This survey provides a comprehensive overview of the study of the binary and Boolean rank from both a mathematical and a computational perspective, with particular emphasis on their relationship to the real rank. We review the basic definitions of these rank functions and present the main alternative formulations of the binary and Boolean rank, together with their computational complexity and their deep connection to the field of communication complexity. We summarize key techniques used to establish lower and upper bounds on the binary and Boolean rank, including methods from linear algebra, combinatorics and graph theory, isolation sets, the probabilistic method, kernelization, communication protocols and the query to communication lifting technique. Furthermore, we highlight the main mathematical properties of these ranks in comparison with those of the real rank, and discuss several non-trivial bounds on the rank of specific families of matrices. Finally, we present algorithmic approaches for computing and approximating these rank functions, such as parameterized algorithms, approximation algorithms, property testing and approximate Boolean matrix factorization (BMF). Together, the results presented outline the current theoretical knowledge in this area and suggest directions for further research.

翻译：本综述从数学与计算两个视角全面概述了二进制秩与布尔秩的研究，特别强调它们与实秩的关系。我们回顾了这些秩函数的基本定义，介绍了二进制秩与布尔秩的主要替代表述形式，并阐述了它们的计算复杂性及其与通信复杂性领域的深刻联系。我们总结了建立二进制秩与布尔秩上下界的关键技术，包括来自线性代数、组合数学和图论的方法，隔离集、概率方法、核化技术、通信协议以及查询到通信的提升技术。此外，我们重点比较了这些秩与实秩的主要数学性质，并讨论了几类特定矩阵族秩的非平凡界。最后，我们介绍了计算和逼近这些秩函数的算法途径，例如参数化算法、近似算法、性质测试以及近似布尔矩阵分解（BMF）。综合所述成果，本文勾勒了该领域当前的理论认知，并指出了未来研究的方向。