Efficient Algorithms for Injectivity and Bounded Surjectivity of One-dimensional Nonlinear Cellular Automata
Chen Wang, Junchi Ma, Defu Lin, Weilin Chen, Chao Wang
TL;DR
The paper addresses the problem of deciding injectivity and surjectivity in nonlinear one-dimensional cellular automata (CAs), where Amoroso–Patt algorithms are prohibitively costly for complex rules. It introduces a streamlined surjectivity decision that extends to fixed, periodic, and reflective boundaries, and presents a new injectivity tree algorithm with quadratic time complexity $O(s^{2m})$ that avoids hash tables. The authors prove new injectivity theorems and demonstrate both theoretical complexity gains and substantial empirical speedups over prior methods, including favorable memory usage. These results enable efficient analysis of larger and more complex CAs, opening up practical applications in simulations, cryptography, and beyond.
Abstract
Nonlinear cellular automata are extensively used in simulations, image processing, cryptography, and so on. The determination of their fundamental properties, injectivity and surjectivity, related to information loss during the evolution, is necessary in various applications. Currently, people still use Amoroso's algorithms for injectivity and surjectivity determinations, but this incurs significant computational costs when applied to complex nonlinear cellular automata. We have optimized Amoroso's surjectivity algorithm, improving its operational efficiency greatly and extended its applicability to various boundaries. Furthermore, we have introduced new theorems and algorithms for determining injectivity, which offer substantial improvements over Amoroso's algorithm in both time and space. With these new algorithms, we are equipped to determine the properties of larger and more complex cellular automata, thereby employing more advanced cellular automata to achieve increasingly complex functionalities.