Applications of L systems to group theory

Laura Ciobanu*, Murray Elder, Michal Ferov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)
56 Downloads (Pure)


L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.

Original languageEnglish
Pages (from-to)309-329
Number of pages21
JournalInternational Journal of Algebra and Computation
Issue number2
Publication statusPublished - 22 Feb 2018


  • co-word problem
  • EDT0L language
  • ET0L language
  • Free group
  • Grigorchuk group
  • indexed language
  • normal form
  • primitive

ASJC Scopus subject areas

  • Mathematics(all)


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