Document Open Access Logo

Varieties of Data Languages (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Henning Urbat, Stefan Milius



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.130.pdf
  • Filesize: 0.49 MB
  • 14 pages

Document Identifiers

Author Details

Henning Urbat
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Stefan Milius
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Cite AsGet BibTex

Henning Urbat and Stefan Milius. Varieties of Data Languages (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 130:1-130:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.130

Abstract

We establish an Eilenberg-type correspondence for data languages, i.e. languages over an infinite alphabet. More precisely, we prove that there is a bijective correspondence between varieties of languages recognized by orbit-finite nominal monoids and pseudovarieties of such monoids. This is the first result of this kind for data languages. Our approach makes use of nominal Stone duality and a recent category theoretic generalization of Birkhoff-type theorems that we instantiate here for the category of nominal sets. In addition, we prove an axiomatic characterization of weak pseudovarieties as those classes of orbit-finite monoids that can be specified by sequences of nominal equations, which provides a nominal version of a classical theorem of Eilenberg and Schützenberger.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Nominal sets
  • Stone duality
  • Algebraic language theory
  • Data languages

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Jiří Adámek, Stefan Milius, Robert S.R. Myers, and Henning Urbat. Generalized Eilenberg Theorem I: Local Varieties of Languages. In Anca Muscholl, editor, Proc. Foundations of Software Science and Computation Structures (FoSSaCS), volume 8412 of Lecture Notes Comput. Sci., pages 366-380. Springer, 2014. Google Scholar
  2. Jiří Adámek, Stefan Milius, Robert S.R. Myers, and Henning Urbat. Varieties of Languages in a Category. In Catuscia Palamidessi, editor, Proc. 30th Annual Symposium on Logic in Computer Science (LICS'15), pages 414-425. IEEE Computer Society, 2015. Google Scholar
  3. Jiří Adámek, Stefan Milius, Robert S.R. Myers, and Henning Urbat. Generalized Eilenberg Theorem: Varieties of Languages in a Category. ACM Trans. Comput. Log., 20(1):3:1-3:47, 2019. Google Scholar
  4. Mikołaj Bojańczyk. Nominal Monoids. Theory of Computing Systems, 53(2):194-222, 2013. Google Scholar
  5. Mikołaj Bojańczyk, Bartek Klin, and Sławomir Lasota. Automata theory in nominal sets. Log. Methods Comput. Sci., 10(3:4):44 pp., 2014. Google Scholar
  6. Samuel Eilenberg. Automata, Languages, and Machines Vol. B. Academic Press, 1976. Google Scholar
  7. Samuel Eilenberg and Marcel-Paul Schützenberger. On pseudovarieties. Advances Math., 10:413-418, 1976. Google Scholar
  8. Murdoch J. Gabbay, Tadeusz Litak, and Daniela Petrişan. Stone Duality for Nominal Boolean Algebras with New. In A. Corradini, B. Klin, and C. Cîrstea, editors, Proc. CALCO'11, volume 6859 of LNCS, pages 192-207. Springer, 2011. Google Scholar
  9. Mai Gehrke, Serge Grigorieff, and Jean-Éric Pin. Duality and equational theory of regular languages. In Proc. ICALP'08, Part II, volume 5126 of LNCS, pages 246-257. Springer, 2008. Google Scholar
  10. Michael Kaminski and Nissim Francez. Finite-memory automata. Theoret. Comput. Sci., 134(2):329-363, 1994. Google Scholar
  11. Alexander Kurz and Daniela Petrisan. On universal algebra over nominal sets. Mathematical Structures in Computer Science, 20(2):285-318, 2010. Google Scholar
  12. Robert McNaughton and Seymour A. Papert. Counter-Free Automata (M.I.T. Research Monograph No. 65). The MIT Press, 1971. Google Scholar
  13. Stefan Milius and Henning Urbat. Equational Axiomatization of Algebras with Structure. Full version; available online at https://arxiv.org/abs/1812.02016, 2018.
  14. Stefan Milius and Henning Urbat. Equational axiomatization of algebras with structure. In Proc. 22nd International Conference on Foundations of Software Science and Computation Structures, 2019. To appear. Google Scholar
  15. Daniela Petrişan. Investigations into Algebra and Topology over Nominal Sets. PhD thesis, University of Leicester, 2012. Google Scholar
  16. Andrew M. Pitts. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, 2013. Google Scholar
  17. Gabriele Puppis, Thomas Colcombet, and Clemens Ley. Logics with rigidly guarded data tests. Log. Methods Comput. Sci., 11(3:10):56 pp., 2015. Google Scholar
  18. Julian Salamanca. Unveiling Eilenberg-type Correspondences: Birkhoff’s Theorem for (finite) Algebras + Duality, February 2017. URL: http://arxiv.org/abs/1702.02822.
  19. Marcel-Paul Schützenberger. On finite monoids having only trivial subgroups. Inform. and Control, 8:190-194, 1965. Google Scholar
  20. Thomas Schwentick. Automata for XML - A survey. J. Comput. System Sci., 73(3):289-315, 2007. Google Scholar
  21. Luc Segoufin. Automata and Logics for Words and Trees over an Infinite Alphabet. In Computer Science Logic, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL, Szeged, Hungary, September 25-29, 2006, Proceedings, pages 41-57, 2006. Google Scholar
  22. Henning Urbat, Jiří Adámek, Liang-Ting Chen, and Stefan Milius. Eilenberg Theorems for Free. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, Proc. 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), volume 83 of LIPIcs. Schloss Dagstuhl, 2017. EATCS Best Paper Award. Google Scholar
  23. Henning Urbat and Stefan Milius. Varieties of Data Languages. CoRR, abs/1903.08053, 2019. URL: http://arxiv.org/abs/1903.08053.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail