###
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.