Skip to main content
Springer Nature Link
Log in

Control Mechanisms for Array Grammars on Cayley Grids

  • Conference paper
  • First Online:
Machines, Computations, and Universality (MCU 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10881))

Included in the following conference series:

  • 253 Accesses

Abstract

In this paper, the computational power of several control mechanisms for specific variants of (sequential, isometric) array grammars generating arrays on Cayley grids of finitely presented groups is investigated. Using \(\#\)-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules or activation and blocking of rules the same computational power is obtained as when using arbitrary array productions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizawa, K., Nakamura, A.: Grammars on the hexagonal array. In: Wang, P.S.P. (ed.) Array Grammars, Patterns and Recognizers. Series in Computer Science, vol. 18, pp. 144–152. World Scientific, River Edge (1989)

    Google Scholar 

  2. Alhazov, A., Fernau, H., Freund, R., Ivanov, S., Siromoney, R., Subramanian, K.G.: Contextual array grammars with matrix control, regular control languages, and tissue P systems control. Theor. Comput. Sci. 682, 5–21 (2017). https://doi.org/10.1016/j.tcs.2017.03.012

    Article  MathSciNet  MATH  Google Scholar 

  3. Alhazov, A., Freund, R., Ivanov, S.: Introducing the concept of activation and blocking of rules in the general framework for regulated rewriting in sequential grammars. In: Proceedings of BWMC 2018 (2018)

    Google Scholar 

  4. Alhazov, A., Freund, R., Ivanov, S.: P systems with activation and blocking of rules. In: Verlan, S. (ed.) Proceedings of UCNC 2018. Lecture Notes in Computer Science. Springer (2018)

    Google Scholar 

  5. Alhazov, A., Freund, R., Ivanov, S.: Sequential grammars with activation and blocking of rules. In: Durand-Lose, J., Verlan, S. (eds.) MCU 2018. LNCS, vol. 10881, pp. 51–68. Springer, Cham (2018)

    Google Scholar 

  6. Cook, C.R., Wang, P.S.P.: A Chomsky hierarchy of isotonic array grammars and languages. Comput. Graph. Image Process. 8, 144–152 (1978)

    Article  Google Scholar 

  7. Csuhaj-Varjú, E., Mitrana, V.: Array grammars on Cayley grids, private communication

    Google Scholar 

  8. Dassow, J., Păun, Gh.: Regulated Rewriting in Formal Language Theory. EATCS Monographs in Theoretical Computer Science, vol. 18. Springer, Heidelberg (1989)

    Chapter  Google Scholar 

  9. Fernau, H., Freund, R.: Bounded parallelism in array grammars used for character recognition. In: Perner, P., Wang, P., Rosenfeld, A. (eds.) SSPR 1996. LNCS, vol. 1121, pp. 40–49. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61577-6_5

    Chapter  Google Scholar 

  10. Fernau, H., Freund, R.: Accepting array grammars with control mechanisms. In: Păun, Gh., Salomaa, A. (eds.) New Trends in Formal Languages. LNCS, vol. 1218, pp. 95–118. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62844-4_7

    Chapter  Google Scholar 

  11. Fernau, H., Freund, R., Holzer, M.: Character recognition with k-head finite array automata. In: Amin, A., Dori, D., Pudil, P., Freeman, H. (eds.) SSPR /SPR 1998. LNCS, vol. 1451, pp. 282–291. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0033246

    Chapter  Google Scholar 

  12. Fernau, H., Freund, R., Holzer, M.: Regulated array grammars of finite index. Part I: theoretical investigations. In: Păun, Gh., Salomaa, A. (eds.) Grammatical Models of Multi-Agent Systems. Topics in Computer Mathematics, vol. 8, pp. 157–181. Gordon and Breach Science Publishers, London (1999)

    Google Scholar 

  13. Fernau, H., Freund, R., Holzer, M.: Regulated array grammars of finite index. Part II: syntactic pattern recognition. In: Păun, Gh., Salomaa, A. (eds.) Grammatical Models of Multi-Agent Systems. Topics in Computer Mathematics, vol. 8, pp. 284–296. Gordon and Breach Science Publishers, London (1999)

    Google Scholar 

  14. Fernau, H., Freund, R., Ivanov, S., Schmid, M.L., Subramanian, K.G.: Array insertion and deletion P systems. In: Mauri, G., Dennunzio, A., Manzoni, L., Porreca, A.E. (eds.) UCNC 2013. LNCS, vol. 7956, pp. 67–78. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39074-6_8

    Chapter  MATH  Google Scholar 

  15. Fernau, H., Freund, R., Siromoney, R., Subramanian, K.G.: Non-isometric contextual array grammars and the role of regular control and local selectors. Fundam. Inform. 155(1–2), 209–232 (2017). https://doi.org/10.3233/FI-2017-1582

    Article  MathSciNet  MATH  Google Scholar 

  16. Freund, R.: Control mechanisms on \(\#\)-context-free array grammars. In: Păun, Gh. (ed.) Mathematical Aspects of Natural and Formal Languages, pp. 97–137. World Scientific Publ., Singapore (1994)

    Chapter  Google Scholar 

  17. Freund, R.: Array grammar systems. J. Autom. Lang. Comb. 5(1), 13–30 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Freund, R.: P systems working in the sequential mode on arrays and strings. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds.) DLT 2004. LNCS, vol. 3340, pp. 188–199. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30550-7_16

    Chapter  MATH  Google Scholar 

  19. Freund, R., Haberstroh, B.: Attributed elementary programmed graph grammars. In: Schmidt, G., Berghammer, R. (eds.) WG 1991. LNCS, vol. 570, pp. 75–84. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55121-2_7

    Chapter  MATH  Google Scholar 

  20. Freund, R., Ivanov, S., Oswald, M., Subramanian, K.G.: One-dimensional array grammars and P systems with array insertion and deletion rules. In: Neary, T., Cook, M. (eds.) Proceedings Machines, Computations and Universality 2013, MCU 2013, Zürich, Switzerland, 9–11 September 2013. EPTCS, vol. 128, pp. 62–75 (2013). https://doi.org/10.4204/EPTCS.128

    Article  MathSciNet  Google Scholar 

  21. Freund, R., Kogler, M., Oswald, M.: A general framework for regulated rewriting based on the applicability of rules. In: Kelemen, J., Kelemenová, A. (eds.) Computation, Cooperation, and Life. LNCS, vol. 6610, pp. 35–53. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20000-7_5

    Chapter  Google Scholar 

  22. Freund, R., Oswald, M.: Array automata on Cayley grids. In: Neary, T., Cook, M. (eds.) Proceedings Machines, Computations and Universality 2013 Zürich, Switzerland, 9–11 September 2013. EPTCS, vol. 128, pp. 27–28 (2013). https://doi.org/10.4204/EPTCS.128

  23. Freund, R., Oswald, M.: Array grammars and automata on Cayley grids. J. Autom. Lang. Comb. 19(1–4), 67–80 (2014). https://doi.org/10.25596/jalc-2014-067

    Article  MathSciNet  MATH  Google Scholar 

  24. Freund, R., Păun, Gh.: One-dimensional matrix array grammars. Elektronische Informationsverarbeitung und Kybernetik 29(6), 357–374 (1993)

    Google Scholar 

  25. Holt, D.F., Eick, B., O’Brien, E.A.: Handbook of Computational Group Theory. CRC Press, Hoboken (2005)

    Book  Google Scholar 

  26. Krithivasan, K., Siromoney, R.: Array automata and operations on array languages. Int. J. Comput. Math. 4(1), 3–30 (1974). https://doi.org/10.1080/00207167408803078

    Article  MathSciNet  MATH  Google Scholar 

  27. Păun, Gh.: Computing with Membranes. J. Comput. Syst. Sci. 61, 108–143 (1998)

    Article  MathSciNet  Google Scholar 

  28. Păun, Gh., Rozenberg, G., Salomaa, A.: The Oxford Handbook of Membrane Computing. Oxford University Press Inc., New York (2010)

    Google Scholar 

  29. Rosenfeld, A.: Picture Languages. Academic Press, Reading (1979)

    MATH  Google Scholar 

  30. Rosenfeld, A., Siromoney, R.: Picture languages - a survey. Lang. Des. 1, 229–245 (1993)

    Google Scholar 

  31. Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages: Volume 3 Beyond Words. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59126-6

    Book  Google Scholar 

  32. Salomaa, A.: Formal Languages. Academic Press, New York (1973)

    MATH  Google Scholar 

  33. Wang, P.S.P.: An application of array grammars to clustering analysis for syntactic patterns. Pattern Recogn. 17, 441–451 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

The current paper has been inspired by the discussions and the exchange of ideas with many of my friends and co-authors, yet I can only mention some of them being most influential for my work: I already became interested in the concepts of regulated rewriting during my studies even a decade before the book of Jürgen Dassow and Gheorghe Păun Regulated Rewriting in Formal Language Theory appeared in 1989, see [8], and collected the main results on control mechanisms for the string case in a comprehensive way.

Yet since the beginning, I have always tried to also apply these control mechanisms to grammars dealing with other objects as graphs, for instance, see [19], or d-dimensional arrays, for example, see [24], the first paper with Gheorghe, as well as [16]. Already more than twenty-five years ago, together with Jürgen I started the investigation of a general framework, yet it took a long time before the initial ideas condensed in A General Framework for Regulated Rewriting Based on the Applicability of Rules, see [21].

Array grammars have also been an interesting topic in picture generation and acceptance, and the first paper together with Jürgen and Gheorghe on Cooperating Array Grammar Systems was exploring some possibilities of how to use controlled array grammars for picture generation. Cooperating systems (of grammars) are another interesting way of controlling the application of rules by allowing different sets of rules to be applied according to a given strategy as, for example, to work as long as possible or a certain number of steps on the underlying object (an array in this case), also see [17].

Variants of array grammars were investigated to be applied for character recognition, for example, see [9], which paper started a very fruitful cooperation with Henning Fernau. The topic of character recognition was continued together with Markus Holzer, see [11,12,13]. Recently the cooperation with Henning (and his team in Trier) has been resumed with several papers dealing with control mechanisms for several variants of array grammars, for example, see [2, 14, 15].

With the concept of membrane systems introduced by Gheorghe Păun, see [27, 28], the Handbook of Membrane Computing, a new model emerged, controlling the applicability of rules by using the hierarchical membrane structure where together with the application of a rule the current array can be sent to the outer or an inner membrane, which makes other sets of rules applicable, for example, see [18], the first paper on that topic for the sequential derivation mode with arrays as underlying objects, as well as [14, 20], the contributions to MCU 2013.

From the beginning of this century, Marion Oswald, my colleague and friend working with me at the TU Wien for a very long period, had become my favorite co-author, especially in the area of P systems, but she also co-authored several papers on regulated rewriting, especially [21] on A General Framework for Regulated Rewriting Based on the Applicability of Rules, and on variants of array grammars, especially the main papers introducing Array Grammars and Automata on Cayley Grids, see [22, 23].

More recently, my friends from Moldova Artiom Alhazov and Sergiu Ivanov have become the motivating force to continue the research on new control mechanisms, not only in the area of P systems, but also with respect to new control mechanisms, especially also for array grammars, for instance, see [2, 14, 20]. The new concept of activation and blocking of rules in its final form has been developed during the Brainstorming Week on Membrane Computing this year in Sevilla mainly together with Sergiu, and different facets of this new control mechanism are presented in several papers, see [3,4,5], including this paper.

Both the list of references and the list of colleagues who contributed to my research on control mechanisms and/or on array grammars are far from being complete. I want to express my deep respect and my gratitude to all my colleagues and friends who helped me to develop new ideas and concepts, some of them presented in this paper.

Author information

Authors and Affiliations

  1. Faculty of Informatics, TU Wien, Favoritenstraße 9–11, 1040, Vienna, Austria

    Rudolf Freund

Authors
  1. Rudolf Freund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rudolf Freund .

Editor information

Editors and Affiliations

  1. Université d'Orléans, Orleans, France

    Jérôme Durand-Lose

  2. Université Paris Est, Creteil, France

    Sergey Verlan

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Freund, R. (2018). Control Mechanisms for Array Grammars on Cayley Grids. In: Durand-Lose, J., Verlan, S. (eds) Machines, Computations, and Universality. MCU 2018. Lecture Notes in Computer Science(), vol 10881. Springer, Cham. https://doi.org/10.1007/978-3-319-92402-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-92402-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-92401-4

  • Online ISBN: 978-3-319-92402-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics