# Line complexity asymptotics of polynomial cellular automata

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## Abstract

Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols are integers modulo some prime *p*. We consider the asymptotic behavior of the *line complexity sequence* \(a_T(k)\), which counts, for each *k*, the number of coefficient strings of length *k* that occur in the automaton. We begin with the modulo 2 case. For a polynomial \(T(x)=c_0+c_1x+\dots +c_nx^n\) with \(c_0,c_n\ne ~0\), we construct *odd* and *even* parts of the polynomial from the strings \(0c_1c_3c_5\cdots c_{1+2\lfloor (n-1)/2\rfloor }\) and \(c_0c_2c_4\cdots c_{2\lfloor n/2\rfloor }\), respectively. We prove that \(a_T(k)\) satisfies recursions of a specific form if the odd and even parts of *T* are relatively prime. We also define the *order* of such a recursion and show that the property of “having a recursion of some order” is preserved when the transition rule is raised to a positive integer power. Extending to a more general setting, we consider an abstract generating function \(\phi (z)=\sum _{k=1}^\infty \alpha (k)z^k\) which satisfies a functional equation relating \(\phi (z)\) and \(\phi (z^p)\). We show that there is a continuous, piecewise quadratic function *f* on [1 / *p*, 1] for which \(\lim _{k\rightarrow \infty }(\alpha (k)/k^2-~f(p^{-\langle \log _p k\rangle })) = 0\) (here \(\langle y\rangle =y-\lfloor y\rfloor \)). We use this result to show that for certain positive integer sequences \(s_k(x)\rightarrow \infty \) with a parameter \(x\in [1/p,1]\), the ratio \(\alpha (s_k(x))/s_k(x)^2\) tends to *f*(*x*), and that the limit superior and inferior of \(\alpha (k)/k^2\) are given by the extremal values of *f*.

## Keywords

Cellular automata Line complexity Additive transition rules Asymptotic estimates## Mathematics Subject Classification

05A15 05A16## Notes

### Acknowledgements

I would like to thank Mr. Chiheon Kim for mentoring this project and for providing many helpful insights and suggestions. I would like to thank Prof. Pavel Etingof for suggesting this project, and Prof. Richard Stanley for suggesting the original topic. I would also like to thank the Center for Excellence in Education, the Massachusetts Institute of Technology, and the MIT Math Department for making RSI possible. I would like to thank Mr. Antoni Rangachev and Dr. Tanya Khovanova for their advice regarding this paper. I would also like to thank my sponsors, Mr. Steven Ferrucci of the American Mathematical Society, Dr. Donald McClure of the American Mathematical Society, Mr. and Mrs. Raymond C. Kubacki, Mr. Piotr Mitros, Mr. and Mrs. Steven Scott, Prof. Tom Leighton of Akamai Technologies, and Prof. Bonnie Berger of MIT. Finally, I would like to thank the MIT Math Department, the UROP+ program, and the Class of 1994 UROP fund for making possible the continuation of this project in the summer of 2015.

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