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Word Structures and Their Automatic Presentations

Authors Xiaoyang Gong , Bakh Khoussainov , Yuyang Zhuge

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LIPIcs.MFCS.2025.51.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Logic
Keywords
  • Automatic structures
  • the isomorphism problem
  • decidability
  • undecidability
  • regular relations

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Abstract

We study automatic presentations of the structures (ℕ; S), (ℕ; E_S), (ℕ; ≤), and their expansions by a unary predicate U. Here S is the successor function, E_S is the undirected version of S, and ≤ is the natural order. We call these structures word structures. Our goal is three-fold. First, we study the isomorphism problem for automatic word structures by focusing on the following three problems. The first problem asks to design an algorithm that, given an automatic structure A, decides if A is isomorphic to (ℕ; S). The second asks to design an algorithm that, given two automatic presentations of (ℕ; S, U₁) and (ℕ; S, U₂), where U₁ and U₂ are unary predicates, decides if these structures are isomorphic. The third problem investigates if there is an algorithm that, given two automatic presentations of (ℕ; ≤, U₁) and (ℕ; ≤, U₂), decides whether U₁ ∩ U₂ ≠ ∅. We show that these problems are undecidable. Next, we study intrinsic regularity of the function S in the structure Path_ω = (ℕ; E_S). We build an automatic presentation of Path_ω in which S is not regular. This implies that S is not intrinsically regular in Path_ω. For U ⊆ ℕ, let d_U be the function that computes the distances between the consecutive elements of U. We build automatic presentations of (ℕ; ≤, U) where d_U can realise logarithmic, radical, intermediate, and exponential functions.

Cite As Get BibTex

Xiaoyang Gong, Bakh Khoussainov, and Yuyang Zhuge. Word Structures and Their Automatic Presentations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.51

Author Details

Xiaoyang Gong
  • School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China
Bakh Khoussainov
  • School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China
Yuyang Zhuge
  • School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China

Funding

The authors acknowledge the support of the NSFC grant 62172077, 62350710215 and Sichuan S&T Department grant 2025HJPJ0006.

References

  1. Vince Bárány. Automatic presentations of infinite structures. Phd thesis, RWTH Aachen University, Germany, 2007. URL: http://darwin.bth.rwth-aachen.de/opus3/volltexte/2007/2019/.
  2. Vince Bárány, Erich Grädel, and Sasha Rubin. Automata-based presentations of infinite structures. In Javier Esparza, Christian Michaux, and Charles Steinhorn, editors, Finite and Algorithmic Model Theory, volume 379 of London Mathematical Society Lecture Note Series, pages 1-76. Cambridge University Press, 2011. Google Scholar
  3. C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Development, 17(6):525-532, 1973. URL: https://doi.org/10.1147/rd.176.0525.
  4. Dmitry Berdinsky and Khoussainov Bakh. Cayley automatic representations of wreath products. International Journal of Foundations of Computer Science, 27(02):147-159, 2016. URL: https://doi.org/10.1142/S0129054116400049.
  5. Dmitry Berdinsky, Sanjay Jain, Bakh Khoussainov, and Frank Stephan. String compression in fa-presentable structures. Theoretical Computer Science, 947:113705, 2023. URL: https://doi.org/10.1016/j.tcs.2023.113705.
  6. Achim Blumensath and Erich Grädel. Automatic structures. In 15th Annual IEEE Symposium on Logic in Computer Science, Santa Barbara, California, USA, June 26-29, 2000, pages 51-62. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/LICS.2000.855755.
  7. Achim Blumensath and Erich Grädel. Finite presentations of infinite structures: Automata and interpretations. Theory Comput. Syst., 37(6):641-674, 2004. URL: https://doi.org/10.1007/s00224-004-1133-y.
  8. J. Richard Büchi. Weak second-order arithmetic and finite automata. Mathematical Logic Quarterly, 6(1-6):66-92, 1960. URL: https://doi.org/10.1002/malq.19600060105.
  9. Cristian Calude, Peter Hertling, Bakh Khoussainov, and Yongge Wang. Recursively enumerable reals and chaitin omega numbers. Theor. Comput. Sci., 255(1-2):125-149, 2001. URL: https://doi.org/10.1016/S0304-3975(99)00159-0.
  10. Christian Delhommé. Automaticité des ordinaux et des graphes homogènes. Comptes Rendus Mathematique, 339(1):5-10, 2004. URL: https://doi.org/10.1016/j.crma.2004.03.035.
  11. Berdinsky Dmitry and Bakh Khoussainov. On automatic transitive graphs. In Arseny M. Shur and Mikhail V. Volkov, editors, Developments in Language Theory, volume 8633 of Lecture Notes in Computer Science, pages 1-12, Cham, 2014. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-09698-8_1.
  12. David B. A. Epstein, M. S. Paterson, J. W. Cannon, D. F. Holt, S. V. Levy, and W. P. Thurston. Word Processing in Groups. A. K. Peters, Ltd., USA, 1992. Google Scholar
  13. G. A. Freĭman. Addition of finite sets. Sov. Math., Dokl., 5:1366-1370, 1964. Google Scholar
  14. Moses Ganardi and Bakh Khoussainov. Automatic equivalence structures of polynomial growth. In Maribel Fernández and Anca Muscholl, editors, 28th EACSL Annual Conference on Computer Science Logic, CSL 2020, January 13-16, 2020, Barcelona, Spain, volume 152 of LIPIcs, pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CSL.2020.21.
  15. Alex Gavryushkin, Bakh Khoussainov, and Frank Stephan. Reducibilities among equivalence relations induced by recursively enumerable structures. Theor. Comput. Sci., 612:137-152, 2016. URL: https://doi.org/10.1016/j.tcs.2015.11.042.
  16. Erich Grädel. Automatic structures: Twenty years later. In Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller, editors, LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020, pages 21-34. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394734.
  17. Michael Gromov. Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Publications Mathématiques de l'IHÉS, 53:53-78, 1981. URL: https://doi.org/10.1007/BF02698687.
  18. Matthew Harrison-Trainor. An introduction to the scott complexity of countable structures and a survey of recent results. Bull. Symb. Log., 28(1):71-103, November 2022. URL: https://doi.org/10.1017/bsl.2021.62.
  19. Greg Hjorth, Bakh Khoussainov, Antonio Montalbán, and André Nies. From automatic structures to borel structures. In Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, 24-27 June 2008, Pittsburgh, PA, USA, pages 431-441. IEEE, IEEE Computer Society, 2008. URL: https://doi.org/10.1109/LICS.2008.28.
  20. Bernard R. Hodgson. Théories décidables par automate fini. PhD thesis, Université de Montréal, 1976. Google Scholar
  21. Sanjay Jain, Bakh Khoussainov, Philipp Schlicht, and Frank Stephan. Tree-automatic scattered linear orders. Theoretical Computer Science, 626:83-96, 2016. URL: https://doi.org/10.1016/j.tcs.2016.02.008.
  22. Sanjay Jain, Bakh Khoussainov, Philipp Schlicht, and Frank Stephan. The isomorphism problem for tree-automatic ordinals with addition. Information Processing Letters, 149:19-24, 2019. URL: https://doi.org/10.1016/j.ipl.2019.05.004.
  23. Sanjay Jain, Bakh Khoussainov, and Frank Stephan. Finitely generated semiautomatic groups. Comput., 7(2-3):273-287, 2018. URL: https://doi.org/10.3233/COM-180089.
  24. Sanjay Jain, Bakh Khoussainov, Frank Stephan, Dan Teng, and Siyuan Zou. Semiautomatic structures. Theory Comput. Syst., 61(4):1254-1287, 2017. URL: https://doi.org/10.1007/s00224-017-9792-7.
  25. Hartley Rogers Jr. Theory of recursive functions and effective computability (Reprint from 1967). MIT Press, Cambridge, MA, USA, 1987. URL: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3182.
  26. Lukasz Kaiser, Sasha Rubin, and Vince Bárány. Cardinality and counting quantifiers on omega-automatic structures. In Susanne Albers and Pascal Weil, editors, STACS 2008, 25th Annual Symposium on Theoretical Aspects of Computer Science, Bordeaux, France, February 21-23, 2008, Proceedings, volume 1 of LIPIcs, pages 385-396. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany, 2008. URL: https://doi.org/10.4230/LIPIcs.STACS.2008.1360.
  27. Olga Kharlampovich, Bakh Khoussainov, and Alexei Miasnikov. From automatic structures to automatic groups. Groups, Geometry, and Dynamics, 8(1):157-198, 2014. URL: https://doi.org/10.4171/GGD/221.
  28. Bakh Khoussainov and Nerode Anil. Automatic presentations of structures. In Daniel Leivant, editor, Logic and Computational Complexity, pages 367-392, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/3-540-60178-3_93.
  29. Bakh Khoussainov, Steffen Lempp, and Theodore A. Slaman. Computably enumerable algebras, their expansions, and isomorphisms. Int. J. Algebra Comput., 15(3):437-454, 2005. URL: https://doi.org/10.1142/S0218196705002281.
  30. Bakh Khoussainov, Jiamou Liu, and Mia Minnes. Unary automatic graphs: An algorithmic perspective. In Manindra Agrawal, Ding-Zhu Du, Zhenhua Duan, and Angsheng Li, editors, Theory and Applications of Models of Computation, 5th International Conference, TAMC 2008, Xi'an, China, April 25-29, 2008. Proceedings, volume 4978 of Lecture Notes in Computer Science, pages 542-553, Berlin, Heidelberg, 2008. Springer. URL: https://doi.org/10.1007/978-3-540-79228-4_47.
  31. Bakh Khoussainov and Mia Minnes. Three lectures on automatic structures. In Proceedings of Logic Colloquium, 2007. Google Scholar
  32. Bakh Khoussainov and Mia Minnes. Model-theoretic complexity of automatic structures. Annals of Pure and Applied Logic, 161(3):416-426, 2009. Papers presented at the Symposium on Logical Foundations of Computer Science 2007. URL: https://doi.org/10.1016/j.apal.2009.07.012.
  33. Bakh Khoussainov and Anil Nerode. Automata Theory and its Applications, volume 21. Springer Science & Business Media, 2001. URL: https://doi.org/10.1007/978-1-4612-0171-7.
  34. Bakh Khoussainov, André Nies, Sasha Rubin, and Frank Stephan. Automatic structures: Richness and limitations. Log. Methods Comput. Sci., 3(2), 2007. URL: https://doi.org/10.2168/LMCS-3(2:2)2007.
  35. Bakh Khoussainov and Sasha Rubin. Graphs with automatic presentations over a unary alphabet. J. Autom. Lang. Comb., 6(4):467-480, April 2001. URL: https://doi.org/10.25596/jalc-2001-467.
  36. Bakh Khoussainov and Sasha Rubin. Automatic structures: overview and future directions. J. Autom. Lang. Comb., 8(2):287-301, April 2003. URL: https://doi.org/10.25596/jalc-2003-287.
  37. Bakh Khoussainov, Sasha Rubin, and Frank Stephan. On automatic partial orders. In Proceedings of the Eighteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2003), pages 168-177. IEEE Computer Society Press, June 2003. URL: https://doi.org/10.1109/LICS.2003.1210056.
  38. Bakh Khoussainov, Sasha Rubin, and Frank Stephan. Definability and regularity in automatic structures. In Volker Diekert and Michel Habib, editors, STACS 2004, 21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings, volume 2996 of Lecture Notes in Computer Science, pages 440-451. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24749-4_39.
  39. Bakh Khoussainov, Theodore A. Slaman, and Pavel Semukhin. P^0_1-presentations of algebras. Arch. Math. Log., 45(6):769-781, 2006. URL: https://doi.org/10.1007/s00153-006-0013-3.
  40. Dietrich Kuske, Jiamou Liu, and Markus Lohrey. The isomorphism problem on classes of automatic structures with transitive relations. Transactions of the American Mathematical Society, 365(10):5103-5151, October 2013. URL: https://doi.org/10.1090/S0002-9947-2013-05766-2.
  41. Dietrich Kuske and Markus Lohrey. First-order and counting theories of omega-automatic structures. In Luca Aceto and Anna Ingólfsdóttir, editors, Foundations of Software Science and Computation Structures, 9th International Conference, FOSSACS 2006, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2006, Vienna, Austria, March 25-31, 2006, Proceedings, volume 3921 of Lecture Notes in Computer Science, pages 322-336, Berlin, Heidelberg, 2006. Springer. URL: https://doi.org/10.1007/11690634_22.
  42. Yuri Vladimirovich Matiyasevich. Hilbert’s tenth problem. MIT press, 1993. Google Scholar
  43. André Nies and Pavel Semukhin. Finite automata presentable abelian groups. In Sergei N. Artemov and Anil Nerode, editors, Logical Foundations of Computer Science, pages 422-436, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-72734-7_29.
  44. André Nies, Frank Stephan, Markus Lohrey, Pavel Semukhin, Dimitry Berdinsky, and et al. Personal communications (over many years). Google Scholar
  45. André Nies and Richard M. Thomas. Fa-presentable groups and rings. Journal of Algebra, 320(2):569-585, 2008. Computational Algebra. URL: https://doi.org/10.1016/j.jalgebra.2007.04.015.
  46. Graham P. Oliver and Richard M. Thomas. Automatic presentations for finitely generated groups. In Volker Diekert and Bruno Durand, editors, STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany, February 24-26, 2005, Proceedings, volume 3404 of Lecture Notes in Computer Science, pages 693-704. Springer, 2005. URL: https://doi.org/10.1007/978-3-540-31856-9_57.
  47. Michael O Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the american Mathematical Society, 141:1-35, 1969. Google Scholar
  48. Downey Rodney and Melnikov Alexander. Computable Structure Theory: A Unified Approach. Springer, 2024. Google Scholar
  49. Sasha Rubin. Automatic structures. PhD thesis, University of Auckland New Zealand, 2004. Google Scholar
  50. Sasha Rubin. Automata presenting structures: A survey of the finite string case. The Bulletin of Symbolic Logic, 14(2):169-209, 2008. URL: https://doi.org/10.2178/bsl/1208442827.
  51. Dana Scott. Logic with denumerably long formulas and finite strings of quantifiers. Journal of Symbolic Logic, 36(1):1104-329, 1965. URL: https://doi.org/10.2307/2271545.
  52. Frank Stephan. Automatic structures — recent results and open questions. Journal of Physics: Conference Series, 622(1):012013, May 2015. URL: https://doi.org/10.1088/1742-6596/622/1/012013.
  53. Todor Tsankov. The additive group of the rationals does not have an automatic presentation. Journal of Symbolic Logic, 76(4):1341-1351, 2011. URL: https://doi.org/10.2178/jsl/1318338853.
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