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

我近年来的研究一直是抽象数据类型计算理论的发展。我在许多出版物中主要与我的同事和学生合作完成了这项工作。由图灵、克莱恩、丘奇等人发展的经典可计算性理论涉及对“离散”结构的计算，例如自然数、整数或由有限字母表生成的字符串（“单词”）。但是，当计算“连续”结构（例如实数）时。经典的技术并不总是有效，经典的结果也不总是成立。例如，测试两个实数之间的相等性并不总是可判定的。开发实数计算理论显然极其重要，因为科学数据通常以实数（或它们的有理近似值）形式给出。“数字”和“模拟”计算之间可以做出重要区分。我也一直在研究这两种范式之间的关系。对离散结构（例如整数）的计算通常是数字的，而对连续结构的计算可以是数字的或模拟的。简而言之，数字计算通过算法进行工作，表示为计算机中的程序，而模拟计算通过模拟时间进行工作，并且是“精确的”。在模拟计算中，数据由物理量（例如电压）表示，并由组件网络处理。我还想研究“混合”系统的理论，该理论研究模拟数字接口。由于混合动力系统的使用越来越普遍，这样做的实际好处是显而易见的。

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.