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Subshifts Defined by Nondeterministic and Alternating Plane-Walking Automata

Authors Benjamin Hellouin de Menibus , Pacôme Perrotin

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Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Tree languages
  • Theory of computation → Regular languages
Keywords
  • Formal languages
  • Finite automata
  • Subshifts
  • Symbolic dynamics
  • Tilings

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Abstract

Plane-walking automata were introduced by Salo & Törma to recognise languages of two-dimensional infinite words (subshifts), the counterpart of 4-way finite automata for two-dimensional finite words. We extend the model to allow for nondeterminism and alternation of quantifiers. We prove that the recognised subshifts form a strict subclass of sofic subshifts, and that the classes corresponding to existential and universal nondeterminism are incomparable and both larger that the deterministic class. We define a hierarchy of subshifts recognised by plane-walking automata with alternating quantifiers, which we conjecture to be strict.

Cite As Get BibTex

Benjamin Hellouin de Menibus and Pacôme Perrotin. Subshifts Defined by Nondeterministic and Alternating Plane-Walking Automata. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.48

Author Details

Benjamin Hellouin de Menibus
  • Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France
Pacôme Perrotin
  • Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France

Funding

The authors acknowledge financial support from the ANR-22-CE40-0011 project Inside Zero Entropy Systems.

Acknowledgements

Many people have contributed to this article through discussions, suggestions and failed attempts; we wish to thank (in alphabetical order) Florent Becker, Amélia Durbec, Pierre Guillon and Guillaume Theyssier. We are grateful to Ville Salo for providing the proof of Theorem 8 and pointing us towards the Kari-Moore rectangles of Section 5.3, and to Denis Kuperberg and Thomas Colcombet for pointing us towards works on tree-walking automata cited in Section 5.2.

References

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