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

近年来，我的研究主要集中在抽象数据类型计算理论的发展上。我在很多出版物中与我的同事和学生合作完成了这项工作。由图灵、克莱因、丘奇等人发展的经典可计算性理论，关注“离散”结构的计算，如自然数、整数或由有限字母生成的字符串（“词”）。然而，当计算“连续”结构，如实数。经典的技巧并不总是有效，经典的结果也并不总是有效。例如，检验两个实数之间是否相等并不总是可以确定的。发展实数计算理论显然是极其重要的，因为科学数据通常是实数（或它们的有理数近似值）。“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。简而言之，数字计算是通过算法进行的，用计算机中的程序表示，而模拟计算是在模拟时间内进行的，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，它研究模拟-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.