Model Theory Seminar..: "Analog Computability in Differential Equations", Dioga Pocas
Department of Mathematics and Statistics
Title: Analog Computability in Differential Equations
Abstract: Most of the physical processes arising in nature are modeled by differential equations, either ordinary (example: the spring/mass/damper system) or partial (example: heat diffusion). From the point of view of analog computability, the existence of an effective way to obtain solutions (either exact or approximate) of these systems is essential. We present a pioneering model of analog computation called General Purpose Analog Computer (GPAC), introduced by Shannon in 1941. The GPAC is capable of manipulating real-valued data streams. Its power is characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address two limitations of this model. The first is its fundamental inability to reason about functions of more than one independent variable (the `time' variable). In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. The second concerns the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC.