Computability of Operators on Continuous and Discrete Time Streams

A stream is a sequence of data indexed by time. The behaviour of natural and artificial systems can be modelled bystreams and stream transformations. There are two distinct types of data stream: streams based on continuous time and streamsbased on discrete time. Having investigated case studies of both kinds separately, we have begun to combine their study in aunified theory of stream transformers, specified by equations. Using only the standard mathematical techniques of topology, wehave proved continuity properties of stream transformers. Here, in this sequel, we analyse their computability. We use the theoryof computable functions on algebras to design two distinct methods for defining computability on continuous and discrete timestreams of data from a complete metric space. One is based on low-level concrete representations, specifically enumerations, andthe other is based on high-level programming, specifically ‘while’ programs, over abstract data types. We analyse when thesemethods are equivalent. We demonstrate the use of the methods by showing the computability of an analog computing system.We discuss the idea that continuity and computability are important for models of physical systems to be “well-posed”

Approximability in the GPAC

Most of the physical processes arising in nature are modeled by either ordinary or partial differential equations. From the point of view of analog computability, the existence of an effective way to obtain solutions of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon as a model of the Differential Analyzer and improved by Pour-El, Lipshitz and Rubel, Costa and Gra\c{c}a and others. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, concerning the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. In particular, the Shannon GPAC cannot be used to generate non-differentially algebraic functions which can be approximately computed in other models of computation. We extend the class of data types using networks with channels which carry information on a general complete metric space $X$; for example $X=C(R,R)$, the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and we add \emph{(continuous or discrete) limit} modules which have one input and one output. We then define an L-GPAC to be a network built with $X$-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. We study these analog systems and their associated operators, and show how some classically non-generable functions, such as the gamma function and the zeta function, can be captured with the L-GPAC

A model of systems with modes and mode transitions

We propose a method of classifying the operation of a system into finitely many modes. Each mode has its own objectives for the system's behaviour and its own algorithms designed to accomplish its objectives. A central problem is deciding when to transition from one mode to some other mode, a decision that may be contested and involve partial or inconsistent information. We propose some general principles and model mathematically their conception of modes for a system. We derive a family of data types for analysing mode transitions; these are simplicial complexes, both abstract and concretely realised as geometric spaces in euclidean space . In the simplicial complex, a mode is represented by a simplex and each state of a system can be evaluated by mapping it into one or more simplices. This evaluation measures the extent to which different modes are appropriate for the state and can decide on a transition. To illustrate the general model in some detail, we work though a case study of an autonomous racing car