Abstract

Analog computation attempts to capture any type of computation, that can be realized by any type of physical system or physical process, including but not limited to computation over continuous measurable quantities. A pioneering model is the General Purpose Analog Computer (GPAC), initially presented by Shannon in 1941. The GPAC is capable of manipulating real-valued data streams; however, it has been shown to be strictly less powerful than other models of computation on the reals, such as computable analysis.In previous work, we proposed an extension of the Shannon GPAC, denoted LGPAC, designed to overcome its limitations. Not only is the LGPAC model capable of expressing computation over general data spaces $\mathcal{X}$, but it also directly incorporates approximating computations by means of a limit module. An important feature of this work is the generalisation of the framework of the computation theory from Banach to Fréchet spaces.In this paper, we compare the LGPAC with a digital model of computation based on effective representations (tracking computability). We establish general conditions under which LGPAC-generable functions are tracking computable.

中文翻译：

---

跟踪GPAC可生成函数的可计算性

摘要

模拟计算尝试捕获可以通过任何类型的物理系统或物理过程实现的任何类型的计算，包括但不限于连续可测量量的计算。最早的模型是通用模拟计算机（GPAC），最初由Shannon在1941年提出。GPAC能够处理实值数据流；它可以处理实际值数据流。但是，它已被证明比实际的其他计算模型（例如可计算分析）严格地功能更弱。在先前的工作中，我们提出了Shannon GPAC的扩展，表示为LGPAC，旨在克服其局限性。LGPAC模型不仅可以表达对通用数据空间$ \ mathcal {X} $的计算，而且还可以通过限制模块直接合并近似计算。这项工作的重要特征是对从Banach空间到Fréchet空间的计算理论框架的概括。在本文中，我们将LGPAC与基于有效表示（跟踪可计算性）的数字计算模型进行了比较。我们建立了可跟踪LGPAC可生成函数的一般条件。