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On Hardest Languages for One-Dimensional Cellular Automata

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Language and Automata Theory and Applications (LATA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12638))

Abstract

Since the famous construction of “the hardest context-free language” by Greibach (1973), the existence of hardest languages under homomorphic reductions has been investigated for quite a few language families. This paper shows that for one-way real-time cellular automata, also known as trellis automata, there is no hardest language, whereas for linear-time cellular automata, the hardest language is constructed.

Research supported by Russian Science Foundation, project 18-11-00100.

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References

  1. Autebert, J.: Non-principalité du cylindre des langages à compteur. Math. Syst. Theory 11, 157–167 (1977)

    Article  MathSciNet  Google Scholar 

  2. Boasson, L., Nivat, M.: Le cylindre des langages linéaires. Math. Syst. Theory 11, 147–155 (1977)

    Article  Google Scholar 

  3. Bucher, W., Culik, K.: On real time and linear time cellular automata. RAIRO Theor. Inform. Appl. 18(4), 307–325 (1984)

    Article  MathSciNet  Google Scholar 

  4. Buchholz, T., Kutrib, M.: On time computability of functions in one-way cellular automata. Acta Informatica 35(4), 329–352 (1998)

    Article  MathSciNet  Google Scholar 

  5. Choffrut, C., Culik II, K.: On real-time cellular automata and trellis automata. Acta Informatica 21, 393–407 (1984)

    Article  MathSciNet  Google Scholar 

  6. Culik II, K.: Variations of the firing squad problem and applications. Inf. Process. Lett. 30(3), 153–157 (1989)

    Google Scholar 

  7. Culik II, K., Maurer, H.A.: On simple representations of language families. RAIRO Theor. Informatics Appl. 13(3), 241–250 (1979)

    MathSciNet  MATH  Google Scholar 

  8. Greibach, S.A.: The hardest context-free language. SIAM J. Comput. 2(4), 304–310 (1973)

    Article  MathSciNet  Google Scholar 

  9. Greibach, S.A.: Jump PDA’s and hierarchies of deterministic context-free languages. SIAM J. Comput. 3(2), 111–127 (1974)

    Article  MathSciNet  Google Scholar 

  10. Harju, T., Karhumäki, J., Kleijn, H.C.M.: On morphic generation of regular languages. Discrete Appl. Math. 15(1), 55–60 (1986)

    Article  MathSciNet  Google Scholar 

  11. Ibarra, O.H., Jiang, T.: Relating the power of cellular arrays to their closure properties. Theoret. Comput. Sci. 57, 225–238 (1988)

    Article  MathSciNet  Google Scholar 

  12. Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theoret. Comput. Sci. 29, 123–153 (1984)

    Article  MathSciNet  Google Scholar 

  13. Ibarra, O.H., Palis, M.A., Kim, S.M.: Some results concerning linear iterative (systolic) arrays. J. Parallel Distrib. Comput. 2(2), 182–218 (1985)

    Article  Google Scholar 

  14. Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theoret. Comput. Sci. 50, 183–238 (1987)

    Article  MathSciNet  Google Scholar 

  15. Okhotin, A.: Conjunctive grammars. J. Autom. Lang. Comb. 6(4), 519–535 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Okhotin, A.: Boolean grammars. Inf. Comput. 194(1), 19–48 (2004)

    Article  MathSciNet  Google Scholar 

  17. Okhotin, A.: On the equivalence of linear conjunctive grammars and trellis automata. RAIRO Theor. Inform. Appl. 38(1), 69–88 (2004)

    Article  MathSciNet  Google Scholar 

  18. Okhotin, A.: State complexity of linear conjunctive languages. J. Autom. Lang. Comb. 9(2/3), 365–381 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Okhotin, A.: A tale of conjunctive grammars. In: Hoshi, M., Seki, S. (eds.) DLT 2018. LNCS, vol. 11088, pp. 36–59. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98654-8_4

    Chapter  Google Scholar 

  20. Okhotin, A.: Hardest languages for conjunctive and Boolean grammars. Inf. Comput. 266, 1–18 (2019)

    Article  MathSciNet  Google Scholar 

  21. Smith III, A.R.: Real-time language recognition by one-dimensional cellular automata. J. Comput. Syst. Sci. 6(3), 233–253 (1972)

    Article  MathSciNet  Google Scholar 

  22. Terrier, V.: On real time one-way cellular array. Theoret. Comput. Sci. 141(1&2), 331–335 (1995)

    Article  MathSciNet  Google Scholar 

  23. Terrier, V.: Language not recognizable in real time by one-way cellular automata. Theoret. Comput. Sci. 156(1&2), 281–287 (1996)

    Article  MathSciNet  Google Scholar 

  24. Terrier, V.: Closure properties of cellular automata. Theoret. Comput. Sci. 352(1–3), 97–107 (2006)

    Article  MathSciNet  Google Scholar 

  25. Terrier, V.: Language recognition by cellular automata. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds.) Handbook of Natural Computing, pp. 123–158. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-540-92910-9_4

    Chapter  Google Scholar 

  26. Terrier, V.: Recognition of poly-slender context-free languages by trellis automata. Theoret. Comput. Sci. 692, 1–24 (2017)

    Article  MathSciNet  Google Scholar 

  27. Yu, S.: A property of real-time trellis automata. Discrete Appl. Math. 15(1), 117–119 (1986)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, St. Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg, 199034, Russia

    Mikhail Mrykhin & Alexander Okhotin

Authors
  1. Mikhail Mrykhin
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  2. Alexander Okhotin
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Correspondence to Mikhail Mrykhin .

Editor information

Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Alberto Leporati

  2. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  3. Ariel University, Ariel, Israel

    Dana Shapira

  4. University of Milano-Bicocca, Milan, Italy

    Claudio Zandron

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Mrykhin, M., Okhotin, A. (2021). On Hardest Languages for One-Dimensional Cellular Automata. In: Leporati, A., Martín-Vide, C., Shapira, D., Zandron, C. (eds) Language and Automata Theory and Applications. LATA 2021. Lecture Notes in Computer Science(), vol 12638. Springer, Cham. https://doi.org/10.1007/978-3-030-68195-1_10

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  • DOI: https://doi.org/10.1007/978-3-030-68195-1_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68194-4

  • Online ISBN: 978-3-030-68195-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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