Fractal dimension for fractal structures: Applications to the domain of words

Keywords: Fractal structure Fractal dimension Box-counting dimension Domain of words Language Regular expression Binary-coded decimal Search tree a b s t r a c t A fractal structure is a tool that is used to study the fractal behavior of a space. In this paper, we show how to apply a new concept of fractal dimension for fractal structures, extending the use of the box-counting dimension to new contexts. In particular, we define a fractal structure on the domain of words and show how to use the new fractal dimension to study the fractal pattern of a language generated by a regular expression, how to calculate the efficiency of an encoding language and how to estimate the number of nodes of a given depth in a search tree. The study and analysis of fractal patterns have become more and more important in the last years due to the large number of applications to diverse scientific fields where fractals have been identified. In this way, one of the main tools that has been used is the fractal dimension since it is a single quantity which shows some useful information about the complexity that a certain system presents. In addition to that, the introduction of fractal structures and a fractal dimension for them allow the study of fractal patterns in new topics as well as new applications. On the other hand, in recent years, there has been a new interest (see for example [8,9,11–14]) in the use of quasi-metrics, and in general Asymmetric Topology, in Computer Science and, in particular, in the study of the domain of words, which appears when modeling the streams of information in Kahn's parallel computation model (see [7,10]). In this paper, we use the fractal structure induced by a non-Archimedean quasi-metric on the domain of words (Section 3.1) to study the fractal dimension of a language. This fractal dimension was introduced by the authors in [5, Definition 3.2] and thus, this is an example of how the fractal dimension for a fractal structure can be applied to new non-Euclidean contexts. In particular, we show how to use the fractal dimension to study the fractal pattern of a language generated by a regular expression (Section 3.2), how to calculate the efficiency of an encoding language (Section 3.3) and how to estimate the number of nodes of a given depth in a search tree (Section 3.4). As an application …