Pseudo-inversion: closure properties and decidability

Abstract

We consider a pseudo-inversion operation inspired by biological events, such as DNA sequence transformations, where only parts of a string are reversed. We define the pseudo-inversion of a string \(w = uxv\) to be the set of all strings \(v^Rxu^R\), where \(uv \ne \lambda \) and consider the operation from a formal language theoretic viewpoint. We show that regular languages are closed under the pseudo-inversion operation whereas context-free languages are not. Furthermore, we study the iterated pseudo-inversion operation and show that the iterated pseudo-inversion of a context-free language is recognized by a nondeterministic reversal-bounded multicounter machine. Finally, we introduce the notion of pseudo-inversion-freeness and examine closure properties and decidability problems for regular and context-free languages. We demonstrate that pseudo-inversion-freeness is decidable in polynomial time for regular languages and undecidable for context-free languages.

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Notes

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    Unrestricted two-counter machines accept all recursively enumerable languages (Ginsburg 1975).

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Acknowledgments

We wish to thank the referees for the careful reading of the paper and many valuable suggestions. Cho, Han, Kang and Ko were supported by the Basic Science Research Program through NRF funded by MEST (2012R1A1A2044562), the International Cooperation Program managed by NRF of Korea (2014K2A1A2048512) and Yonsei University Future-leading Research Initiative of 2014, Kim was supported by NRF-2013-Global Ph.D. Fellowship Program and Salomaa was supported by the Natural Sciences and Engineering Research Council of Canada Grant OGP0147224.

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Correspondence to Yo-Sub Han.

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A preliminary version appeared in Proceedings of Unconventional Computation & Natural Computation 2014, UCNC 2014, LNCS, vol. 8553, Springer-Verlag, 2014, pp. 93–104.

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Cho, DJ., Han, YS., Kang, SD. et al. Pseudo-inversion: closure properties and decidability. Nat Comput 15, 31–39 (2016). https://doi.org/10.1007/s11047-015-9502-9

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Keywords

  • Pseudo-inversion
  • Bio-inspired operation
  • Closure properties
  • Decidability
  • Formal languages
  • Reversal-bounded multicounter machines