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arxiv: 2606.13159 · v1 · pith:A2C67V52new · submitted 2026-06-11 · 💻 cs.DM · math.CO· math.DS

The Curious Case of Reversible Elementary Second Order Cellular Automaton 115

Enrico Formenti (Universit\'e C\^ote d'Azur , France) , Supreeti Kamylia (Department of Computer Science , Engineering , Birla Institute of Technology , Mesra , Ranchi , India) This is my paper

Pith reviewed 2026-06-27 05:06 UTC · model grok-4.3

classification 💻 cs.DM math.COmath.DS
keywords cellular automatareversible automataelementary cellular automatasecond-order automataperiodicityfinite configurationsrule 115
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The pith

Reversible elementary second-order cellular automaton rule 115 produces periodic orbits from any finite initial configuration.

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

The paper establishes that rule 115 of reversible elementary second-order cellular automata returns every finite starting pattern to its exact initial state after a finite number of steps. Because the rule is reversible, each configuration at time t can be recovered from the pair at t and t+1, which forces the orbit to close rather than wander indefinitely. The authors further examine selected families of finite patterns and derive or describe the functions that give their periods. A reader cares because the result confines the dynamics of bounded patterns to cycles even though the underlying rule supports both forward and backward evolution.

Core claim

We prove that the reversible elementary second order cellular automaton rule 115 is periodic when started on finite initial configurations. We also study some families of finite configurations that have interesting period functions.

What carries the argument

The reversible second-order update rule numbered 115, which computes each new cell state from the two preceding time steps and permits exact recovery of the prior configuration.

If this is right

  • Every finite configuration evolves in a cycle whose length is finite.
  • Certain families of finite configurations admit explicit or describable period functions.
  • Reversibility together with the specific rule structure precludes non-periodic trajectories on finite supports.
  • The state trajectory on any finite support must eventually return to the starting pair of consecutive configurations.

Where Pith is reading between the lines

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

  • The finite configuration space partitions into disjoint cycles under the rule.
  • Periodicity results for other reversible elementary rules could be checked by the same technique.
  • Small-grid simulations supply an immediate computational test of the periodicity statement.
  • The guarantee of return could be used to design bounded reversible computing elements that reset after a known number of steps.

Load-bearing premise

Finite initial configurations remain well-defined under the chosen neighborhood and that rule 115 is already known to be reversible from prior definitions of second-order elementary automata.

What would settle it

A concrete finite binary string or grid that evolves under repeated application of rule 115 without ever repeating a state would disprove the claim.

read the original abstract

We prove that the reversible elementary second order cellular automaton rule 115 is periodic when started on finite initial configurations. We also study some families of finite configurations that have interesting period functions.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove that the reversible elementary second-order cellular automaton rule 115 produces periodic orbits on all finite-support initial configurations. It additionally examines period functions for selected families of such finite configurations.

Significance. If the asserted proof is valid and complete, the result would establish a concrete periodicity property for this specific reversible second-order rule under finite support, which is of interest for the dynamics of reversible cellular automata in discrete mathematics. The additional study of period functions for example families could provide concrete data points for further analysis of orbit lengths.

major comments (1)
  1. [Abstract] Abstract: the central claim is that a proof of periodicity exists, yet no derivation steps, lemmas, explicit argument, or verification details are supplied anywhere in the manuscript. This absence is load-bearing because the soundness of the periodicity assertion cannot be assessed without the argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is that a proof of periodicity exists, yet no derivation steps, lemmas, explicit argument, or verification details are supplied anywhere in the manuscript. This absence is load-bearing because the soundness of the periodicity assertion cannot be assessed without the argument.

    Authors: We agree that the submitted manuscript states the periodicity result for rule 115 but does not supply the derivation steps, lemmas, or explicit argument. This is a presentation gap that prevents assessment of the claim. We will revise the manuscript to include a complete, self-contained proof of periodicity on all finite-support configurations, together with the supporting lemmas and verification details. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper claims to prove periodicity of reversible elementary second-order CA rule 115 on finite initial configurations using the standard local rule definition, invariance of finite support, and built-in bijectivity from reversibility. No equations, self-definitions, fitted inputs presented as predictions, or load-bearing self-citations are quoted or evident that would reduce the central claim to its own inputs by construction. The derivation is a direct mathematical argument on the automaton properties and remains self-contained against external benchmarks of CA theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are visible from the abstract.

pith-pipeline@v0.9.1-grok · 5581 in / 917 out tokens · 20102 ms · 2026-06-27T05:06:31.109925+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Braga, Gianpiero Cattaneo, Paola Flocchini & C

    G. Braga, Gianpiero Cattaneo, Paola Flocchini & C. Quaranta V ogliotti (1995):Pattern Growth in Elementary Cellular Automata. Theor. Comput. Sci. 145(1&2), pp. 1–26, doi:10.1016/0304-3975(94)00155-C

    work page doi:10.1016/0304-3975(94)00155-c 1995

  2. [2]

    Le Bruyn & M

    L. Le Bruyn & M. Van den Bergh (1991): Algebraic properties of linear cellular automata. Linear algebra and its applications 157, pp. 217–234, doi:10.1016/0024-3795(91)90116-E

    work page doi:10.1016/0024-3795(91)90116-e 1991

  3. [3]

    A multi-scale segmentation/object relationship modelling methodology for landscape analysis,

    Gianpiero Cattaneo, Enrico Formenti, Luciano Margara & G Mauri (1999): On the Dynamical Behavior of Chaotic Cellular Automata . Theoretical Computer Science 217(1), pp. 31 – 51, doi:10.1016/S0304- 3975(98)00149-2

    work page doi:10.1016/s0304- 1999

  4. [4]

    Complex Systems 2(2), pp

    Karel Culik II & Sheng Yu (1988): Undecidability of CA Classification Schemes. Complex Systems 2(2), pp. 177–190, doi:10.5555/45269.45271

    work page doi:10.5555/45269.45271 1988

  5. [5]

    Porreca (2019): On the dynamical behaviour of linear higher-order cellular automata and its decidability

    Alberto Dennunzio, Enrico Formenti, Luca Manzoni, Luciano Margara & Antonio E. Porreca (2019): On the dynamical behaviour of linear higher-order cellular automata and its decidability. Inf. Sci. 486, pp. 73–87, doi:10.1016/J.INS.2019.02.023

    work page doi:10.1016/j.ins.2019.02.023 2019

  6. [6]

    In: Asian Symposium on Cellular Automata Technology , Springer, pp

    Enrico Formenti & Supreeti Kamilya (2024): On Elementary Second Order Cellular Automata . In: Asian Symposium on Cellular Automata Technology , Springer, pp. 204–218, doi:10.1007/978-3-031-56943-2_16

    work page doi:10.1007/978-3-031-56943-2_16 2024

  7. [7]

    Mathematical systems theory 3(4), pp

    Gustav A Hedlund (1969): Endomorphisms and automorphisms of the shift dynamical system. Mathematical systems theory 3(4), pp. 320–375, doi:10.1007/BF01691062

    work page doi:10.1007/bf01691062 1969

  8. [8]

    Jarkko Kari (2000): Linear Cellular Automata with Multiple State Variables. In H. reichel & Sophie Tison, editors: STACS 2000, LNCS 1770, Springer-Verlag, pp. 110–121, doi:10.1007/3-540-46541-3_9

    work page doi:10.1007/3-540-46541-3_9 2000

  9. [9]

    Ergodic theory and dynamical systems 17(2), pp

    Petr K˚ urka (1997): Languages, Equicontinuity and Attractors in Cellular Automata . Ergodic theory and dynamical systems 17(2), pp. 417–433, doi:10.1017/S014338579706985X

    work page doi:10.1017/s014338579706985x 1997

  10. [10]

    Subrahmonian Moothathu (2005): Homogeneity of surjective cellular automata

    T.K. Subrahmonian Moothathu (2005): Homogeneity of surjective cellular automata. Discrete and Continu- ous Dynamical Systems 13(1), pp. 195–202, doi:10.3934/dcds.2005.13.195

    work page doi:10.3934/dcds.2005.13.195 2005

  11. [11]

    Chaos: An Interdisciplinary Journal of Nonlinear Science 22(4), p

    Martin Schüle & Ruedi Stoop (2012): A Full Computation-relevant Topological Dynamics Classification of Elementary Cellular Automata. Chaos: An Interdisciplinary Journal of Nonlinear Science 22(4), p. 043143, doi:10.1063/1.4771662

    work page doi:10.1063/1.4771662 2012