Decidability for Sturmian Words

Authors Philipp Hieronymi, Dun Ma, Reed Oei, Luke Schaeffer, Christian Schulz, Jeffrey Shallit



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Author Details

Philipp Hieronymi
  • Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA
  • Mathematisches Institut, Universität Bonn, Germany
Dun Ma
  • Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA
Reed Oei
  • Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA
Luke Schaeffer
  • Institute for Quantum Computing, University of Waterloo, Canada
Christian Schulz
  • Department of Mathematics, University of Illinois at Urbana-Champaign, IL, USA
Jeffrey Shallit
  • School of Computer Science, University of Waterloo, Canada

Acknowledgements

Part of this work was done in the research project "Building a theorem-prover" at the Illinois Geometry Lab in Spring 2020. P.H. and C.S. were partially supported by NSF grant DMS-1654725. We thank Mary Angelica Gramcko-Tursi for carefully reading a draft of this paper.

Cite AsGet BibTex

Philipp Hieronymi, Dun Ma, Reed Oei, Luke Schaeffer, Christian Schulz, and Jeffrey Shallit. Decidability for Sturmian Words. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CSL.2022.24

Abstract

We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly ω-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Decidability
  • Sturmian words
  • Ostrowski numeration systems
  • Automated theorem proving

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References

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