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

The algebraic specification of spatial data types with applications to constructive volume geometry.

Spatial objects are modelled as total functions, mapping a topological space of points to a topological algebra of data attributes. High-level operations on these spatial objects form algebras of spatial objects, which model spatial data types. This thesis presents a comprehensive account of the theory of spatial data types. The motivation behind the general theory is Constructive Volume Geometry (CVG). CVG is an algebraic framework for the specification, representation and manipulation of graphics objects in 3D. By using scalar fields as the basic building blocks, CVG gives an abstract representation of spatial objects, with the goal of unifying the many representations of objects used in 3D computer graphics today. The general theory developed in this thesis unifies discrete and continuous spatial data, and the many examples where such data is used - from computer graphics to hardware design. Such a theory is built from the algebraic and topological properties of spatial data types. We examine algebraic laws, approximation methods, and finiteness and computability for general spatial data types. We show how to apply the general theory to modelling (i) hardware and (ii) CVG. We pose the question "Which spatial objects can be represented in the algebraic framework developed for spatial data types?". To answer such a question, we analyse the expressive power of our algebraic framework. Applying our results to the CVG framework yields a new result: We show any CVG spatial object can be approximated by way of CVG terms, to arbitrary accuracy