Saturating Sorting without Sorts

Pamina Georgiou, Márton Hajdu, Laura Kovács
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Abstract

We present a first-order theorem proving framework for establishing the correctness of functional programs implementing sorting algorithms with recursive data structures. We formalize the semantics of recursive programs in many-sorted first-order logic and integrate sortedness/permutation properties within our first-order formalization. Rather than focusing on sorting lists of elements of specific first-order theories, such as integer arithmetic, our list formalization relies on a sort parameter abstracting (arithmetic) theories and hence concrete sorts. We formalize the permutation property of lists in first-order logic so that we automatically prove verification conditions of such algorithms purely by superpositon-based first-order reasoning. Doing so, we adjust recent efforts for automating inducion in saturation. We advocate a compositional approach for automating proofs by induction required to verify functional programs implementing and preserving sorting and permutation properties over parameterized list structures. Our work turns saturation-based first-order theorem proving into an automated verification engine by (i) guiding automated inductive reasoning with manual proof splits and (ii) fully automating inductive reasoning in saturation. We showcase the applicability of our framework over recursive sorting algorithms, including Mergesort and Quicksort.
无排序饱和排序
我们提出了一个一阶定理证明框架,用于确定使用递归数据结构实现排序算法的函数式程序的正确性。我们用多排序一阶逻辑形式化了递归程序的语义,并在一阶形式化中整合了排序性/迭代属性。我们并不关注特定一阶理论(如整数算术)元素列表的排序,我们的列表形式化依赖于抽象(算术)理论的排序参数,因此也依赖于具体的排序。我们在一阶逻辑中形式化了列表的置换属性,这样我们就可以纯粹通过基于上ositon 的一阶推理来自动证明此类算法的验证条件。通过这样做,我们调整了最近在饱和中自动诱导方面所做的努力。我们提倡一种组合方法,用于自动化归纳证明,这是验证在参数化列表结构上实现并保持排序和置换属性的函数式程序所必需的。我们的工作通过(i)用手动证明拆分指导自动归纳推理,以及(ii)在饱和中完全自动化归纳推理,将基于饱和的一阶定理证明转化为自动化验证引擎。我们展示了我们的框架在递归排序算法(包括 Mergesort 和 Quicksort)上的适用性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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