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A topology for P-systems with active membranes

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Abstract

This paper proposes a study of deterministic P systems with active membranes in the context of discrete time dynamical systems. First of all, we prove that, for a fixed set of objects and labels, the set of all P system configuration is countable and that the dynamical behaviors defining a chaotic system are not possible. Then, we define a notion of distance between membrane configurations encoding the intuitive concept of “dissimilarity” between configurations. We prove that all functions defined by evolution, communication, and division rules are continuous under that distance and that the resulting topological space is discrete but not complete. Furthermore, we adapt in a natural way the classical notions of sensitivity to initial conditions and topological transitivity to P systems, and we show that P systems exhibiting those new properties exist. Finally, we prove that the proposed distance is efficiently computable, i.e., its computation only requires polynomial time with respect to the size of the input configurations.

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Data Structure 1:
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Algorithm 1
Algorithm 2

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Notes

  1. Even when dissolution rules are considered, they only allow the membrane structure to decrease in depth.

  2. To avoid the problem of multiple representations for the same multiset, we consider as a representation the smaller string in lexicographic order.

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Acknowledgements

This work was partially supported by PRIN PNRR P2022MPFRT CASCA (“Cellular Automata Synthesis for Cryptography Applications”) and by MUR under the grant “Dipartimenti di Eccellenza 2023-2027” of the Department of Informatics, Systems and Communication of the University of Milano-Bicocca, Italy.

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Authors and Affiliations

  1. Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Viale Sarca 336, 20126, Milan, Italy

    Alberto Dennunzio

  2. Université Côte d’Azur, CNR,S I3S, Nice, France

    Enrico Formenti

  3. Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste, Via Alfonso Valerio 12/1, 34127, Trieste, Italy

    Luca Manzoni & Giuliamaria Menara

  4. Department of Computer Science and Engineering, University of Bologna, Cesena Campus, Via Sacchi 3, Cesena, Italy

    Luciano Margara

Authors
  1. Alberto Dennunzio
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  2. Enrico Formenti
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  3. Luca Manzoni
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  4. Luciano Margara
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Correspondence to Alberto Dennunzio.

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Dennunzio, A., Formenti, E., Manzoni, L. et al. A topology for P-systems with active membranes. J Membr Comput 5, 193–204 (2023). https://doi.org/10.1007/s41965-023-00132-x

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  • DOI: https://doi.org/10.1007/s41965-023-00132-x

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