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

Olivier Bournez (1) , Manuel L. Campagnolo (2) , Daniel S. Graça (2) , Emmanuel Hainry (3)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 IST - Instituto Superior Técnico, Universidade Técnica de Lisboa
3 CARTE - Theoretical adverse computations, and safety

In the last decade, the field of analog computation has experienced renewed interest. In particular, there have been several attempts to understand which relations exist between the many models of analog computation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equivalent models of analog computation. Using this approach, it has been shown recently that the Gamma function becomes computable by a GPAC \cite{Gra04}. 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.

Domains: Computer Science [cs]; Computational Complexity [cs.CC]

Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, Emmanuel Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity, 2007, 23 (3), pp.317--335. ⟨inria-00102947v2⟩