Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences

2026-02-22

AI Summary

This study resolves the long-standing open problem of classifying the computational complexity of the prediction problem for fungal automata under arbitrary update sequences involving both horizontal (H) and vertical (V) directions. By embedding arbitrary Boolean circuits into fungal automata with mixed H/V update schedules, the authors prove that the prediction problem is P-complete whenever the update sequence contains at least one H and one V update. This result provides the first complete characterization of the complexity boundary across all such hybrid update sequences, unifying and generalizing prior findings restricted to specific patterns—such as alternating HV or periodic H⁴V⁴ schedules—and thereby filling a critical gap in the complexity-theoretic understanding of this model.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• 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
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

The sandpile automata of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett., 1987) are a simple model for the diffusion of particles in space. A fundamental problem related to the complexity of the model is predicting its evolution in the parallel setting. Despite decades of effort, a classification of this problem for two-dimensional sandpile automata remains outstanding. Fungal automata were recently proposed by Goles et al. (Phys. Lett. A, 2020) as a spin-off of the model in which diffusion occurs either in horizontal $(H)$ or vertical $(V)$ directions according to a so-called update scheme. Goles et al. proved that the prediction problem for this model with the update scheme $H^4V^4$ is $\textbf{P}$-complete. This result was subsequently improved by Modanese and Worsch (Algorithmica, 2024), who showed the problem is $\textbf{P}$-complete also for the simpler updatenscheme $HV$. In this work, we fill in the gaps and prove that the prediction problem is $\textbf{P}$-complete for any update scheme that contains both $H$ and $V$ at least once.

Problem

• fungal automata
• prediction problem
• P-completeness
• update scheme
• Boolean circuits

Innovation

• fungal automata
• P-completeness
• update scheme
• Boolean circuits
• computational complexity