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Reliable Cellular Automata with Self-Organization

Peter Gacs

TL;DR

The paper tackles reliable memory and computation in probabilistic cellular automata under transient faults, proving that 1D asynchronous self-organizing constructions can store information and perform computation despite positive fault rates. It develops a rigorous hierarchical framework of colonies, block codes, and generalized media to simulate higher-level automata, culminating in amplifiers that boost reliability across multiple levels. A key contribution is the demonstration that a one-dimensional asynchronous fault-tolerant CA can self-organize its hierarchy over time, enabling infinite-string memory and universal computation in higher dimensions via robust simulations. The work offers a concrete, scalable theory for reliable distributed computation with unreliable components, with potential implications for fault-tolerant parallel architectures and self-stabilizing systems.

Abstract

In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help much to solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our more complex technique organizes the cells in blocks that perform a reliable simulation of a second (generalized) cellular automaton. The cells of the latter automaton are also organized in blocks, simulating even more reliably a third automaton, etc. Since all this (a possibly infinite hierarchy) is organized in ``software'', it must be under repair all the time from damage caused by errors. A large part of the problem is essentially self-stabilization recovering from a mess of arbitrary size and content. The present paper constructs an asynchronous one-dimensional fault-tolerant cellular automaton, with the further feature of ``self-organization''. The latter means that the initial configuration does not have to encode an infinite hierarchy -- this will be built up over time. This is a corrected and strengthened version of the journal paper of 2001.

Reliable Cellular Automata with Self-Organization

TL;DR

The paper tackles reliable memory and computation in probabilistic cellular automata under transient faults, proving that 1D asynchronous self-organizing constructions can store information and perform computation despite positive fault rates. It develops a rigorous hierarchical framework of colonies, block codes, and generalized media to simulate higher-level automata, culminating in amplifiers that boost reliability across multiple levels. A key contribution is the demonstration that a one-dimensional asynchronous fault-tolerant CA can self-organize its hierarchy over time, enabling infinite-string memory and universal computation in higher dimensions via robust simulations. The work offers a concrete, scalable theory for reliable distributed computation with unreliable components, with potential implications for fault-tolerant parallel architectures and self-stabilizing systems.

Abstract

In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help much to solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our more complex technique organizes the cells in blocks that perform a reliable simulation of a second (generalized) cellular automaton. The cells of the latter automaton are also organized in blocks, simulating even more reliably a third automaton, etc. Since all this (a possibly infinite hierarchy) is organized in ``software'', it must be under repair all the time from damage caused by errors. A large part of the problem is essentially self-stabilization recovering from a mess of arbitrary size and content. The present paper constructs an asynchronous one-dimensional fault-tolerant cellular automaton, with the further feature of ``self-organization''. The latter means that the initial configuration does not have to encode an infinite hierarchy -- this will be built up over time. This is a corrected and strengthened version of the journal paper of 2001.
Paper Structure (118 sections, 72 theorems, 429 equations, 11 figures)

This paper contains 118 sections, 72 theorems, 429 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Historical remarks
  3. Hierarchical constructions
  4. New features
  5. Asynchrony
  6. Self-organization
  7. Proof method simplification
  8. Overview of the paper
  9. Cellular automata
  10. Deterministic cellular automata
  11. Fields of a local state
  12. Probabilistic cellular automata
  13. Continuous-time probabilistic cellular automata
  14. Perturbation
  15. Discrete time
  16. ...and 103 more sections

Key Result

Theorem 1

For any constant $c_{1}>1$ there is a one-dimensional probabilistic transition matrix that remembers a field, in a self-organizing way; the function $f(t)$ in eq:remember-field can be chosen as $t^{-c_{1}}$. If the space is finite with size $N$ then in any cell, at time $t$, the probability of forge

Figures (11)

  • Figure 1: Three neighbor colonies with their tracks
  • Figure 2: Fields of a cell simulated by a colony
  • Figure 3: Error-correcting code in a shared field
  • Figure 4: The Toom rule's effect on a large triangular island
  • Figure 5: A system of disjoint random local conditions
  • ...and 6 more figures

Theorems & Definitions (303)

  • Remark 1.1
  • Definition 2.2: Space and states
  • Definition 2.3: Space-time
  • Definition 2.4: Deterministic cellular automata
  • Definition 2.5: Capacity
  • Definition 2.6: Fields
  • Example 2.7
  • Definition 2.8: Legality
  • Definition 2.9
  • Definition 2.11: Events
  • ...and 293 more