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Single semi-contextual insertion-deletion systems

Abstract

In this paper we consider the model of single insertion-deletion systems that at each step insert or delete a single symbol in a context-free manner (i.e. at any position in the word). The corresponding operation is performed if the word contains a set of permitting (that have to be present in the word) and/or forbidding (that must not be present in the word) strings of some size. The main result of this paper states that if forbidding strings of size 2 and permitting strings of size 1 are used then computational completeness can be achieved; moreover, checking for a single permitting symbol is sufficient. We also show that in the case of systems having rules with forbidding conditions only, all regular languages can be obtained. Finally, we show the computational non-completeness in the case of systems using rules with forbidding strings of size 1 (single symbols) and permitting strings of any finite size.

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References

  1. Adli M (2018) The CRISPR tool kit for genome editing and beyond. Nat Commun 9(1):1–13

    Article  Google Scholar 

  2. Alhazov A, Krassovitskiy A, Rogozhin Y, Verlan S (2011) P systems with minimal insertion and deletion. Theor Comput Sci 412(1–2):136–144

    MathSciNet  Article  Google Scholar 

  3. Benne R (ed) (1993) RNA editing: the alteration of protein coding sequences of rna. series in molecular biology. Ellis Horwood, Chichester, UK

  4. Biegler F, Burrell MJ, Daley M (2007) Regulated RNA rewriting: modelling RNA editing with guided insertion. Theor Comput Sci 387(2):103–112

    MathSciNet  Article  Google Scholar 

  5. Castellanos J, Martín-Vide C, Mitrana V, Sempere JM (2001) Solving NP-complete problems with networks of evolutionary processors. In:  Mira J, Prieto A (eds.) Connectionist Models of Neurons, Learning Processes and Artificial Intelligence, 6th International Work-Conference on Artificial and Natural Neural Networks, IWANN 2001 Granada, Spain, June 13-15, 2001, Proceedings, Part I, Lecture Notes in Computer Science, vol. 2084, pp. 621–628. Springer. https://doi.org/10.1007/3-540-45720-8_74

  6. Fernau H, Kuppusamy L, Raman I (2016) Generative power of matrix insertion-deletion systems with context-free insertion or deletion. In: Amos M,  Condon A (eds.) Unconventional Computation and Natural Computation - 15th International Conference, UCNC 2016, Manchester, UK, July 11-15, 2016, Proceedings, Lecture Notes in Computer Science, vol. 9726, pp. 35–48. Springer. https://doi.org/10.1007/978-3-319-41312-9_4

  7. Fernau H, Kuppusamy L, Raman I (2018) Computational completeness of simple semi-conditional insertion-deletion systems. In: Stepney S, Verlan S (eds.) Unconventional Computation and Natural Computation - 17th International Conference, UCNC 2018, Fontainebleau, France, June 25-29, 2018, Proceedings, Lecture Notes in Computer Science, vol. 10867, pp. 86–100. Springer. https://doi.org/10.1007/978-3-319-92435-9_7

  8. Fernau H, Kuppusamy L, Raman I (2018) Investigations on the power of matrix insertion-deletion systems with small sizes. Nat Comput 17(2):249–269. https://doi.org/10.1007/s11047-017-9656-8

    MathSciNet  Article  Google Scholar 

  9. Fernau H, Kuppusamy L, Raman I (2019) Computational completeness of simple semi-conditional insertion-deletion systems of degree (2, 1). Nat Comput 18(3):563–577. https://doi.org/10.1007/s11047-019-09742-w

    MathSciNet  Article  MATH  Google Scholar 

  10. Fernau H, Kuppusamy L, Raman I (2020) On the power of generalized forbidding insertion-deletion systems. In: Descriptional Complexity of Formal Systems (DCFS 2020)

  11. Fernau H, Kuppusamy L, Verlan S (2017) Universal matrix insertion grammars with small size. In: Patitz MJ, Stannett M (eds.) Unconventional Computation and Natural Computation - 16th International Conference, UCNC 2017, Fayetteville, AR, USA, June 5-9, 2017, Proceedings, Lecture Notes in Computer Science. 10240: 182–193. Springer. https://doi.org/10.1007/978-3-319-58187-3_14

  12. Freund R, Kogler M, Rogozhin Y, Verlan S (2010) Graph-controlled insertion-deletion systems. In: McQuillan I, Pighizzini G (eds.) Proceedings Twelfth Annual Workshop on Descriptional Complexity of Formal Systems, Electronic Proceedings in Theoretical Computer Science, 31: pp. 88–98

  13. Galiukschov B (1981) Semicontextual grammars. Matem. Logica i Matem. Lingvistika , 38–50. Tallin University, (in Russian)

  14. Gazdag Z, Tichler K, Csuhaj-Varjú E (2019) A pumping lemma for permitting semi-conditional languages. Int J Found Comput Sci 30(1):73–92. https://doi.org/10.1142/S0129054119400045

    MathSciNet  Article  MATH  Google Scholar 

  15. Haussler D (1983) Insertion languages. Inf Sci 31(1):77–89. https://doi.org/10.1016/0020-0255(83)90023-3

    MathSciNet  Article  MATH  Google Scholar 

  16. Ivanov S, Verlan S (2013) About one-sided one-symbol insertion-deletion P systems. In:  Alhazov A, Cojocaru S,  Gheorghe M, Rogozhin Y,  Rozenberg G,  Salomaa A (eds.) Membrane Computing - 14th International Conference, CMC 2013, Chişinău, Republic of Moldova, August 20-23, 2013, Revised Selected Papers, Lecture Notes in Computer Science, vol. 8340, pp. 225–237. Springer

  17. Ivanov S, Verlan S (2015) Random context and semi-conditional insertion-deletion systems. Fundam Inf 138(1–2):127–144. https://doi.org/10.3233/FI-2015-1203

    MathSciNet  Article  MATH  Google Scholar 

  18. Ivanov S, Verlan S (2017) Universality and computational completeness of controlled leftist insertion-deletion systems. Fundam Inform 155(1–2):163–185. https://doi.org/10.3233/FI-2017-1580

    MathSciNet  Article  MATH  Google Scholar 

  19. Jančar P, Mráz F, Plátek M, Vogel J (1995) Restarting automata. In: Reichel H (ed.) Fundamentals of Computation Theory, FCT, LNCS, vol. 965, pp. 283–292

  20. Kari L (1991) On insertion and deletion in formal languages. Ph.D. thesis, University of Turku

  21. Kari L (1994) Deletion operations: closure properties. Int J Comput Math 52:23–42

    Article  Google Scholar 

  22. Kari L, Păun Gh, Thierrin G, Yu S (1999) At the crossroads of DNA computing and formal languages: Characterizing recursively enumerable languages using insertion-deletion systems. In: Rubin H, Wood DH (eds.) DNA Based Computers III, DIMACS Series in Discrete Mathematics and Theretical Computer Science, vol. 48, pp. 329–338. AMS

  23. Kari L, Sosík P (2008) On the weight of universal insertion grammars. Theor Comput Sci 396(1–3):264–270

    MathSciNet  Article  Google Scholar 

  24. Kari L, Thierrin G (1996) Contextual insertions/deletions and computability. Inf Comput 131(1):47–61

    MathSciNet  Article  Google Scholar 

  25. Krassovitskiy A, Rogozhin Y, Verlan S (2008) Further results on insertion-deletion systems with one-sided contexts. In:  Martín-Vide C, Otto F,  Fernau (eds.) Language and Automata Theory and Applications, Second International Conference, LATA 2008, Tarragona, Spain, March 13-19, 2008. Revised Papers, Lecture Notes in Computer Science, Springer, Berlin

  26. Krassovitskiy A, Rogozhin Y, Verlan S (2011) Computational power of insertion-deletion (P) systems with rules of size two. Nat Comput 10(2):835–852

    MathSciNet  Article  Google Scholar 

  27. Kuppusamy L, Mahendran A, Krishna SN (2011) Matrix insertion-deletion systems for bio-molecular structures. In: Natarajan R, Ojo AK (eds.) Distributed Computing and Internet Technology - 7th International Conference, ICDCIT 2011, Bhubaneshwar, India, February 9-12, 2011. Proceedings, Lecture Notes in Computer Science, Springer, Berlin

  28. Marcus S (1969) Contextual grammars. In: Third International Conference on Computational Linguistics, COLING 1969, Stockholm, Sweden, September 1-4, 1969

  29. Margenstern M, Păun Gh, Rogozhin Y, Verlan S (2005) Context-free insertion-deletion systems. Theor Comput Sci 330(2):339–348

    MathSciNet  Article  Google Scholar 

  30. Matveevici A, Rogozhin Y, Verlan S (2007) Insertion-deletion systems with one-sided contexts. In: Durand-Lose J O, Margenstern M (eds.) Machines, Computations, and Universality, 5th International Conference, MCU, LNCS, Springer, Berlin

  31. Minsky M (1967) Computations: finite and infinite machines. Prentice Hall, Englewood Cliffts, NJ

    MATH  Google Scholar 

  32. Motwani R, Panigrahy R, Saraswat V, Ventkatasubramanian S (2000) On the decidability of accessibility problems (extended abstract). In: Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, STOC, pp. 306–315. ACM

  33. Mutyam M, Krithivasan K, Reddy AS (2005) On characterizing recursively enumerable languages by insertion grammars. Fundam Inform 64(1–4):317–324

    MathSciNet  MATH  Google Scholar 

  34. Onodera K (2003) A note on homomorphic representation of recursively enumerable languages with insertion grammars. Transac Inf Process Soc Japan 44(5):1424–1427

    MathSciNet  Google Scholar 

  35. Păun G, Pérez-Jiménez MJ, Yokomori T (2008) Representations and characterizations of languages in chomsky hierarchy by means of insertion-deletion systems. Int J Found Comput Sci 19(4):859–871. https://doi.org/10.1142/S0129054108006005

    MathSciNet  Article  MATH  Google Scholar 

  36. Petre I, Verlan S (2012) Matrix insertion-deletion systems. Theor Comput Sci 456:80–88. https://doi.org/10.1016/j.tcs.2012.07.002

    MathSciNet  Article  MATH  Google Scholar 

  37. Potapov I, Prianychnykova O, Verlan S (2016) Insertion-deletion systems over relational words. In: K.G. Larsen, I. Potapov, J. Srba (eds.) Reachability Problems - 10th International Workshop, RP 2016, Aalborg, Denmark, September 19-21, 2016, Proceedings, Lecture Notes in Computer Science, Springer, Berlin

  38. Păun Gh (1985) A variant of random context grammars: semi-conditional grammars. Theor Comput Sci 41:1–17. https://doi.org/10.1016/0304-3975(85)90056-8

    Article  MATH  Google Scholar 

  39. Păun Gh, Rozenberg G, Salomaa A (1998) DNA computing: new computing paradigms. Springer, Berlin

    Book  Google Scholar 

  40. Ran FA, Hsu PD, Wright J, Agarwala V, Scott DA, Zhang F (2013) Genome engineering using the CRISPR-Cas9 system. Nat Protocols 8(11):2281–2308. https://doi.org/10.1038/nprot.2013.143

    Article  Google Scholar 

  41. Rozenberg G, Salomaa A (eds) (1997) Handbook of Formal Languages. Springer-Verlag, Berlin

  42. Takahara A, Yokomori T (2003) On the computational power of insertion-deletion systems. Nat Comput 2(4):321–336

    MathSciNet  Article  Google Scholar 

  43. Verlan S (2007) On minimal context-free insertion-deletion systems. J Automata, Lang Comb 12(1–2):317–328

    MathSciNet  MATH  Google Scholar 

  44. Verlan S (2010) Recent developments on insertion-deletion systems. Comput Sci J Moldova 18(2):210–245

    MathSciNet  MATH  Google Scholar 

  45. Wood D (1987) Theory of computation. Harper and Row, NewYork

    MATH  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments and suggestions.

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Correspondence to Sergey Verlan.

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Ivanov, S., Verlan, S. Single semi-contextual insertion-deletion systems. Nat Comput (2021). https://doi.org/10.1007/s11047-021-09861-3

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Keywords

  • Insertion-deletion
  • Forbidding grammars
  • Descriptional complexity
  • Computational completeness
  • Regular languages

Mathematics Subject Classification

  • 68Q42
  • 68Q45
  • 68Q05