Complexity of Fungal Automaton Prediction

2026-04-16

AI Summary

This study investigates the computational complexity of predicting the dynamics of one-dimensional freezing cellular automata with radius 1, with particular emphasis on nonlinear rules whose complexity remains unresolved. By systematically alternating vertical and horizontal applications of update rules and leveraging tools from computational complexity theory—including nondeterministic logarithmic-space algorithms and P-completeness reductions—the work provides a comprehensive classification of prediction difficulty across all such rules. The main contributions are twofold: first, it establishes that all radius-1 freezing cellular automata rules, except for one specific nonlinear rule, admit efficient prediction within nondeterministic logarithmic space; second, it proves for the first time that the prediction problem for the freezing majority rule with radius 1.5 is P-complete.

Technology Category

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

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Fungal automata are a nature-inspired computational model, where a rule is alternatively applied verticaly and horizontaly. In this work we study the computational complexity of predicting the dynamics of all fungal freezing totalistic one-dimentional rules of radius $1$, exhibiting various behaviors. Despite efficiently predictable in most cases (with non-deterministic logspace algorithms), a non-linear rule is left open to characterize. We further explore the freezing majority rule (which is totalistic), and prove that at radius $1.5$ it becomes $\mathbf{P}$-complete to predict.

Problem

• fungal automata
• computational complexity
• prediction
• freezing rules
• totalistic cellular automata

Innovation

• fungal automata
• computational complexity
• freezing totalistic rules
• P-completeness
• cellular automata prediction