Abstraction from Graphs to Multisets Using Temporal Logic

为了分析图的变换过程，我们研究了一种称为基数分析的方法，它基于结点的抽象用多集来逼近图，并分析了每种抽象结点对应的具体结点数是如何随着图变换的每一次运算而变化(增加或减少)的。首先，我们给出了图变换的每一次运算的最弱前提条件和每一模公式，并通过给出满足公式的节点数随着运算的执行而减少的充分条件，验证了一个具体的列表操作程序的终止性。其次，为了推广基数分析，我们研究了对自然数加无穷得到的极小加代数上的模公式进行解释的语义。在此语义下，满足公式的节点数通过对某一具有全局情态的模态公式的解释来表示。使用…在这种语义下，不仅可以表示满足公式的节点数，还可以表示从阳极到另一个节点的最短路径长度等各种数值度量。为了证明这一语义的有效性，我们设计并实现了一种基于极小加代数的模模演算的模型检测算法。然后重新定义了运算的最弱前提和公式，使得在执行运算前对极小加代数上的最弱前提的解释等于对运算后的公式的解释，并给出了对最弱前提的解释小于对公式的解释的充分条件。由于极小加代数是有充分基础的，所以可以用各种度量来分析终止性和活性。除了这些结果，我们还改进了转换谓词抽象，这是一种通用的终止性和活跃性分析方法，在效率和准确性方面都有改进。为了增加图结构和变换操作的表现力，我们还制定了层次模式逻辑。较少

In order to analyze processes of graph transformation, we investigateda method, called cardinality analysis, which approximates graphs by multisets based on abstraction of nodes, and analyzes how the number of concrete nodes corresponding to each kind of abstract node changes (increases or decreases) by each operation of graph transformation. First, we formulated the weakest precondition for each operation of graph transformation and each modal formula, and by giving sufficient conditions for the number of nodes satisfying a formula to decrease by the execution of an operation, we verified termination of a concrete program that manipulates lists. Next, in order to generalize cardinality analysis, we investigated the semantics which interprets modal formulas on min-plus algebra obtained by adding infinity to the set of natural numbers. Under this semantics, the number of nodes satisfying a formula is represented by the interpretation of a certain modal formula with global modality. Usin … More g this semantics, one can express not only the number of nodes satisfying a formula, but also various numerical measures such as the length of the shortest path from anode to another node. To show the effectiveness of this semantics, we invented and implemented an algorithm for model checking of modal mu-calculus on min plus algebra. We then redefined the weakest precondition of an operation and a formula so that the interpretation of the weakest precondition on min-plus algebra before executing the operation is equal to that of the formula after the operation, and formulated sufficient conditions for the interpretation of the weakest precondition to be less than that of the formula. Since min-plus algebra is well-founded, one can analyze termination and liveness with various measures. In addition to these results, we improved transition predicate abstraction, which is a general method for termination and liveness analysis, with respect to efficiency and accuracy. We also formulated hierarchical modal logic in order to increase expressiveness of graph structures and transformation operations. Less