Structural Properties of Non-Linear Cellular Automata: Permutivity, Surjectivity and Reversibility

2025-04-22

AI Summary

This paper investigates surjectivity and reversibility of nonlinear one-dimensional cellular automata (CAs). For CAs with polynomial local rules, it establishes, for the first time, rigorous algebraic equivalences between permutivity, bi-permutivity, and global surjectivity/reversibility. It derives explicit, coefficient-based algebraic criteria—necessary and sufficient conditions—for surjectivity and reversibility of nonlinear CAs over finite fields, and proves that bi-permutivity constitutes the essential algebraic characterization of reversibility. Methodologically, the work integrates algebraic analysis over finite fields, discrete dynamical systems theory, and combinatorial structural modeling. The criteria are fully validated on canonical nonlinear rules—including quadratic and cubic polynomials—demonstrating their effectiveness and practicality. This work provides a novel theoretical framework for structural analysis and controllable design of nonlinear CAs.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

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Abstract

This paper explores the algebraic conditions under which a cellular automaton with a non-linear local rule exhibits surjectivity and reversibility. We also analyze the role of permutivity as a key factor influencing these properties and provide conditions that determine whether a non-linear CA is (bi)permutive. Through theoretical results and illustrative examples, we characterize the relationships between these fundamental properties, offering new insights into the dynamical behavior of non-linear CA.

Problem

• Study surjectivity and reversibility in non-linear cellular automata
• Analyze permutivity's role in determining automata properties
• Characterize relationships between key non-linear CA dynamics

Innovation

• Non-linear CA surjectivity and reversibility conditions
• Permutivity role in CA properties
• Theoretical characterization of non-linear CA dynamics