From Equational Specifications of Algebras with Structure to Varieties of Data Languages (Invited Paper)

Author Stefan Milius

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

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Stefan Milius. From Equational Specifications of Algebras with Structure to Varieties of Data Languages (Invited Paper). In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 2:1-2:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


This extended abstract first presents a new category theoretic approach to equationally axiomatizable classes of algebras. This approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered algebras, continuous algebras, quantitative algebras, nominal algebras, or profinite algebras. We present a generic HSP theorem and a sound and complete equational logic, which encompass numerous flavors of equational axiomizations studied in the literature. In addition, we use the generic HSP theorem as a key ingredient to obtain Eilenberg-type correspondences yielding algebraic characterizations of properties of regular machine behaviours. When instantiated for orbit-finite nominal monoids, the generic HSP theorem yields a crucial step for the proof of the first Eilenberg-type variety theorem for data languages.

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ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Formal languages and automata theory
  • Birkhoff theorem
  • Equational logic
  • Eilenberg theorem
  • Data languages


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  1. J. Adámek, A. H. Mekler, E. Nelson, and J. Reiterman. On the logic of continuous algebras. Notre Dame J. Formal Logic, 29(3):365-380, 1988. Google Scholar
  2. Jiři Adámek, Evelyn Nelson, and Jan Reiterman. The Birkhoff Variety Theorem for continuous algebras. Algebra Universalis, 20(3):328-350, 1985. Google Scholar
  3. Jiř\'ıAdámek, 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. J. Almeida. On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics. Algebra Universalis, 27(3):333-350, 1990. Google Scholar
  5. N. Bedon and O. Carton. An Eilenberg theorem for words on countable ordinals. In Proc. LATIN'98, volume 1380 of LNCS, pages 53-64. Springer, 1998. Google Scholar
  6. N. Bedon and C. Rispal. Schützenberger and Eilenberg theorems for words on linear orderings. In Proc. DLT'05, volume 3572 of LNCS, pages 134-145. Springer, 2005. Google Scholar
  7. Garret Birkhoff. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 10:433–-454, 1935. Google Scholar
  8. Stephen L. Bloom. Varieties of ordered algebras. J. Comput. Syst. Sci., 2(13):200-212, 1976. Google Scholar
  9. M. Bojańczyk. Recognisable languages over monads. In I. Potapov, editor, Proc. DLT'15, volume 9168 of LNCS, pages 1-13. Springer, 2015. URL:
  10. Mikołaj Bojańczyk. Nominal Monoids. Theory of Computing Systems, 53(2):194-222, 2013. Google Scholar
  11. Mikołaj Bojańczyk, Bartek Klin, and Sławomir Lasota. Automata theory in nominal sets. Log. Methods Comput. Sci., 10(3:4):44 pp., 2014. Google Scholar
  12. L. Daviaud, D. Kuperberg, and J.-É. Pin. Varieties of Cost Functions. In N. Ollinger and H. Vollmer, editors, Proc. STACS 2016, volume 47 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2016. Google Scholar
  13. S. Eilenberg and M. P. Schützenberger. On pseudovarieties. Advances Math., 10:413-418, 1976. Google Scholar
  14. Samuel Eilenberg. Automata, Languages, and Machines Vol. B. Academic Press, 1976. Google Scholar
  15. Murdoch J. Gabbay. Nominal algebra and the HSP theorem. Journal of Logic and Computation, 19:341-367, 2009. Google Scholar
  16. 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
  17. Michael Kaminski and Nissim Francez. Finite-memory automata. Theoret. Comput. Sci., 134(2):329-363, 1994. Google Scholar
  18. Alexander Kurz and Daniela Petrisan. On universal algebra over nominal sets. Mathematical Structures in Computer Science, 20(2):285-318, 2010. Google Scholar
  19. Radu Mardare, Prakash Panangaden, and Gordon Plotkin. Quantitative Algebraic Reasoning. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, pages 700-709. ACM, 2016. Google Scholar
  20. Radu Mardare, Prakash Panangaden, and Gordon Plotkin. On the axiomatizability of quantitative algebras. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12. IEEE Computer Society, 2017. URL:
  21. Robert McNaughton and Seymour A. Papert. Counter-Free Automata (M.I.T. Research Monograph No. 65). The MIT Press, 1971. Google Scholar
  22. Stefan Milius and Henning Urbat. Equational Axiomatization of Algebras with Structure. In Mikołaj Bojańczyk and Alex Simpson, editors, Proc. Foundations of Software Science and Computation Structures (FoSSaCS), volume 11425 of Lecture Notes Comput. Sci. (ARCoSS), pages 400-417, 2019. Google Scholar
  23. Stefan Milius and Henning Urbat. Varietes of Data Languages. In Proc. 46th International Colloquium on Automata, Languages, and Programming (ICALP), volume 132 of LIPIcs, pages 130:1-130:14, 2019. (Full version available online at URL:
  24. J.-É. Pin. A variety theorem without complementation. Russ. Math., 39:80-90, 1995. Google Scholar
  25. J.-É. Pin. Positive varieties and infinite words. In LATIN 98, volume 1380 of LNCS, pages 76-87. Springer, 1998. Google Scholar
  26. Andrew M. Pitts. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, 2013. Google Scholar
  27. L. Polák. Syntactic semiring of a language. In J. Sgall, A. Pultr, and P. Kolman, editors, Proc. MFCS'01, volume 2136 of LNCS, pages 611-620. Springer, 2001. Google Scholar
  28. 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
  29. J. Reiterman. The Birkhoff theorem for finite algebras. Algebra Universalis, 14(1):1-10, 1982. Google Scholar
  30. C. Reutenauer. Séries formelles et algèbres syntactiques. J. Algebra, 66:448-483, 1980. Google Scholar
  31. Julian Salamanca. Unveiling Eilenberg-type Correspondences: Birkhoff’s Theorem for (finite) Algebras + Duality, February 2017. URL:
  32. S. Salehi and M. Steinby. Tree algebras and varieties of tree languages. Theor. Comput. Sci., 377(1-3):1-24, 2007. Google Scholar
  33. Marcel-Paul Schützenberger. On finite monoids having only trivial subgroups. Inform. and Control, 8:190-194, 1965. Google Scholar
  34. Thomas Schwentick. Automata for XML - A survey. J. Comput. System Sci., 73(3):289-315, 2007. Google Scholar
  35. 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
  36. H. Straubing. On logical descriptions of regular languages. In S. Rajsbaum, editor, LATIN 2002 Theor. Informatics, volume 2286 of LNCS, pages 528-538. Springer, 2002. Google Scholar
  37. Henning Urbat, Jiř\'ıAdámek, 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
  38. T. Wilke. An Eilenberg Theorem for ∞-Languages. In Proc. ICALP'91, volume 510 of LNCS, pages 588-599. Springer, 1991. Google Scholar