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