Computation on many-sorted topological algebras: digital and analog paradigms

多分类拓扑代数的计算：数字和模拟范式

• 批准号: 46670-2013
• 负责人: Zucker, Jeffery
• 金额: $ 1.09万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634812
• 关键词: Computation; many; sorted; topological; algebras

My research in recent years has been in the development of the theory of computation on abstract data types. I have done this largely in collaboration with my colleagues and students in many publications.Classical computability theory, as developed by Turing, Kleene, Church and others, concerned computation over "discrete" structures, such the natural numbers, or integers, or strings ("words") generated by a finite alphabet. However, when computing on "continuous" structures such as the real numbers. the classical techniques do not always work, and the classical results do not always hold. For example, testing for equality between two real numbers is not always decidable.Developing theories of computation on the reals is clearly extremely important, since scientific data are generally given as real numbers (or rational approximations to them).An important distinction can be made between "digital" and "analog" computation. I have also been investigating the relationship between these two paradigms. Computation on discrete structures such as the integers is generally digital, whereas computation on continuous structures can be digital or analog. Briefly, digital computation works by algorithms, represented as programs in a computer, and analog computation works with simulation time, and is "exact". In analog computation, data are represented by physical quantities (e.g. voltage) which are processed by networks of components.I also want to investigate the theory of "hybrid" systems, which investigates analog-digital interfaces. The practical benefit of this is clear, because of the increasingly common use of hybrid systems.

近年来，我的研究一直是抽象数据类型计算理论的发展。我在许多出版物中与我的同事和学生合作完成了这一工作。经典的可计算性理论，如图灵、克莱因、丘奇和其他人所发展的，涉及对“离散”结构的计算，如自然数、整数或由有限字母表生成的字符串（“字”）。然而，当计算“连续”结构，如真实的数字。经典技术并不总是有效，经典结果也不总是成立。例如，检验两个真实的数是否相等并不总是可判定的。发展关于实数的计算理论显然是极其重要的，因为科学数据通常是以真实的数（或它们的有理近似值）给出的。“数字”计算和“模拟”计算之间有一个重要的区别。我也一直在研究这两种范式之间的关系。离散结构（如整数）上的计算通常是数字的，而连续结构上的计算可以是数字的或模拟的。简单地说，数字计算通过算法工作，表示为计算机中的程序，模拟计算通过模拟时间工作，并且是“精确的”。在模拟计算中，数据由物理量（例如电压）表示，这些物理量由组件网络处理。我还想研究“混合”系统的理论，该理论研究模拟-数字接口。这样做的实际好处是显而易见的，因为混合动力系统的使用越来越普遍。