Infiniteness and Boundedness in 0L, DT0L, and T0L Systems

  • Tim Smith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7810)

Abstract

We investigate the boundary between finiteness and infiniteness in three types of L systems: 0L, DT0L, and T0L. We establish necessary and sufficient conditions for 0L, DT0L, and T0L systems to be infinite, and characterize the boundedness of finite classes of such systems. First, we give a pumping lemma for these systems, proving that the language of a system is infinite iff the system is pumpable. Next, we show that the number of steps needed to derive any string in any finite 0L or DT0L system is bounded by a function depending only on the size of the alphabet, and not on the production rules or start string. This alphabet boundedness does not hold for finite T0L systems in general. Finally, we show that every infinite 0L system has an infinite D0L subsystem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Jungers, R.M., Protasov, V., Blondel, V.D.: Efficient algorithms for deciding the type of growth of products of integer matrices. Linear Algebra and its Applications 428, 2296–2311 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Kari, L., Rozenberg, G., Salomaa, A.: L systems. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 253–328. Springer-Verlag New York, Inc., New York (1997)CrossRefGoogle Scholar
  3. 3.
    Nishida, T.: Quasi-Deterministic 0L Systems. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 65–76. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  4. 4.
    Nishida, T.Y., Salomaa, A.: Slender 0L languages. Theoretical Computer Science 158(1-2), 161–176 (1996)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Rabkin, M.: Ogden’s Lemma for ET0L Languages. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 458–467. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Rozenberg, G., Salomaa, A.: Mathematical Theory of L Systems. Academic Press, Inc., Orlando (1980)MATHGoogle Scholar
  7. 7.
    Vitányi, P.: On the Size of D0L Languages. In: Rozenberg, G., Salomaa, A. (eds.) L Systems. LNCS, vol. 15, pp. 78–92. Springer, Heidelberg (1974)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tim Smith
    • 1
  1. 1.College of Computer and Information ScienceNortheastern UniversityBostonUSA

Personalised recommendations