Line Complexity Asymptotics of Polynomial Cellular Automata Published Nov. 2017 in The Ramanujan Journal (Springer)
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 \(c_0,c_n\neq~\!\!0\), 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 \(\lim_{k\to\infty}(\alpha(k)/k^2-~\!\!f(p^{-\langle\log_p k\rangle})) = 0\) (here \(\langle y\rangle=y-\lfloor y\rfloor\)). We use this result to show that for certain positive integer sequences \(s_k(x)\to\infty\) 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 \(f\).