Abstract
We introduce an algebraic approach for the analysis and composition of finite, discrete-time dynamical systems based on the category-theoretical operations of product and sum (coproduct). This allows us to define a semiring structure over the set of dynamical systems (modulo isomorphism) and, consequently, to express many decomposition problems in terms of polynomial equations. We prove that these equations are, in general, algorithmically unsolvable, but we identify a solvable subclass. Finally, we describe an implementation of the semiring operations for the case of finite cellular automata.
Enrico Formenti has been partially supported by PACA APEX FRI project. Luca Manzoni was partially supported by “Premio giovani talenti 2017” of Università degli Studi di Milano-Bicocca and Accademia dei Lincei.
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Notes
- 1.
While the existence of integer roots of a polynomial in \(\mathbb {Z}[X]\) is undecidable, the existence of roots in the larger ring of real numbers is decidable, and the problem becomes even trivial for complex roots (due to the fundamental theorem of algebra).
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Dennunzio, A., Dorigatti, V., Formenti, E., Manzoni, L., Porreca, A.E. (2018). Polynomial Equations over Finite, Discrete-Time Dynamical Systems. In: Mauri, G., El Yacoubi, S., Dennunzio, A., Nishinari, K., Manzoni, L. (eds) Cellular Automata. ACRI 2018. Lecture Notes in Computer Science(), vol 11115. Springer, Cham. https://doi.org/10.1007/978-3-319-99813-8_27
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DOI: https://doi.org/10.1007/978-3-319-99813-8_27
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