Abstract not found

Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2-dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.

We consider the general problem of comparing and integrating computational models of cardiac tissue at di#erent levels of physiological detail. We use a general theory of synchronous concurrent algorithms to model spatially extended biological systems, and expand the theory to create hierarchical models by relating observable behaviour at di#erent levels. The general concepts and methods are illustrated by a detailed study of electrical behaviour in cardiac tissue, in which models based on coupled systems of ordinary di#erential equations, partial di#erential equations and cellular automata are compared and combined. 1 1

We present a general theory for the computation of stream transformers of the form F: (R-- B)-- (T-- A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains can be applied to transformations of continuous streams. A stream transformer is continuous in the compact-open topology on continuous streams if and only if it has a continuous lifting to a standard algebraic domain representation of such streams. We also examine the important problem of representing discontinuous streams, such as signals T-- A, where time T is continuous and data A is discrete.

. First, we study the general idea of a spatially extended system (SES) and argue that many mathematical models of systems in computing and natural science are examples of SESs. We examine the computability and the equational definability of SESs and show that, in the discrete case, there is a natural sense in which an SES is computable if, and only if, it is definable by equations. We look at a simple idea of hierarchical structure for SESs and, using respacings and retimings, we define how one SES abstracts, approximates, or is implemented by another SES. Secondly, we study a special kind of SES called a synchronous concurrent algorithm (SCA). We define the simplest kind of SCA with a global clock and unit delay which are computable and equationally definable by primitive recursive equations over time. We focus on two examples of SCAs: a systolic array for convolution and a non-linear model of cardiac tissue. We investigate the hierarchical structure of SCAs by applyin...