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Hierarchies and undecidability results for iterative arrays with sparse communication

  • Andreas MalcherEmail author
Article
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Abstract

Iterative arrays with restricted internal inter-cell communication are investigated. A quantitative measure for the communication is defined by counting the number of uses of the links between cells and it is differentiated between the sum of all communications of an accepting computation and the maximum number of communications per cell occurring in accepting computations. The computational complexity of both classes of devices is investigated and put into relation. In addition, a strict hierarchy depending on the maximum number of communications per cell is established. Furthermore, it is shown that almost all commonly studied decidability questions are not semidecidable for iterative arrays with restricted communication. Finally, non-recursive trade-offs are proved among the iterative arrays providing the strict hierarchy depending on the maximum number of communications per cell.

Keywords

Cellular automata Iterative arrays Communication bounds Computational capacity Decidability questions Descriptional complexity 

Notes

Acknowledgements

The author would like to thank the anonymous referees for their careful reading of the manuscript and for giving many valuable comments that helped much to improve the presentation of the paper.

References

  1. Chang JH, Ibarra OH, Palis MA (1987) Parallel parsing on a one-way array of finite-state machines. IEEE Trans Comput C–36:64–75CrossRefGoogle Scholar
  2. Choffrut C, Culik K II (1984) On real-time cellular automata and trellis automata. Acta Inform 21:393–407MathSciNetCrossRefGoogle Scholar
  3. Cole SN (1969) Real-time computation by \(n\)-dimensional iterative arrays of finite-state machines. IEEE Trans Comput C–18(4):349–365MathSciNetCrossRefGoogle Scholar
  4. Fischer PC (1965) Generation of primes by a one-dimensional real-time iterative array. J ACM 12:388–394MathSciNetCrossRefGoogle Scholar
  5. Holzer M, Kutrib M (2010) Descriptional complexity—an introductory survey. In: Martín-Vide C (ed) Scientific applications of language methods. Imperial College Press, London, pp 1–58zbMATHGoogle Scholar
  6. Ibarra OH, Palis MA (1985) Some results concerning linear iterative (systolic) arrays. J Parallel Distrib Comput 2:182–218CrossRefGoogle Scholar
  7. Ibarra OH, Palis MA (1988) Two-dimensional iterative arrays: characterizations and applications. Theor Comput Sci 57:47–86MathSciNetCrossRefGoogle Scholar
  8. Kutrib M (2008) Cellular automata—a computational point of view. In: Bel-Enguix G, Jiménez-López MD, Martín-Vide C (eds) New developments in formal languages and applications, chapter 6. Springer, Berlin, pp 183–227CrossRefGoogle Scholar
  9. Kutrib M (2009) Cellular automata and language theory. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, Berlin, pp 800–823CrossRefGoogle Scholar
  10. Kutrib M, Malcher A (2009) Computations and decidability of iterative arrays with restricted communication. Parallel Process Lett 19(2):247–264MathSciNetCrossRefGoogle Scholar
  11. Kutrib M, Malcher A (2009) On one-way one-bit \({O}(1)\)-message cellular automata. Electron Notes Theor Comput Sci 252:77–91MathSciNetCrossRefGoogle Scholar
  12. Kutrib M, Malcher A (2010) Cellular automata with sparse communication. Theor Comput Sci 411(38–39):3516–3526MathSciNetCrossRefGoogle Scholar
  13. Kutrib M, Malcher A (2010) One-way cellular automata, bounded languages, and minimal communication. J Autom Lang Comb 15(1/2):135–153MathSciNetzbMATHGoogle Scholar
  14. Kutrib M, Malcher A (2011) Cellular automata with limited inter-cell bandwidth. Theor Comput Sci 412(30):3917–3931MathSciNetCrossRefGoogle Scholar
  15. Kutrib M, Malcher A (2018) Cellular automata: descriptional complexity and decidability. In: Adamatzky A (ed) Reversibility and universality, emergence, complexity and computation, vol 30. Springer, Berlin, pp 129–168CrossRefGoogle Scholar
  16. Malcher A (2004) On the descriptional complexity of iterative arrays. IEICE Trans Inf Syst E87–D:721–725Google Scholar
  17. Mazoyer J, Terrier V (1999) Signals in one-dimensional cellular automata. Theor Comput Sci 217(1):53–80MathSciNetCrossRefGoogle Scholar
  18. Smith AR III (1972) Real-time language recognition by one-dimensional cellular automata. J Comput Syst Sci 6(3):233–253MathSciNetCrossRefGoogle Scholar
  19. Umeo H, Kamikawa N (2002) A design of real-time non-regular sequence generation algorithms and their implementations on cellular automata with 1-bit inter-cell communications. Fundam Inform 52:257–275MathSciNetzbMATHGoogle Scholar
  20. Umeo H, Kamikawa N (2003) Real-time generation of primes by a 1-bit-communication cellular automaton. Fundam Inform 58:421–435MathSciNetzbMATHGoogle Scholar
  21. Worsch T (2000) Linear time language recognition on cellular automata with restricted communication. In: Gonnet GH, Panario D, Viola A (eds) Theoretical informatics (LATIN 2000), LNCS, vol 1776. Springer, Berlin, pp 417–426CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

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