Polynomial differential equations compute all real computable functions on computable compact intervals

Abstract

In the last decade, there have been several attempts to understand the relations between the many models of analog computation. Unfortunately, most models are not equivalent. Euler's Gamma function, which is computable according to computable analysis, but that cannot be generated by Shannon's General Purpose Analog Computer (GPAC), has often been used to argue that the GPAC is less powerful than digital computation. However, when computability with GPACs is not restricted to real-time generation of functions, it has been shown recently that Gamma becomes computable by a GPAC. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models.