Existence, Continuity, and Computability of Unique Fixed Points in Analog Network Models

Abstract

<p>The thesis consists of three research projects concerning mathematical models for analog computers, originally developed by John Tucker and Jeff Zucker. The models are capable of representing systems that essentially “diverge,” exhibiting no valid behaviour---much the way that digital computers are capable of running programs that never halt. While there is no solution to the general Halting Problem, there are certainly theorems that identify large collections of instances that are guaranteed to halt. For example, if we use a simplified language featuring only assignment, branching, algebraic operations, and loops whose bounds must be fixed in advance (i.e. at “compile time”), we know that all instances expressible in this language will halt.</p> <p>In this spirit, one of the major objectives of all three thesis projects is identify a large class of instances of analog computation (analog computer + input) that are guaranteed to “converge.” In our semantic models, this convergence is assured if a certain operator (representing the computer and its input) has a unique fixed point. The first project is based on an original fixed point construction, while the second and third projects are based on Tucker and Zucker's construction. The second project narrows the scope of the model to a special case in order to concretely identify a class of operators with well-behaved fixed points, and considers some applications. The third project goes the opposite way: widening the scope of the model in order to generalize it.</p>