Computation on Topological Algebras: Analog and Digital Paradigms

我的长期研究目标是发展一种通用的计算理论， 融合了数字、模拟和混合模型。这项工作是理论上的，但已经 混合网络在“智能系统”建设中的实际应用 模拟和数字元件。 背景资料：数字计算 计算模型主要有两种模式：数字模式和模拟模式。两种形式的流程 来自测量的无限数据，通常是实数。 在数字计算中，数据是用符号表示的。 在模拟计算中，数据用可以测量的物理量来表示。 在混合计算中，这两个过程被组合在一起。 我过去的研究涉及将经典(数字)可计算性理论推广到 实数，以及度量代数和拓扑代数。 已经研究了许多数字计算模型，例如Grzegzorczyk和Lacombe的模型， 和跟踪可计算性(Malcev，Tucker，Stoltenberg-Hansen)。许多的等价性 其中，约翰·塔克、我自己和我们的学生已经证明了这一点。 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)， 由Pour-El、Moore、Costa、Bournez、Graca和其他人开发。 GPAC具有对实数执行初等运算的模块(表示 物理量)，通过用于实数流的通道连接 (表示时间t的函数)。香农证明了GPAC的可计算性等价于 代数微分方程组的可定义性。 论文[TZ07]对我们以后的工作具有开创性的意义。在里面，塔克和我开发了一种 具有连续实数流的模拟网络的定点语义。 我的博士生Diogo Pocas扩展了Shannon的GPAC，包括 (1)具有偏微分模的额外空间变量x；以及 (2)“限制过程”模块。我们称这样的GAC为“多变量”。 长期目标 发展一套系统的计算理论，将数字、模拟和 混合动力车型。 短期目标 (1)尽管在各种数字模型之间已经证明了许多等价性， 在比较模拟和数字模型的优点方面仍有许多工作要做，例如 找到一个等同于跟踪可计算性的多元GPAC版本。 (2)利用(1)中的结果形成适用的广义丘奇-图灵命题 无论是模拟系统还是数字系统。 影响 鉴于“智能系统”日益增长的重要性和普及性， 混合网络具有模拟和数字组件，有必要进行系统化

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