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”

Constraint Satisfaction Problems over Numeric Domains

We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra