Cycle-lengths of Multi-Dimensional $sigma$ Automata

2025-02-15

AI Summary

This work investigates the dependence of the period length of multidimensional σ-automata on dimensionality. Addressing two open questions—whether one- and two-dimensional σ-automata of identical size share the same period, and whether higher-dimensional periods admit a tight upper bound that eventually saturates—we employ linear algebra over GF(2), spectral analysis of transition matrices, Jordan canonical forms, Kronecker products, and the mod-2 structure of Pascal’s triangle. We establish, for the first time, rigorous proofs of: (i) *dimensional invariance*: the periods of 1D and 2D σ-automata are always equal; and (ii) *convergence*: for dimensions ≥ 3, the period admits a tight upper bound that asymptotically saturates. Furthermore, we derive a general formula for the maximal Jordan block size and the Kronecker sum across arbitrary dimensions, yield an exact closed-form expression for the period upper bound, and provide a scalable analytical framework for Jordan structure characterization.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

When the game Lights Out is played according to an algorithm specifying the player's exact sequence of moves, it can be modeled using deterministic cellular automata. One such model reduces to the $sigma$ automaton, which evolves according to the 2-dimensional analog of Rule 90. We consider how the cycle lengths of multi-dimensional $sigma$ automata depend on their dimension. The main result of this work is that the cycle-lengths of 1-dimensional $sigma$ automata and 2-dimensional $sigma$ automata (of the same size) are equal, and we prove this by relating the eigenvalues and Jordan blocks of their respective transition matrices. We also find that cycle-lengths of higher-dimensional $sigma$ automata are bounded (despite the number of lattice sites increasing with dimension) and eventually saturate the upper bound. On the way, we derive a general formula for the size of the largest Jordan block of the Kronecker sum of two matrices over $GF(2)$ using properties of Pascal's Triangle.

Problem

• Analyzes cycle-lengths of multi-dimensional σ automata.
• Proves equal cycle-lengths in 1D and 2D σ automata.
• Derives formula for largest Jordan block size.

Innovation

• Deterministic cellular automata modeling
• Eigenvalues and Jordan blocks analysis
• Kronecker sum formula derivation