Abstract
Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing k-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type \(AX^k=B\) for connected X. These results are some of the important final steps for solving more general polynomial equations on FDDS.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bernot, G., Comet, J.P., Richard, A., Chaves, M., Gouzé, J.L., Dayan, F.: Modeling and analysis of gene regulatory networks. In: Cazals, F., Kornprobst, P. (eds.) Modeling in Computational Biology and Biomedicine, pp. 47–80. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31208-3_2
Dennunzio, A., Dorigatti, V., Formenti, E., Manzoni, L., Porreca, A.E.: Polynomial equations over finite, discrete-time dynamical systems. In: Mauri, G., El Yacoubi, S., Dennunzio, A., Nishinari, K., Manzoni, L. (eds.) ACRI 2018. LNCS, vol. 11115, pp. 298–306. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99813-8_27
Dennunzio, A., Formenti, E., Margara, L., Montmirail, V., Riva, S.: Solving equations on discrete dynamical systems. In: Cazzaniga, P., Besozzi, D., Merelli, I., Manzoni, L. (eds.) CIBB 2019. LNCS, vol. 12313, pp. 119–132. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-63061-4_12
Dennunzio, A., Formenti, E., Margara, L., Riva, S.: An algorithmic pipeline for solving equations over discrete dynamical systems modelling hypothesis on real phenomena. J. Comput. Sci. 66, 101932 (2023)
Dennunzio, A., Formenti, E., Margara, L., Riva, S.: A note on solving basic equations over the semiring of functional digraphs. arXiv e-prints (2024). https://arxiv.org/abs/2402.16923
Doré, F., Formenti, E., Porreca, A.E., Riva, S.: Decomposition and factorisation of transients in functional graphs. Theoret. Comput. Sci. 999, 114514 (2024)
Ehrenfeucht, A., Rozenberg, G.: Reaction systems. Fund. Inform. 75, 263–280 (2007)
Formenti, E., Régin, J.-C., Riva, S.: MDDs boost equation solving on discrete dynamical systems. In: Stuckey, P.J. (ed.) CPAIOR 2021. LNCS, vol. 12735, pp. 196–213. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-78230-6_13
Gadouleau, M., Riis, S.: Graph-theoretical constructions for graph entropy and network coding based communications. IEEE Trans. Inf. Theory 57(10), 6703–6717 (2011)
Gershenson, C.: Introduction to random Boolean networks. arXiv e-prints (2004). https://doi.org/10.48550/arXiv.nlin/0408006
Goles, E., Martìnez, S.: Neural and Automata Networks: Dynamical Behavior and Applications. Kluwer Academic Publishers (1990)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. Discrete Mathematics and Its Applications, 2nd edn. CRC Press (2011)
Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs (preliminary report). In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 172–184 (1974)
Naquin, E., Gadouleau, M.: Factorisation in the semiring of finite dynamical systems. Theoret. Comput. Sci. 998, 114509 (2024)
Strozecki, Y.: Enumeration complexity: incremental time, delay and space. HDR thesis, Université de Versailles Saint-Quentin-en-Yvelines (2021)
Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42(3), 563–585 (1973)
Thomas, R., D’Ari, R.: Biological Feedback. CRC Press (1990)
Acknowledgment
SR was supported by the French Agence Nationale pour la Recherche (ANR) in the scope of the project “REBON” (grant number ANR-23-CE45-0008), and KP, AEP and MR by the EU project MSCA-SE-101131549 “ACANCOS”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 IFIP International Federation for Information Processing
About this paper
Cite this paper
Doré, F., Perrot, K., Porreca, A.E., Riva, S., Rolland, M. (2024). Roots in the Semiring of Finite Deterministic Dynamical Systems. In: Gadouleau, M., Castillo-Ramirez, A. (eds) Cellular Automata and Discrete Complex Systems. AUTOMATA 2024. Lecture Notes in Computer Science, vol 14782. Springer, Cham. https://doi.org/10.1007/978-3-031-65887-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-65887-7_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-65886-0
Online ISBN: 978-3-031-65887-7
eBook Packages: Computer ScienceComputer Science (R0)
