On Strongest Algebraic Program Invariants

IF 2.3 2区计算机科学 Q2 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE

Journal of the ACM Pub Date : 2023-08-08 DOI:10.1145/3614319

E. Hrushovski, J. Ouaknine, Amaury Pouly, J. Worrell

引用次数: 4

Abstract

A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, our main tool is an algebraic result of independent interest: Given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

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关于最强代数规划不变量

多项式程序是这样一种程序，其中所有赋值都由多项式表达式给出，并且所有分支都是非确定的(与条件相反)。给定这样一个程序，代数不变量是由在每个程序位置的程序变量上的多项式方程定义的不变量。m ller- olm和Seidl提出了是否可以计算给定多项式规划的最强代数不变量的问题。在本文中，我们证明，虽然最强代数不变量在一般情况下是不可计算的，但它们可以在仿射规划的特殊情况下计算，即具有完全线性赋值的规划。对于后一个结果，我们的主要工具是一个独立的代数结果:给定一组相同维数的有限有理方阵，我们展示如何计算它们生成的半群的Zariski闭包。

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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来源期刊

Journal of the ACM 工程技术-计算机：理论方法

CiteScore

7.50

自引率

0.00%

发文量

审稿时长

3 months

期刊介绍： The best indicator of the scope of the journal is provided by the areas covered by its Editorial Board. These areas change from time to time, as the field evolves. The following areas are currently covered by a member of the Editorial Board: Algorithms and Combinatorial Optimization; Algorithms and Data Structures; Algorithms, Combinatorial Optimization, and Games; Artificial Intelligence; Complexity Theory; Computational Biology; Computational Geometry; Computer Graphics and Computer Vision; Computer-Aided Verification; Cryptography and Security; Cyber-Physical, Embedded, and Real-Time Systems; Database Systems and Theory; Distributed Computing; Economics and Computation; Information Theory; Logic and Computation; Logic, Algorithms, and Complexity; Machine Learning and Computational Learning Theory; Networking; Parallel Computing and Architecture; Programming Languages; Quantum Computing; Randomized Algorithms and Probabilistic Analysis of Algorithms; Scientific Computing and High Performance Computing; Software Engineering; Web Algorithms and Data Mining