Freezing Majority Cellular Automata

• Freezing majority cellular automata are irreversible discrete systems where cells update by a local majority rule and permanently fix in state +1.
• The study details how neighborhood geometry, such as the Toom L-shaped pattern, shapes global dynamics and computational complexity outcomes.
• Analyses reveal that restricted connectivity allows efficient parallel algorithms while expanded neighborhoods induce P-completeness, impacting consensus and distributed computation.

Freezing majority cellular automata are a class of irreversible discrete dynamical systems where each cell updates according to a local majority rule, and crucially, once a cell reaches a designated "frozen" state—typically corresponding to the majority—it remains fixed in that state forever. The transition rules, neighborhood geometry, and freezing dynamics induce rich global behaviors and sharp computational complexity dichotomies, with links to bootstrap percolation, consensus dynamics, and circuit evaluation.

1. Formal Definition and Dynamics

The core objects are two-dimensional cellular automata on a periodic grid (torus), where each cell $u=(i,j)$ has state $x_u \in \{-1, +1\}$, updated in discrete time. The freezing majority rule is defined as:

$$F(x)_u = \begin{cases} +1 & \text{if } x_u = +1 \text{ or } \sum_{v \in N(u)} x_v > 0 \ -1 & \text{otherwise} \end{cases}$$

Here, $N(u)$ is the set of neighboring cells—most commonly defined by geometric patterns such as von Neumann, Moore, or L-shaped arrangements. The freezing condition enforces that once $x_u = +1$, it is invariant: $\forall t > t^* : x_u^{(t)} = +1$. For cells in state $-1$, the update is irreversible: a transition to $+1$ cannot be undone.

The neighborhood geometry has profound consequences. The simplest "Toom" L-shaped neighborhood (north and east neighbors) is minimal; more general L-shaped neighborhoods (multiple consecutive or spaced sites in north and east) or other neighborhood patterns yield different computational structures (Concha-Vega et al., 19 Sep 2025).

2. Neighborhood Structure and Freezing Dynamics

An L-shaped neighborhood is algebraically defined:

$$N(i,j) = \{ (i, j+k) : k \in S_N \} \cup \{ (i+k, j) : k \in S_E \}$$

For $S_N = S_E = \{1\}$, the neighborhood comprises immediate north and east neighbors. When the neighborhood is enlarged (e.g., $|S_N| = |S_E| = 2$), or when neighbors are non-contiguous, the topology of local interactions and the propagation of "freezing" events change, affecting both local and global stabilization.

The freezing mechanism means the automaton always evolves toward an absorbing state: as time progresses, more cells enter and remain in state $+1$; no cell can switch back to $-1$. This leads to an eventual "frozen" pattern throughout the lattice unless blocked by local constraints.

3. Prediction Problem and Computational Complexity

The central decision problem is: given initial configuration $x$, a cell $v$, and time $t$, does $F^t(x)_v \neq x_v$? The answer depends on the neighborhood:

• Toom Neighborhood ($|S_N| = |S_E| = 1$): Prediction is in $\mathsf{NC}$, i.e., it is efficiently decidable in polylogarithmic parallel time (Concha-Vega et al., 19 Sep 2025). The essential method is to construct a dependency digraph $G_x$ on cells in state $-1$; a cell remains $-1$ iff it is on or reachable to a cycle in $G_x$. Determining this can be parallelized via matrix powering and prefix sum techniques over the $O(n^2)$ grid.
• Larger/Noncontiguous L-shapes ($|S_N|,|S_E|\geq 2$): Prediction becomes $\mathsf{P}$-complete, signifying inherent sequentiality. Reductions show that for such neighborhoods, the freezing majority rule can simulate the monotone circuit value problem, classically $\mathsf{P}$-complete.
• General L-shaped neighborhoods: The complexity transition is sharp. All simple L-shaped neighborhoods of size two remain in $\mathsf{NC}$, but any generalization can cross into $\mathsf{P}$-completeness.

This complexity dichotomy is summarized in the following table:

Neighborhood Type	Prediction Complexity
Toom L-shape ($|S_N|=1$, $|S_E|=1$)	$\mathsf{NC}$
Any $|S_N|,|S_E| > 1$	$\mathsf{P}$-complete
Noncontiguous, $|S_N|,|S_E|=2$	$\mathsf{P}$-complete

4. Structural Mechanisms and Graph-Theoretic Insights

The paper introduces a dependency digraph $G_x$:

• Vertices: positions $(i, j)$ with $x_{i,j} = -1$
• Edges: a directed edge from $(i, j)$ to any of its L-shaped neighbors in state $-1$

Key lemma: a cell remains at $-1$ iff it can reach a cycle in $G_x$ via a directed path. The dynamics reduce to graph reachability and cycle-detection, enabling efficient parallel algorithms when the digraph is sparse (as in the Toom case) but growing in complexity for denser neighborhoods.

Matrix powering (computing $A^k$ of the adjacency matrix) and the parallel prefix sum are explicitly used to test for cycles and reachability, foundational for $\mathsf{NC}$ algorithms (Concha-Vega et al., 19 Sep 2025).

5. Implications for Distributed Dynamics and Consensus

Freezing majority automata are prototypical models for irreversible consensus and percolation-type phenomena. The freezing majority rule directly models situations where an adopted decision (state $+1$) becomes permanent—mirroring irreversible infection, adoption, or agreement processes.

• Consensus Models: Predicting whether consensus is eventually reached equates to determining if all cells will freeze to $+1$. This is tractable in simple L-shaped cases but generally computationally hard when local dependency structures are richer.
• Distributed Computation: The connection to circuit value problems demonstrates that freezing majority automata can, in principle, compute arbitrary monotone, planar circuit values, illuminating their computational universality in sufficiently general geometries.
• Parallel Computation: The presence of $\mathsf{NC}$ algorithms for simple neighborhoods implies efficient parallelization is attainable only for networks with limited local dependency, crucial for applications like fault-tolerant distributed protocols, error-correcting codes, and decentralized voting procedures.

6. Extensions, Generalizations, and Open Questions

The paper generalizes beyond nearest-neighbor L-shapes, exploring contiguous and noncontiguous variants and their effect on complexity. Open questions highlighted include:

• What is the exact complexity for L-shapes with $|S_E| > 1$ and $|S_N| = 1$?
• How do other asymmetric neighborhood patterns (T-shaped, etc.) or modified update rules affect both the freezing paradigm and computational tractability?
• Can the same techniques be extended to asynchronous update schemes or higher dimensions, which are of interest in modeling realistic distributed processes?

A plausible implication is that as the effective local dependency "width" increases, the system's global prediction and stabilization become less amenable to parallel computation and more reflective of the computational universality and intractability of the underlying circuit class.

7. Theoretical and Practical Significance

The detailed complexity classifications in freezing majority cellular automata demonstrate how subtle changes in local interaction and freezing rules lead to drastic shifts in global predictability and computability. This suggests that “cold dynamics” models—theoretical frameworks where all orbits stabilize due to irreversible local rule application—are powerful abstractions for exploring the interface of local-to-global emergence, computational complexity, and irreversible consensus in both artificial and natural systems. The insights provided extend to applications in statistical physics (irreversible percolation, Ising-like models), distributed computing (fault-tolerant consensus), and theoretical biology (irreversible developmental processes).

This synthesis reflects the main contributions and findings of (Concha-Vega et al., 19 Sep 2025) for the freezing majority rule with L-shaped neighborhoods, placing it in context with broader freezing CA literature.