Computation on Topological Algebras: Analog and Digital Paradigms

拓扑代数计算：模拟和数字范式

• 批准号: RGPIN-2019-07063
• 负责人: Zucker, Jeffery
• 金额: $ 2.48万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716558
• 关键词: Computation; Topological; Algebras; Analog; Digital

My long-term research goal is to develop a generalized theory of computation, incorporating digital, analog and hybrid models. This work is theoretical, but has practical applications in the construction of "smart systems" with hybrid networks of analog and digital components. Background: Digital computation Computation models exist in two main paradigms: digital and analog. Both forms process infinite data, typically real numbers, originating from measurements. In digital computation, the data are represented symbolically. In analog computation, the data are represented by physical quantities which can be measured. In hybrid computation, both processes are combined. My past research involved the generalization of classical (digital) computability theory to the real numbers, and to metric and topological algebras. Many models of digital computation have been studied, e.g. those of Grzegzorczyk and Lacombe, Weihrauch, and tracking computability (Malcev, Tucker, Stoltenberg-Hansen). The equivalence of many of these has been proved by (among others) John Tucker, myself and our students. Most of the above has involved digital computation models. We turn to analog computation. Background: Analog computation The modern theory begins with Shannon's GPAC ("General Purpose Analog Computer"), as developed by Pour-El, Moore, Costa, Bournez, Graca and others. The GPAC has modules for performing elementary operations on real numbers (representing physical quantities), connected by channels for streams of real numbers (representing functions of time t). Shannon proved GPAC computability to be equivalent to definability by a system of algebraic differential equations. The paper [TZ07] was seminal for our later work. In it Tucker and I developed a fixed point semantics for analog networks with continuous streams of reals. My PhD student Diogo Pocas extended Shannon's GPAC by including (1) an extra space variable x with a partial differential module; and (2) a "limiting process" module. We call such GPACs "multivariate". Long term goal Developing a systematic theory of computation, incorporating digital, analog and hybrid models. Short term goals (1) Although many equivalences have been proved among various digital models, much remains to be done in comparing strengths of analog and digital models, e.g finding a version of the multivariate GPAC equivalent to tracking computability. (2) Use the results in (1) to formulate a Generalized Church-Turing Thesis, applicable to analog as well as digital systems. Impact Given the increasing importance and ubiquity of "smart systems", incorporating hybrid networks with analog and digital components, the necessity for a systematic

我的长期研究目标是发展一个通用的计算理论， 包括数字、模拟和混合模型。 这项工作是理论性的，但 混合网络“智能系统”建设的实际应用 模拟和数字组件。 背景：数字计算 计算模型存在于两种主要范式中：数字和模拟。 两种形式的过程 源于测量的无限数据，通常为真实的数。 在数字计算中，数据用符号表示。 在模拟计算中，数据由可测量的物理量表示。 在混合计算中，这两个过程被结合在一起。 我过去的研究涉及经典（数字）可计算性理论的推广， 真实的数，以及度量和拓扑代数。 人们研究了许多数字计算模型，例如Grzegzorczyk和拉科姆贝的模型， Weihrauch和跟踪可计算性（Malcev，Tucker，Stoltenberg Hansen）。 许多的等价物 这些已经被约翰·塔克（John Tucker）、我和我们的学生证明了。 上述大多数都涉及数字计算模型。 我们转向模拟计算。 背景：模拟计算 现代理论始于香农的GPAC（“通用模拟计算机”）， 如Pour-El、摩尔、Costa、Bournez、Graca和其他人开发的。 GPAC具有用于对真实的数（表示 物理量），由真实的数字流的通道连接 （表示时间t的函数）。 香农证明了GPAC的可计算性等价于 代数微分方程系统的可定义性。 该文件[TZ 07]是开创性的，为我们后来的工作。 在这本书中，塔克和我开发了一种 具有连续实数流的模拟网络的不动点语义。 我的博士生Diogo Pocas扩展了Shannon的GPAC， (1)具有偏微分模的额外空间变量X;以及 (2)一个“限制过程”模块。 我们称这样的GPAC为“多元”。 长期目标 发展系统的计算理论，将数字，模拟和 混合模型。 短期目标 (1)虽然已经证明了各种数字模型之间的许多等价性， 在比较模拟和数字模型的优势方面还有很多工作要做， 找到一个版本的多元GPAC相当于跟踪可计算性。 (2)使用（1）中的结果来表述广义丘奇-图灵命题，适用于 模拟和数字系统。 影响 鉴于“智能系统”的重要性和普遍性日益增加， 模拟和数字组件的混合网络，系统的必要性