通过计算样本点的消失理想生成循环不变量
Generating Loop Invariants by Computing Vanishing Ideals of Sample Points
摘要：
循环不变量在证明程序正确性方面起着非常重要的作用。在本文中，我们解决的问题，生成多项式循环程序的不变量。提出了一种通过计算样本点的消失理想来生成多项式循环程序的多项式方程不变量的新方法。我们应用有理函数插值，提前终止技术的基础上，生成具有符号初值的循环程序的不变量。我们的方法避免了一阶量词消去和柱形代数分解（CAD）。提出了一种生成多项式不变量的算法，并举例说明了该算法。此外，我们证明了一组循环程序的符号初始值，我们的算法可以产生多项式不变量与度高达15。
英文摘要：
Loop invariants play a very important role in proving correctness of programs. In this paper, we address the problem of generating invariants of polynomial loop programs. We present a new approach, for generating polynomial equation invariants of polynomial loop programs through computing vanishing ideals of sample points. We apply rational function interpolation, based on early termination technique, to generate invariants of loop programs with symbolic initial values. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition(CAD). An algorithm for generating polynomial invariants is proposed and some examples are given to illustrate the algorithm. Furthermore, we demonstrate on a set of loop programs with symbolic initial values that our algorithm can yield polynomial invariants with degrees high up to 15.
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