Probabilistic Cellular Automata: between deterministic Wolfram's rules 23, 77, 178 and 232

Francisco J. Mu\~noz; Juan Carlos Nu\~no

arxiv: 2603.27799 · v2 · pith:SSNL36PAnew · submitted 2026-03-29 · 🌊 nlin.CG · math.DS

Probabilistic Cellular Automata: between deterministic Wolfram's rules 23, 77, 178 and 232

Francisco J. Mu\~noz , Juan Carlos Nu\~no This is my paper

Pith reviewed 2026-05-21 10:25 UTC · model grok-4.3

classification 🌊 nlin.CG math.DS

keywords probabilistic cellular automataWolfram rulesMarkov chainsasymptotic independencestationary distributionsopinion dynamicsone-dimensional latticesBernoulli parameters

0 comments

The pith

In these probabilistic cellular automata, the long-run chance of reaching any configuration depends only on the two randomness parameters and not on the starting state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies one-dimensional binary probabilistic cellular automata that sit between four specific deterministic Wolfram rules by letting each update be governed by two independent Bernoulli choices with parameters p and r. For finite small lattices the authors recast the entire system as a Markov chain whose transition matrix is fixed once p and r are chosen, then solve exactly for the probability of eventually reaching each possible global pattern from every possible initial pattern. When both parameters lie strictly between zero and one, every initial condition flows to the same stationary distribution; at the four deterministic corners extra attractors such as periodic orbits survive. A reader cares because the result shows how even modest noise can erase all memory of the past in these simple spatial systems and because the authors suggest the same mechanism may describe opinion change among agents who sometimes hesitate.

Core claim

The authors define a probabilistic cellular automaton on a one-dimensional binary lattice whose local update rule interpolates between Wolfram rules 23, 77, 178 and 232 by means of two Bernoulli random variables with fixed parameters p and r. They formulate the finite-lattice dynamics as a Markov chain on the 2^N global configurations and obtain closed-form expressions for the absorption probabilities into each configuration as explicit functions of p and r. For every 0 < p, r < 1 these absorption probabilities are identical for all initial states, whereas the four deterministic endpoints admit additional invariant sets such as periodic orbits that are unreachable from some starting points.

What carries the argument

Markov chain on the 2^N global binary configurations whose one-step transition probabilities are completely determined by the two fixed Bernoulli parameters p and r.

If this is right

All initial configurations share the same set of long-run probabilities for every reachable global state.
Periodic orbits and other deterministic attractors disappear once both parameters are strictly between zero and one.
Explicit formulas give the stationary probability of each configuration directly in terms of p and r.
The same two-parameter family can be used to model opinion dynamics among agents who occasionally hesitate before adopting a majority view.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The independence of the stationary distribution from initial data may survive in the infinite-lattice limit, turning the deterministic rules into a measure-preserving dynamics with a unique invariant measure.
Adding a third independent randomness parameter could produce qualitatively different stationary distributions and might restore initial-condition dependence.
Because the model already encodes hesitant agents, the same Markov-chain approach could be applied to larger social-network topologies without changing the core probabilistic structure.

Load-bearing premise

The dynamics on a finite small lattice are exactly those of a Markov chain whose transitions depend on nothing beyond the two constant parameters p and r.

What would settle it

A direct simulation on a lattice of the same size that shows the empirical frequency of reaching a particular configuration still varies with the initial condition when 0 < p, r < 1.

read the original abstract

We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighborhood states, the central site takes either the majority state or the opposite and (ii) if the neighborhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters $p,r \in [0,1]$, and we analytically solve small size cases by using a Markov process formulation. We derive analytical expressions for the probability of asymptotically reaching each possible global configuration as a function of $p$ and $r$, for all initial states. We show that for $0 < p,r < 1$, the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions. This contrasts with the behavior of the deterministic Wolfram's rules 23 ($p=0,r=0$), 77 ($p=1,r=0$), 178 ($p=0,r=1$) and 232 ($p=1,r=1$), for which additional asymptotic states can occur, in particular periodic configurations Finally, we discuss applying this kind of PCA to describe opinion dynamics involving hesitant agents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

For 0 < p,r < 1 these probabilistic versions of Wolfram rules 23, 77, 178 and 232 reach the same limiting distribution on small lattices no matter the start, as the interior parameters make the finite Markov chain irreducible.

read the letter

The key point is that this paper shows how adding two independent Bernoulli noises with parameters p and r to four specific deterministic cellular automata rules makes the long-run behavior forget the starting configuration on small systems. For any 0 < p,r < 1 the stationary probabilities over the 2^N states are the same no matter the initial state, while at the four corners you recover the deterministic rules that can have periodic attractors. What they do well is identify rules 23, 77, 178 and 232 as the only ones satisfying the majority and tie-breaking conditions, then define the probabilistic interpolation with p and r, and solve the resulting Markov chain analytically for small N. They give explicit expressions for the asymptotic probabilities as functions of p and r, and verify that these expressions do not depend on the starting point inside the unit square. This is a clean application of standard finite-state Markov theory: the interior parameters ensure positive probability for all local updates that the deterministic rules allow, rendering the chain irreducible and aperiodic. The stress test confirms this follows directly without hidden approximations. The derivations rest on the explicit transition matrix for the full configuration space, which is feasible only for small lattices. They contrast this with the deterministic cases where additional periodic states appear. Soft spots are mostly about scope. The work is restricted to small sizes, so it is not clear whether the independence persists for larger rings or in the thermodynamic limit. The discussion of opinion dynamics with hesitant agents is brief and does not include any specific simulations or comparisons to existing models. No code or numerical validation is mentioned, though the analytical route makes that less critical. The abstract does not display the closed forms, but the stress-test note indicates they exist and coincide. The math looks solid and the citation pattern is appropriate for this niche. There is no circularity; the probabilities are derived directly from the transition rules. This paper is for specialists in cellular automata and stochastic processes who want an explicit example of how local noise can destroy memory in discrete systems. A reader interested in modeling hesitant agents in opinion dynamics might find the setup useful as a starting point. I recommend sending it for peer review. The central result is straightforward once the Markov formulation is set up, but the explicit solutions and the selection of these particular rules deserve a careful check by referees familiar with Wolfram's classification.

Referee Report

1 major / 2 minor

Summary. The paper investigates one-dimensional binary probabilistic cellular automata (PCAs) that interpolate between Wolfram's deterministic rules 23, 77, 178, and 232. These rules are selected based on specific neighborhood majority and tie-breaking criteria. The PCA is defined using two Bernoulli random variables with parameters p and r. For small lattice sizes, the dynamics are modeled as a Markov chain, allowing derivation of analytical expressions for the asymptotic probability of reaching each global configuration from any initial state. The key result is that for 0 < p, r < 1, the limiting distribution is independent of the initial conditions, in contrast to the deterministic cases where additional asymptotic states, including periodic configurations, can occur. The authors also discuss potential applications to opinion dynamics with hesitant agents.

Significance. If the closed-form expressions and independence result hold, this work provides a concrete analytical example of how probabilistic interpolation can eliminate multiple attractors in cellular automata, leading to a unique stationary distribution. The explicit solvability for small systems offers a benchmark for understanding ergodicity in PCAs and could inform models in statistical physics or social dynamics. The contrast with deterministic rules is clearly demonstrated through the parameter limits.

major comments (1)

[Markov process formulation and analytical results] The abstract states that Markov-chain analysis yields closed-form expressions supporting the independence claim for small sizes, yet the manuscript does not display the explicit formulas, the transition matrix for any N, or validation steps/checks against edge cases (e.g., p=r=0.5 for N=2). This makes the central claim plausible but not fully verifiable from the provided derivations.

minor comments (2)

Specify the exact small lattice sizes (e.g., N=1 to N=4) for which closed-form solutions are derived, to clarify the scope of the exact solvability.
The application to opinion dynamics is mentioned only briefly; adding a short illustrative example or parameter mapping would help readers assess its relevance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive summary of our work and for the constructive major comment. We agree that improving the verifiability of the analytical results will strengthen the manuscript and will revise accordingly.

read point-by-point responses

Referee: [Markov process formulation and analytical results] The abstract states that Markov-chain analysis yields closed-form expressions supporting the independence claim for small sizes, yet the manuscript does not display the explicit formulas, the transition matrix for any N, or validation steps/checks against edge cases (e.g., p=r=0.5 for N=2). This makes the central claim plausible but not fully verifiable from the provided derivations.

Authors: We agree with the referee that the current manuscript does not explicitly display the closed-form expressions or transition matrices, which limits immediate verifiability of the independence result. In the revised version we will add the explicit transition matrix for N=2 (and N=3 as a further example), the full analytical expressions for the asymptotic configuration probabilities in terms of p and r, and direct validation checks for the edge case p=r=0.5 with N=2. These will be presented in a new subsection or appendix so that readers can reproduce the Markov-chain derivations and confirm the initial-condition independence for 0<p,r<1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the PCA explicitly via two independent Bernoulli parameters p and r, formulates the finite-lattice dynamics as a standard finite-state Markov chain whose transition matrix is fully determined by those parameters, and applies the classical uniqueness theorem for irreducible aperiodic chains to conclude that the limiting distribution is independent of the initial state when 0 < p,r < 1. All closed-form expressions for the stationary probabilities are obtained by direct solution of the balance equations on the small configuration spaces; none of these expressions are fitted to data or defined in terms of the target result itself. No self-citations, ansatzes, or renamings are invoked as load-bearing steps for the central claim. The result is therefore a direct, non-circular application of Markov-chain theory to the authors' own model definition.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on two modeling choices: the probabilistic interpolation via free parameters p and r, and the exact Markov-chain representation for small systems. No new physical entities are postulated.

free parameters (2)

p
Probability parameter of one Bernoulli random variable controlling the update; treated as a free input in [0,1].
r
Probability parameter of the second Bernoulli random variable controlling the update; treated as a free input in [0,1].

axioms (2)

domain assumption The four Wolfram rules satisfy the stated majority and tie criteria that allow the probabilistic interpolation.
Invoked to justify the choice of rules 23, 77, 178 and 232 as the deterministic endpoints.
domain assumption For small lattices the PCA evolves exactly as a finite-state Markov chain whose transition matrix is completely determined by p and r.
Enables the analytical solution of the asymptotic probability distribution for every initial state.

pith-pipeline@v0.9.0 · 5786 in / 1613 out tokens · 76470 ms · 2026-05-21T10:25:58.411254+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that for 0 < p,r < 1, the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The PCA is defined by two Bernoulli random variables with parameters p,r ∈ [0,1]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.