  The main goal of this paper has a double purpose. On the one hand, we propose
a new definition in order to compute the fractal dimension of a subset respect
to any fractal structure, which completes the theory of classical box-counting
dimension. Indeed, if we select the so called natural fractal structure on each
euclidean space, then we will get the box-counting dimension as a particular
case. Recall that box-counting dimension could be calculated over any euclidean
space, although it can be defined over any metrizable one. Nevertheless, the
new definition we present can be computed on an easy way over any space
admitting a fractal structure. Thus, since a space is metrizable if and only if
it supports a starbase fractal structure, our model allows to classify and
distinguish a much larger number of topological spaces than the classical
definition. On the other hand, our aim consists also of studying some
applications of effective calculation of the fractal dimension over a kind of
contexts where the box-counting dimension has no sense, like the domain of
words, which appears when modeling the streams of information in Kahn's
parallel computation model. In this way, we show how to calculate and
understand the fractal dimension value obtained for a language generated by
means of a regular expression, and also we pay attention to an empirical and
novel application of fractal dimension to natural languages.
