Polynomial logical zonotope: A set representation for reachability analysis of logical systems

IF 4.8 2区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS Automatica Pub Date : 2024-09-13 DOI:10.1016/j.automatica.2024.111896
Amr Alanwar , Frank J. Jiang , Karl H. Johansson
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Abstract

In this paper, we introduce a set representation called polynomial logical zonotopes for performing exact and computationally efficient reachability analysis on logical systems. We prove that through this polynomial-like construction, we are able to perform all of the fundamental logical operations (XOR, NOT, XNOR, AND, NAND, OR, NOR) between sets of points exactly in a reduced space, i.e., generator space with reduced complexity. Polynomial logical zonotopes are a generalization of logical zonotopes, which are able to represent up to 2γ binary vectors using only γ generators. Due to their construction, logical zonotopes are only able to support exact computations of some logical operations (XOR, NOT, XNOR), while other operations (AND, NAND, OR, NOR) result in over-approximations in the generator space. In order to perform all fundamental logical operations exactly, we formulate a generalization of logical zonotopes that is constructed by dependent generators and exponent matrices. While we are able to perform all of the logical operations exactly, this comes with a slight increase in computational complexity compared to logical zonotopes. To illustrate and showcase the computational benefits of polynomial logical zonotopes, we present the results of performing reachability analysis on two use cases: (1) safety verification of an intersection crossing protocol and (2) reachability analysis on a high-dimensional Boolean function. Moreover, to highlight the extensibility of logical zonotopes, we include an additional use case where we perform a computationally tractable exhaustive search for the key of a linear feedback shift register.

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多项式逻辑纵向图:逻辑系统可达性分析的集合表示法
在本文中,我们介绍了一种称为多项式逻辑众数的集合表示法,用于对逻辑系统进行精确且计算效率高的可达性分析。我们证明,通过这种类似多项式的构造,我们能够在缩小的空间(即复杂度降低的生成器空间)中精确地执行点集之间的所有基本逻辑运算(XOR、NOT、XNOR、AND、NAND、OR、NOR)。多项式逻辑众数是逻辑众数的广义化,只用 γ 个产生器就能表示多达 2γ 个二进制向量。由于其结构原因,逻辑众元只能支持某些逻辑运算(XOR、NOT、XNOR)的精确计算,而其他运算(AND、NAND、OR、NOR)则会导致生成器空间的过度逼近。为了精确地执行所有基本逻辑运算,我们提出了一种逻辑众数的广义,它是由从属生成器和指数矩阵构造的。虽然我们能准确执行所有逻辑运算,但与逻辑众数相比,计算复杂度略有增加。为了说明和展示多项式逻辑众数的计算优势,我们介绍了在两个用例中进行可达性分析的结果:(1) 交叉路口协议的安全验证;(2) 高维布尔函数的可达性分析。此外,为了突出逻辑振型的可扩展性,我们还加入了一个额外的用例,即对线性反馈移位寄存器的密钥进行可计算的穷举搜索。
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
Automatica
Automatica 工程技术-工程:电子与电气
CiteScore
10.70
自引率
7.80%
发文量
617
审稿时长
5 months
期刊介绍: Automatica is a leading archival publication in the field of systems and control. The field encompasses today a broad set of areas and topics, and is thriving not only within itself but also in terms of its impact on other fields, such as communications, computers, biology, energy and economics. Since its inception in 1963, Automatica has kept abreast with the evolution of the field over the years, and has emerged as a leading publication driving the trends in the field. After being founded in 1963, Automatica became a journal of the International Federation of Automatic Control (IFAC) in 1969. It features a characteristic blend of theoretical and applied papers of archival, lasting value, reporting cutting edge research results by authors across the globe. It features articles in distinct categories, including regular, brief and survey papers, technical communiqués, correspondence items, as well as reviews on published books of interest to the readership. It occasionally publishes special issues on emerging new topics or established mature topics of interest to a broad audience. Automatica solicits original high-quality contributions in all the categories listed above, and in all areas of systems and control interpreted in a broad sense and evolving constantly. They may be submitted directly to a subject editor or to the Editor-in-Chief if not sure about the subject area. Editorial procedures in place assure careful, fair, and prompt handling of all submitted articles. Accepted papers appear in the journal in the shortest time feasible given production time constraints.
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