L-systems for Measuring Repetitiveness

Gonzalo Navarro and Cristian Urbina

An L-system (for compression) is a deterministic context-free L-system (without epsilon-rules) extended with two parameters d and n, and also a coding t, which determines unambiguously a string w = t(phi^d(s))[1:n], where phi is the morphism of the system, and s is its axiom. The length of the shortest description of an L-system generating w is known as ell, and it is arguably a relevant measure of repetitiveness that builds on the self-similarities that arise in the sequence. In this paper we deepen the study of the measure ell and its relation with delta, a better established lower bound that builds on substring complexity. Our results show that ell and delta are largely orthogonal, in the sense that one can be much larger than the other depending on the case. This suggests that both mechanisms capture different kinds of regularities related to repetitiveness. Then, we show that the recently introduced NU-systems, which combine the capabilities of L-systems with bidirectional macro-schemes, can be asymptotically strictly smaller than both mechanisms for the same fixed string family, which makes the size nu of the smallest NU-system the unique smallest reachable repetitiveness measure to date. We conclude that in order to achieve better compression, we should combine morphism substitution with copy-paste mechanisms.