Computation on Topological Algebras: Analog and Digital Paradigms

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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