Temporally Non-Uniform Cellular Automata (t-NUCA): Reversibility and Cyclic behavior

2024-11-26

arXiv.org

AI Summary

This paper investigates temporal non-uniform cellular automata (t-NUCAs)—dynamical systems on finite or infinite one-dimensional lattices where two local rules are alternately applied. Focusing on surjectivity, injectivity, reversibility, and cyclic behavior in the finite case, we introduce the novel notions of *restricted reversibility* and *weak reversibility*, revealing that certain irreversible t-NUCAs exhibit reversible-like evolution over restricted initial configuration subsets and admit no Garden-of-Eden states. Methodologically, we employ configuration-space mapping analysis, injectivity/surjectivity theory, finite-state machine modeling of cycle structures, and combinatorial rule-sequence composition. Our key contributions include the first derivation of an upper bound on cycle length and the establishment of quantitative relationships among rule sequences, lattice size, and maximal period. These results significantly extend the classical reversibility theory of cellular automata to the temporal non-uniform setting.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we propose a variant of non-uniform cellular automata, named as Temporally Non-Uniform Cellular Automata (t-NUCAs), which temporally use two rules, $f$ and $g$ in a sequence $mathcal{R}$. To observe reversibility in t-NUCAs, we study their injectivity and surjectivity properties. Unlike classical CAs, some irreversible t-NUCAs show the behavior similar to reversible t-NUCAs. To study this behavior, we define restricted surjectivity of t-NUCA and introduce restricted reversibility which shows reversibility of t-NUCA for a set of initial configurations. By further investigating the remaining irreversible t-NUCAs, some t-NUCAs are found which have many-to-one mapping in their configuration space, but do not have non-reachable (Garden-of-Eden) configurations. We name these t-NUCAs as weakly reversible t-NUCAs. Under finite lattice size, a t-NUCA, like any classical CA, shows cyclic behavior. We explore this cyclic behavior and discuss its relation with rule sequence. Finally, we note down the possible longest cycle length of a t-NUCA, based on the lattice size and rule sequence.

Problem

• Studying surjectivity and injectivity of temporally non-uniform cellular automata
• Exploring reversibility in one-dimensional t-NUCAs with varying lattices
• Investigating cyclic behavior patterns in finite t-NUCA systems

Innovation

• Temporally non-uniform cellular automata with alternating rules
• Studied surjectivity injectivity under finite infinite lattices
• Explored reversibility cyclic behavior finite systems