Learning deterministic DEC grammars is learning rational numbers

Below I show that the class of strings that can be learned by a deterministic DEC grammar is exactly the class of rational numbers between 0 and 1. I call this the class of semi-periodic or rational strings. Dynamically Expanding Context (Dec) grammars were introduced by Kohonen in order to model speech signals ([8]). They can be learned in quadratic time in the size of the grammar. They have been used successfully in the automatic generation and analysis of music ([7], [5], [6]).