多项式可达性见证通过Stellensätze

A. Asadi, K. Chatterjee, Hongfei Fu, A. K. Goharshady, Mohammad Mahdavi
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引用次数: 13

摘要

考虑了具有实变量的命令式程序的可达性分析的基本问题。以前解决可达性的工作要么无法处理由一般循环组成的程序(例如符号执行),要么缺乏完整性保证(例如抽象解释),要么不是自动化的(例如逻辑不正确)。相比之下,我们提出了一种新的可达性分析方法,它可以处理一般和复杂的循环,是完整的,并且可以完全自动化用于广泛的程序族。通过归纳可达性见证(irw)的概念,我们的方法将不变生成和终止的思想扩展到可达性分析。我们首先表明,我们基于irw的方法对于命令式程序的可达性分析是健全和完整的。然后,我们将重点放在线性和多项式程序上,并开发自动合成线性和多项式irw的方法。在线性情况下,我们使用Farkas引理遵循众所周知的方法。我们的主要贡献是在多项式情况下,我们提出了一个按钮半完全算法。我们使用实际代数几何中的经典定理的新组合来实现这一点,例如Putinar的正stellensatz和Hilbert的强Nullstellensatz。最后,我们的实验结果表明,我们可以在各种基准上证明复杂的可达性目标,这是以前的方法无法达到的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Polynomial reachability witnesses via Stellensätze
We consider the fundamental problem of reachability analysis over imperative programs with real variables. Previous works that tackle reachability are either unable to handle programs consisting of general loops (e.g. symbolic execution), or lack completeness guarantees (e.g. abstract interpretation), or are not automated (e.g. incorrectness logic). In contrast, we propose a novel approach for reachability analysis that can handle general and complex loops, is complete, and can be entirely automated for a wide family of programs. Through the notion of Inductive Reachability Witnesses (IRWs), our approach extends ideas from both invariant generation and termination to reachability analysis. We first show that our IRW-based approach is sound and complete for reachability analysis of imperative programs. Then, we focus on linear and polynomial programs and develop automated methods for synthesizing linear and polynomial IRWs. In the linear case, we follow the well-known approaches using Farkas' Lemma. Our main contribution is in the polynomial case, where we present a push-button semi-complete algorithm. We achieve this using a novel combination of classical theorems in real algebraic geometry, such as Putinar's Positivstellensatz and Hilbert's Strong Nullstellensatz. Finally, our experimental results show we can prove complex reachability objectives over various benchmarks that were beyond the reach of previous methods.
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