非确定性递归程序的多项式不变量生成

K. Chatterjee, Hongfei Fu, A. K. Goharshady, Ehsan Kafshdar Goharshady
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引用次数: 26

摘要

我们考虑了具有多项式赋值的规划的经典不变量生成问题,并重点研究了作为严格多项式不等式的合并的不变量的综合问题。我们提出了一种基于正stellensaetze的完备半完备方法,即半代数几何中表征半代数集上正多项式的定理。在理论方面,我们方法的最坏情况复杂性是次指数的,而之前的完整方法(Kapur, ACA 2004)的最坏情况复杂性是双指数的。即使限于线性不变量,完全不变量生成的最佳前复杂度也是指数级的(Colon et al, CAV 2003)。在实际应用方面,我们将不变量生成问题简化为二次规划问题(QCLP),这是一个具有许多工业解的经典优化问题。我们通过在几个学术基准上提供实验结果来证明我们方法的适用性。据我们所知,以前唯一为多项式不等式组成的不变量提供完备性保证的不变量生成方法是(Kapur, ACA 2004),它依赖于量词消除,甚至不能处理像我们的运行示例这样的玩具程序。
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Polynomial invariant generation for non-deterministic recursive programs
We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method based on positivstellensaetze, i.e. theorems in semi-algebraic geometry that characterize positive polynomials over a semi-algebraic set. On the theoretical side, the worst-case complexity of our approach is subexponential, whereas the worst-case complexity of the previous complete method (Kapur, ACA 2004) is doubly-exponential. Even when restricted to linear invariants, the best previous complexity for complete invariant generation is exponential (Colon et al, CAV 2003). On the practical side, we reduce the invariant generation problem to quadratic programming (QCLP), which is a classical optimization problem with many industrial solvers. We demonstrate the applicability of our approach by providing experimental results on several academic benchmarks. To the best of our knowledge, the only previous invariant generation method that provides completeness guarantees for invariants consisting of polynomial inequalities is (Kapur, ACA 2004), which relies on quantifier elimination and cannot even handle toy programs such as our running example.
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