Conjugacy of One-Dimensional One-Sided Cellular Automata is Undecidable

@article{Jalonen2018ConjugacyOO,
  title={Conjugacy of One-Dimensional One-Sided Cellular Automata is Undecidable},
  author={Joonatan Jalonen and Jarkko Kari},
  journal={ArXiv},
  year={2018},
  volume={abs/1710.08111}
}
Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs $(F,G)$ of one-dimensional one-sided cellular automata over a full shift are recursively inseparable: (i) pairs where $F$ has strictly larger topological entropy than $G$, and (ii) pairs that are strongly conjugate and have zero topological entropy. Because there is no factor map from a lower entropy system to a higher entropy one, and there is no… 

On the conjugacy problem of cellular automata

On the conjugacy problem of cellular automata. Topological and differentiable equivalence, conjugacy, dynamical systems

  • Mathematics
  • 2022
It is well known that a natural notion of isomorphism in topological dynamics is topological conjugacy. Let f : AZ → AZ be a one-dimensional cellular automaton. In [Topological conjugacies between

On Dynamical Complexity of Surjective Ultimately Right-Expansive Cellular Automata

TLDR
It is shown that there exists a right-expansive reversible cellular automaton that has a non-sofic trace and thus does not have the pseudo-orbit tracing property.

Cellular Automata and Discrete Complex Systems

TLDR
This paper describes a step-by-step gauging procedure to enforce local symmetries upon a given Cellular Automaton to provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata.

Reversible Computation: Extending Horizons of Computing: Selected Results of the COST Action IC1405

TLDR
Many threads of research in the area of foundations of reversible computing are reported on, giving particular emphasis to the results obtained in the framework of the European COST Action IC1405, entitled “Reversible Computation Extending Horizons of Computing”, which took place in the years 2015–2019.

Foundations of Reversible Computation

TLDR
Many threads of research in the area of foundations of reversible computing are reported below, giving particular emphasis to the results obtained in the framework of the European COST Action IC1405, entitled “Reversible Computation - Extending Horizons of Computing”, which took place in the years 2015–2019.

References

SHOWING 1-10 OF 22 REFERENCES

The Nilpotency Problem of One-Dimensional Cellular Automata

  • J. Kari
  • Mathematics
    SIAM J. Comput.
  • 1992
TLDR
The present work proves that it is algorithmically undecidable whether a given one-dimensional cellular automaton is nilpotent, the basis of the proof of Rice's theorem for CA limit sets.

Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata

TLDR
It is shown that left expansivity of a reversible cellular automaton is an undecidable property and the tile set construction gives yet another proof for the universality of one-dimensional reversible Cellular automata.

Classification of Elementary Cellular Automata Up to Topological Conjugacy

TLDR
Different invariants for topological conjugacy are investigated in order to distinguish between non-conjugate systems and it is shown how to compute the cardinality of the set of points with minimal period \(n\) for one-dimensional CA.

The topological entropy of cellular automata is uncomputable

TLDR
A set of aperiodic Wang tiles arising from Penrose's kite and dart tiles is used to demonstrate specific examples of cellular automata with a single periodic point but non-trivial non-wandering sets, which furthermore can be constructed to have arbitrarily high topological entropy.

Topological dynamics of one-dimensional cellular automata

1 Glossary Almost equicontinuous CA has an equicontinuous configuration. Attractor: omega-limit of a clopen invariant set. Blocking word interrupts information flow. Closing CA: distinct asymptotic

Theory of cellular automata: A survey

  • J. Kari
  • Computer Science
    Theor. Comput. Sci.
  • 2005

Periodicity and Immortality in Reversible Computing

TLDR
It turns out that periodicity and immortality problems are both undecidable in all three models of reversible computation: reversible counter machines, reversible Turing machines and reversible one-dimensional cellular automata.