Document Open Access Logo

A Pumping-Like Lemma for Languages over Infinite Alphabets

Author Yoav Danieli

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2026.29.pdf
  • Filesize: 0.83 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • infinite alphabets
  • pumping lemma
  • alternation
  • semi-linearity

Metrics

Abstract

We prove a kind of a pumping lemma for languages accepted by one-register alternating finite-memory automata. As a corollary, we obtain that the set of lengths of words in such languages is semi-linear.

Cite As Get BibTex

Yoav Danieli. A Pumping-Like Lemma for Languages over Infinite Alphabets. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.29

Author Details

Yoav Danieli
  • The Henry and Marilyn Taub Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

References

  1. Rajeev Alur and David L. Dill. A theory of timed automata. Theor. Comput. Sci., 126(2):183-235, 1994. URL: https://doi.org/10.1016/0304-3975(94)90010-8.
  2. Michael Blondin, Alain Finkel, and Jean Goubault-Larrecq. Forward analysis for wsts, part III: karp-miller trees. Log. Methods Comput. Sci., 16(2), 2020. URL: https://doi.org/10.23638/LMCS-16(2:13)2020.
  3. Mikołaj Bojanczyk. Slightly infinite sets. A draft of a book available at https://www.mimuw.edu.pl/~bojan/paper/atom-book, 2019.
  4. Mikolaj Bojanczyk, Bartek Klin, and Sławomir Lasota. Automata theory in nominal sets. Logical Methods in Computer Science, 10(3):1-44, 2014. URL: https://doi.org/10.2168/LMCS-10(3:4)2014.
  5. A.K. Chandra, D.C. Kozen, and L.J. Stockmeyer. Alternation. Journal of the ACM, 28(1):114-133, 1981. URL: https://doi.org/10.1145/322234.322243.
  6. Taolue Chen, Fu Song, and Zhilin Wu. Formal reasoning on infinite data values: An ongoing quest. In Jonathan P. Bowen, Zhiming Liu, and Zili Zhang, editors, Engineering Trustworthy Software Systems - Second International School, SETSS 2016, Chongqing, China, March 28 - April 2, 2016, Tutorial Lectures, volume 10215 of Lecture Notes in Computer Science, pages 195-257, 2016. URL: https://doi.org/10.1007/978-3-319-56841-6_6.
  7. Lorenzo Clemente, Slawomir Lasota, and Radoslaw Piórkowski. Determinisability of register and timed automata. Logical Methods in Computer Science, 18:Paper No. 9, 37, 2022. URL: https://doi.org/10.46298/LMCS-18(2:9)2022.
  8. Yoav Danieli and Michael Kaminski. Bounded languages over infinite alphabets. In Klaus Meer, Alexander Rabinovich, Elena Ravve, and Andrés Villaveces, editors, Model Theory, Computer Science, and Graph Polynomials: Festschrift in Honor of Johann A. Makowsky, pages 193-213. Springer Nature Switzerland, Cham, 2025. URL: https://doi.org/10.1007/978-3-031-86319-6_15.
  9. S. Demri and R. Lazić. LTL with the freeze quantifier and register automata. ACM Transactions on Computational Logic, 10, 2009. Article 16. Google Scholar
  10. L.E. Dickson. Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics, 35:413-422, 1913. Google Scholar
  11. Diego Figueira, Santiago Figueira, Sylvain Schmitz, and Philippe Schnoebelen. Ackermannian and primitive-recursive bounds with Dickson’s lemma. In 2011 IEEE 26th Annual Symposium on Logic in Computer Science, pages 269-278. IEEE, 2011. URL: https://doi.org/10.1109/LICS.2011.39.
  12. Diego Figueira, Piotr Hofman, and Sławomir Lasota. Relating timed and register automata. Mathematical Structures in Computer Science, 26:993-1021, 2016. URL: https://doi.org/10.1017/S0960129514000322.
  13. Alain Finkel and Jean Goubault-Larrecq. Forward analysis for wsts, part II: complete WSTS. Log. Methods Comput. Sci., 8(3), 2012. URL: https://doi.org/10.2168/LMCS-8(3:28)2012.
  14. Alain Finkel and Jean Goubault-Larrecq. Forward analysis for wsts, part I: completions. Math. Struct. Comput. Sci., 30(7):752-832, 2020. URL: https://doi.org/10.1017/S0960129520000195.
  15. D. Genkin, M. Kaminski, and L. Peterfreund. A note on the emptiness problem for alternating finite-memory automata. Theoretical Computer Science, 526:97-107, 2014. URL: https://doi.org/10.1016/J.TCS.2014.01.020.
  16. Graham Higman. Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society, 3(1):326-336, 1952. Google Scholar
  17. Piotr Hofman, Marta Juzepczuk, Sławomir Lasota, and Mohnish Pattathurajan. Parikh’s theorem for infinite alphabets. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470626.
  18. M. Kaminski and N. Francez. Finite-memory automata. In Proceedings of the 31th Annual IEEE Symposium on Foundations of Computer Science, pages 683-688, Los Alamitos, CA, 1990. IEEE Computer Society Press. Google Scholar
  19. M. Kaminski and N. Francez. Finite-memory automata. Theoretical Computer Science, 134:329-363, 1994. URL: https://doi.org/10.1016/0304-3975(94)90242-9.
  20. Ahmet Kara. Logics on data words: Expressivity, satisfiability, model checking. PhD thesis, Technical University of Dortmund, Germany, 2016. URL: https://hdl.handle.net/2003/35216.
  21. R. M. Karp and R. E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3:147-195, 1969. URL: https://doi.org/10.1016/S0022-0000(69)80011-5.
  22. B. Klin. A pumping lemma for automata with atoms. https://atoms.mimuw.edu.pl/?p=43/, 26.03.2014. [accessed 08.08.2025].
  23. Bartek Klin, Sławomir Lasota, and Szymon Toruńczyk. Nondeterministic and co-nondeterministic implies deterministic, for data languages. In Stefan Kiefer and Christine Tasson, editors, Foundations of Software Science and Computation Structures - 24th International Conference, FOSSACS 2021, volume 12650 of Lecture Notes in Computer Science, pages 365-384. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-71995-1_19.
  24. Slawomir Lasota and Igor Walukiewicz. Alternating timed automata. ACM Trans. Comput. Log., 9(2):10:1-10:27, 2008. URL: https://doi.org/10.1145/1342991.1342994.
  25. M. Löb and S. Wainer. Hierarchies of number-theoretic functions I, II: a correction. Archive for Mathematical Logic, 14(3-4):198-199, 1971. Google Scholar
  26. Rindo Nakanishi, Yoshiaki Takata, and Hiroyuki Seki. Pumping lemmas for languages expressed by computational models with registers. IEICE Transactions on Information and Systems, 106:284-293, 2023. URL: https://doi.org/10.1587/TRANSINF.2022FCP0004.
  27. F. Neven, T. Schwentick, and V. Vianu. Finite state machines for strings over infinite alphabets. ACM Transactions on Computational Logic, 5:403-435, 2004. URL: https://doi.org/10.1145/1013560.1013562.
  28. Alexander Okhotin. The dual of concatenation. Theoretical Computer Science, 345(2-3):425-447, 2005. URL: https://doi.org/10.1016/J.TCS.2005.07.019.
  29. M. Rabin and D. Scott. Finite automata and their decision problems. IBM Journal of Research Development, 3:114-125, 1959. URL: https://doi.org/10.1147/RD.32.0114.
  30. H. Sakamoto and D. Ikeda. Intractability of decision problems for finite-memory automata. Theoretical Computer Science, 231:297-308, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00105-X.
  31. Philippe Schnoebelen. Revisiting Ackermann-hardness for lossy counter machines and reset Petri nets. In International Symposium on Mathematical Foundations of Computer Science, 2010. URL: https://api.semanticscholar.org/CorpusID:14086694.
  32. Luc Segoufin. Automata and logics for words and trees over an infinite alphabet. In Zoltán Ésik, editor, Computer Science Logic, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL, volume 4207 of Lecture Notes in Computer Science, pages 41-57. Springer, 2006. URL: https://doi.org/10.1007/11874683_3.
  33. T. Tan. On pebble automata for data languages with decidable emptiness problem. Journal of Computer and System Sciences, 76:778-791, 2010. URL: https://doi.org/10.1016/J.JCSS.2010.03.004.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact