Optimal Encodings

The basic notion of encoding is one of the most important present in computer science. So far, they have not been per se the subject of serious research, because of their apparent simplicity. In this paper we show how the realm of encodings, instead, deserves big attention. In particular, we address the fundamental question of optimality of encodings: data need a certain storage cost, so it seems natural to investigate whether some encodings are better than others, in the sense they waste less space. We give a precise formalization of this analysis, within the context of extendable families of encodings, and show that the structure is so rich that no optimal encoding can be found. viz. one can arbitrarily improve the data packing. Secondly, we raise the subtle point of the effect of encodings on the computational power of the device: although so far this problem has been passed over, it is not obvious at all whether or not an encoding affects the computational power of a machine. The subtlety of the point is formally shown, by proving that for almost all the machines encodings behave nicely. However, things become deeply involved just in the most basic case of the 2-register machine, where only particular encodings are safe. The analysis then reveals than in this context not only there are optimal elements, but even a best one, which rather intriguingly is shown to be the first encoding system ever developed.