的精确表达,即循环不变式。抽象算法又可以通过循环不变式获得。对算法程序中的算子进行提取、抽象就可以得到算法框架,而算法框架可以设计出可重用部件。文章通过对数组段极值问题的求解,展示了形式化推导不仅可以得到正确、高效的算法程序,而且具有软件重用的功能,并进一步给出了利用可重用部件求解数组段极值问题的C++实现。     

Meta-level inference: Two applications

We describe two uses of meta-level inference: to control the search for a proof; and to derive new control information, and illustrate them in the domain of algebraic equation solving. The derivation of control information is the main focus of the paper. It involves the proving of theorems in the Meta-Theory of Algebra. These proofs are guided by meta-meta-level inference. We are developing a meta-meta-language to describe formulae, and proof plans, and have built a program, IMPRESS, which uses these plans to build a proof. We describe one such proof plan in detail. IMPRESS will form part of a self-improving algebra system.     

Proof planning for strategy development

Proof planning extends the tactic-based theorem proving paradigm through the explicit representation of proof strategies. We see three key benefits to the proof planning approach to the development of proof strategies: flexibility, re-usability and synergy. Here we demonstrate these benefits in terms of reasoning about imperative programs where we reuse strategies developed previously for proof by mathematical induction. In particular, we focus upon strategies for automating the discovery of loop invariants. Our approach tightly couples the discovery of invariants with the process of patching proof strategy failures. This revised version was published online in June 2006 with corrections to the Cover Date.     

非线性循环不变式的自动生成

提出了一个自动生成非线性循环不变式的算法。循环不变式可以表示成一个带参数的多项式的形式,根据断言的归纳特性,将循环不变式的生成问题转变成一个约束求解问题,这个约束求解问题的每个解对应于一个循环不变式,如果约束求解问题仅有零解,则说明不存在该参数多项式形式的循环不变式。该算法在Maple中得到了实现,并通过一些实例说明了该算法的有效性。