非线性循环的终止性分析?

单重线性循环程序的终止性问题已被广泛研究,而有关非线性循环终止性判定的结果甚少。利用不动点理论研究了n维单重非线性循环的终止性问题,并建立了相应的符号判定算法。同时,对几类特殊循环的终止性进行了分析,得出了相应的结论。     

有界闭连通域上的非线性循环终止性分析

运用计算机代数中的Groebner基理论,对有界闭连通域上的单重非线性循环程序的终止性问题进行研究,建立了可计算的终止性判定算法. 该算法将这类循环的终止性判定问题归约为有无不动点的判定问题.     

多分支单变量循环程序的终止性分析

对多分支单变量循环程序的终止性问题进行了研究.证明了在适定的条件下,该类循环程序不可终止性的充分必要条件是迭代映射在循环条件形成的区域中有不动点.特别地,当这类循环程序是多项式循环程序时,在给定条件下,其在实数域上的终止性问题是可判定的.     

有界闭区间并上的非线性循环程序的终止性验证

利用符号动力学理论中有关一维离散映射的函数和区间的转换图方法及相关结论,证明一类非线性循环程序不终止的必要条件是在该程序循环区间上有不动点或者周期点存在,并给出相应的终止性验证算法.利用该算法可以验证一维有界闭区间上的非线性循环程序的终止性.最后,给出计算实例演示该算法的算法步骤.     

基于活化路径和条件公式的主动规则集可终止性判定方法

支持主动规则机制已经成为现代数据库系统的一个重要特征.主动规则集的可终止性判定是主动数据库中一个核心问题之一,利用触发图和活化图的方法来判定可终止性都存在不同的保守性.为此,提出了为有效活化路径建立条件公式的思想,在此基础上给出了一个新的判定主动规则集可终止性的方法.分析的结果表明,提出的方法较现有方法可以发现更多的可终止性情形,最后给出了相应的算法及其可终止性、正确性证明.     

有界闭域上的线性赋值循环终止性分析

对有界闭域上的线性赋值循环程序终止性问题进行研究.利用Jordan 标准型技术将原循环程序的终止性问题约减为终止性等价的具有简单结构的循环程序的终止性问题.证明了当线性迭代映射满足一定条件时,该类循环程序不可终止的充分必要条件是：迭代映射在有界闭域上有不动点或周期轨.     

单分支线性约束循环程序的终止性分析

秩函数法是循环终止性分析的主要方法,秩函数的存在表明了循环程序是可终止的.针对单分支线性约束循环程序,提出一种方法对此类循环的终止性进行分析.基于增函数法向空间的计算,该方法将原程序空间上的秩函数计算问题归结为其子空间上的秩函数计算问题.实验结果表明,该方法能有效验证现有文献中大部分循环程序的终止性.     

基于增强条件公式的主动规则集可终止性判定

主动规则集的可终止性判定是一个研究热点问题,现有的基于触发图和活化图的方法没有考虑触发环所有所属规则能否在同一次执行中执行;现有的条件公式判定方法只能包含不可更新或有限次更新变量,当主动规则集只包含可有限次循环执行的触发环时,现有方法不能准确判定它是可终止的。为此,提出了触发环的执行序列的概念和建立包含可更新变量的增强条件公式的方法,新的判定方法将触发环和执行语义有机地结合在一起,较现有方法可以发现更多的可终止性情形,同时给出了新算法的可终止性和正确性证明。     

基于增强条件公式的主动规则集可终止性判定

主动规则集的可终止性判定是一个研究热点问题,现有的基于触发图和活化图的方法没有考虑触发环所有所属规则能否在同一次执行中执行;现有的条件公式判定方法只能包含不可更新或有限次更新变量,当主动规则集只包含可有限次循环执行的触发环时,现有方法不能准确判定它是可终止的。为此,提出了触发环的执行序列的概念和建立包含可更新变量的增强条件公式的方法,新的判定方法将触发环和执行语义有机地结合在一起,较现有方法可以发现更多的可终止性情形,同时给出了新算法的可终止性和正确性证明。     

Active XML文档物化终止性检验算法

Active XML（简记为AXML）文档在XML文档中引入嵌入式Web服务,通过调用这些服务,来获取相应的内涵信息,为AXML文档物化过程。研究了AXML文档物化的终止性检验问题,提出了多项式时间的检验算法,该算法通过构造AXML模式依赖图,检验其无环性来判定AXML文档物化终止性问题,证明了算法的正确性和有效性。     

Proving termination of nonlinear command sequences

We describe a simple and efficient algorithm for proving the termination of a class of loops with nonlinear assignments to variables. The method is based on divergence testing for each variable in the cone-of-influence of the loop’s condition. The analysis allows us to automatically prove the termination of loops that cannot be handled using previous techniques. We also describe a method for integrating our nonlinear termination proving technique into a larger termination proving framework that depends on linear reasoning.     

Recent advances in program verification through computer algebra

In this paper, we summarize the results on program verification through semi-algebraic systemsSASs solving that we have obtained, including automatic discovery of invariants and ranking functions, symbolic decision procedure for the termination of a class of linear loops, termination analysis of nonlinear systems, and so on.     

Recent advances in program verification through computer algebra

In this paper, we summarize the results on program verification through semi-algebraic systems (SASs) solving that we have obtained, including automatic discovery of invariants and ranking functions, symbolic decision procedure for the termination of a class of linear loops, termination analysis of nonlinear systems, and so on.     

Termination of linear programs with nonlinear constraints

Tiwari (2004) proved that the termination problem of a class of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvector computation. Braverman (2006) proved that it is also decidable over the integers. Following their work, we consider the termination problems of three more general classes of programs which are loops with linear updates and three kinds of polynomial loop conditions, i.e., strict constraints, non-strict constraints and both strict and non-strict constraints, respectively. First, we prove that the termination problems of such loops over the integers are all undecidable. Then, for each class we provide an algorithm to decide the termination of such programs over the reals. The algorithms are complete for those programs satisfying a property, Non-Zero Minimum.     

Rely-Guarantee Termination and Cost Analyses of Loops with Concurrent Interleavings

By following a rely-guarantee style of reasoning, we present novel termination and cost analyses for concurrent programs that, in order to prove termination or infer the cost of a considered loop: (1) infer the termination/cost of each loop as if it were a sequential one, imposing assertions on how shared-data is modified concurrently; and then (2) prove that these assertions cannot be violated infinitely many times and, for cost analysis, infer how many times they are violated. At the core of the analysis, we use a may-happen-in-parallel analysis to restrict the set of program points whose execution can interleave. Interestingly, the same kind of reasoning can be applied to prove termination and infer upper bounds on the number of iterations of loops with concurrent interleavings. To the best of our knowledge, this is the first method to automatically bound the cost of such kind of loops. We have implemented our analysis for an actor-based language, and showed its accuracy and efficiency by applying it on several typical applications for concurrent programs and on an industrial case study.     

基于不变式生成的循环停机性验证

循环的停机性验证是程序验证中的一个难点。程序不变式用来描述程序变量的取值关系,其中线性不变式可以帮助描述程序变量间的线性关系,循环不变式能够有效刻画循环中的变量关系。本文基于线性不变式和多项式循环不变式的生成,将循环的停机性验证转化为求解一个最优化问题,给出了一个实用的程序停机性验证框架。基于该框架可以自动地验证程序的停机性,并给出循环的复杂度上界。实验结果说明了该方法的实用性。     

Quantum loop programs

Loop is a powerful program construct in classical computation, but its power is still not exploited fully in quantum computation. The exploitation of such power definitely requires a deep understanding of the mechanism of quantum loop programs. In this paper, we introduce a general scheme of quantum loops and describe its computational process. The function computed by a quantum loop is defined, and a denotational semantics and a weakest precondition semantics of a quantum loop are given. The notions of termination and almost termination are proposed for quantum loops. This paper only consider the case of finite-dimensional state spaces. Necessary and sufficient conditions for termination and almost termination of a general quantum loop on any mixed input state are presented. A quantum loop is said to be (almost) terminating if it (almost) terminates on any input state. We show that a quantum loop is almost terminating if and only if it is uniformly almost terminating. It is observed that a small disturbance either on the unitary transformation in the loop body or on the measurement in the loop guard can make any quantum loop (almost) terminating, provided that some dimension restriction is satisfied. Moreover, a representation of the function computed by a quantum loop is given in terms of finite summations of matrices. To illustrate the notions and results obtained in this paper, two simple classes of quantum loop programs, one qubit quantum loops, and two qubit quantum loops defined by controlled gates, are carefully examined, and to show their expressive power, quantum loops are applied in describing quantum walks.