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L Systems pp 104-126 | Cite as

Growth of strings in context dependent Lindenmayer systems

  • Paul M. B. Vitányi
Growth Functions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 15)

Abstract

Growth functions of context dependent Lindenmayer systems are investigated. Bounds on the fastest and slowest growth in such systems are derived, and a method to obtain (P)D1L growth functions from (P)D2L growth functions is given. Closure of context dependent growth functions under several operations is studied with special emphasis on an application of the firing squad synchronization problem. It is shown that, although all growth functions of DILs using a one letter alphabet are DOL growth functions, there are growth functions of PDILs using a two letter alphabet which are not. Several open problems concerning the decidability of growth equivalence, growth type classification etc. of context dependent growth are shown to be undecidable. As a byproduct we obtain that the language equivalence of PDILs is undecidable and that a problem proposed by Varshavsky has a negative solution.

References

  1. 2.
    For results and terminology concerning these devices see e.g. M. Minsky, Computation: finite and infinite machines. Prentice-Hall, London (1967).Google Scholar
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    See e.g. Minsky, Op. cit., 28–29.Google Scholar
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    Balzer, R., An 8 state minimal solution to the firing squad synchronization problem, Inf. Contr. 10 (1967), 22–42.Google Scholar
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    Minsky, Op. cit.Google Scholar
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    The idea of simulating Tag systems with 1Ls occurs already in the first papers on L systems i.e. [33] and [12].Google Scholar
  6. 9.
    In: Unusual automata theory. Univ. of Aarhus, Comp. Sci. Dept. Tech. Rept. DAIMI PB-15 (1973), 20.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Paul M. B. Vitányi
    • 1
  1. 1.Mathematisch CentrumAmsterdamThe Netherlands

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