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A syntactic congruence for languages of birooted trees

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Abstract

We present a framework for the composition of overlapping objects based on semigroup theory. To do so, we develop a language theory of (labelled) birooted trees, that is, subsets of (extension of) free inverse monoids. In the underlying setting of partial algebras, we define a suitable notion of a syntactic congruence such that (i) having a syntactic congruence of finite index captures \(\mathrm{MSO}\)-definability; (ii) a certain order-bisimulation refinement of the syntactic congruence captures (so called) quasi-recognisability in the same way.

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Notes

  1. Otherwise, there is no need of a zero.

  2. Equivalently for all since the relation \(\simeq \) is closed.

  3. The left (or right) projection of a quasi-recognisable language may not be quasi-recognisable.

  4. Or, by (M6), for all.

References

  1. Berthaut, F., Janin, D., Martin, B.: Advanced synchronization of audio or symbolic musical patterns: an algebraic approach. Int. J. Semant. Comput. 6(4), 409–427 (2012)

    Article  MATH  Google Scholar 

  2. Bojańczyk, M., Walukiewicz, I.: Forest algebras. In: Logic and Automata, pp. 107–132. Amsterdam University Press (2008)

  3. Burmeister, P.: A Model Theoretic Oriented Approach to Partial Algebras. Akademie, Berlin (1986)

    MATH  Google Scholar 

  4. Cornock, C., Gould, V.: Proper two-sided restriction semigroups and partial actions. J. Pure Appl. Algebra 216, 935–949 (2012)

    Article  MATH  Google Scholar 

  5. Courcelle, B., Weil, P.: The recognizability of sets of graphs is a robust property. Theor. Comput. Sci. 342, 173–228 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Danos, V., Regnier, L.: Reversible, irreversible and optimal lambda-machines. Theor. Comput. Sci. 227(1–2), 79–97 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dicky, A., Janin, D.: Modélisation algébrique du diner des philosophes. In Modélisation des Systèmes Réactifs (MSR), in Journal Européen des Systèmes Automatisés (JESA Volume 47 - no 1-2-3/2013) (2013)

  8. Ésik, Z., Weil, P.: On logically defined recognizable tree languages. In Found. of Soft. tech and Theor. Comp. Science (FSTTCS), pp. 195–207 (2003)

  9. Gould, V., Hollings, C.: Actions and partial actions of inductive constellations. Semigr. Forum 82(1), 35–60 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hollings, C.D.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2(1), 21–57 (2009)

    MATH  MathSciNet  Google Scholar 

  11. Janin, D.: Vers une modélisation combinatoire des structures rythmiques simples de la musique. Revue Francophone d’Informatique Musicale (RFIM), 2 (2012)

  12. Janin, D.: Algebras, automata and logic for languages of labeled birooted trees. In: Int. Col. on Aut., Lang. and Programming (ICALP), volume 7966 of LNCS, pp. 318–329. Springer, Berlin (2013)

  13. Janin, D.: On languages of one-dimensional overlapping tiles. In Int. Conf. on Current Trends in Theo. and Prac. of Comp. Science (SOFSEM), volume 7741 of LNCS, pp. 244–256. Springer, Berlin (2013)

  14. Janin, D.: Overlaping tile automata. In: 8th International Computer Science Symposium in Russia (CSR), volume 7913 of LNCS, pp. 431–443. Springer, Berlin (2013)

  15. Janin, D.: Walking automata in the free inverse monoid. Research report RR-1464-12, LaBRI, Université de Bordeaux (2013)

  16. Janin, D.: Towards a higher dimensional string theory for the modeling of computerized systems (invited talk). In: Int. Conf. on Current Trends in Theo. and Prac. of Comp. Science (SOFSEM), volume 8327 of LNCS, pp. 7–20. Springer, Berlin (2014)

  17. Janin, D., Berthaut, F., DeSainteCatherine, M.: Multi-scale design of interactive music systems: the libTuiles experiment. In: Sound and Music Computing (SMC) (2013)

  18. Jongh, D., Troelstra, A.: On the connection of partially ordered sets with some pseudo-boolean algebras. Indag. Math. 28, 317–329 (1966)

    Article  Google Scholar 

  19. Kellendonk, J.: The local structure of tilings and their integer group of coinvariants. Commun. Math. Phys. 187, 115–157 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kellendonk, J., Lawson, M.V.: Tiling semigroups. J. Algebra 224(1), 140–150 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Körtesi, P., Radeleczki, S., Szilágyi, S.: Congruences and isotone maps on partially ordered sets. Math. Pannon. 16(1), 39–55 (2005)

    MATH  MathSciNet  Google Scholar 

  22. Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  23. Lawson, M.V.: McAlister semigroups. J. Algebra 202(1), 276–294 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Margolis, S.W., Pin, J.-E.: Languages and inverse semigroups. In: Int. Col. on Aut., Lang. and Programming (ICALP), volume 172 of LNCS, pp. 337–346. Springer, Berlin (1984)

  25. McAlister, D.B.: Inverse semigroups which are separated over a subsemigroups. Trans. Am. Math. Soc. 182, 85–117 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shelah, S.: The monadic theory of order. Ann. Math. 102, 379–419 (1975)

    Article  MATH  Google Scholar 

  27. Silva, P.V.: On free inverse monoid languages. ITA 30(4), 349–378 (1996)

    MATH  Google Scholar 

  28. Stephen, J.B.: Presentations of inverse monoids. J. Pure Appl. Algebra 63, 81–112 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  29. Thomas, W.: Chap. 7. Languages, automata, and logic. In: Handbook of Formal Languages, vol. III, pp. 389–455. Springer, Berlin (1997)

  30. Thomas, W.: Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In: Structures in Logic and Computer Science, volume 1261 of LNCS, pp. 118–143. Springer, Berlin (1997)

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Authors and Affiliations

  1. Mathematik, AG Logik, TU Darmstadt, 64289, Darmstadt, Germany

    Achim Blumensath

  2. Université de Bordeaux, LaBRI UMR 5800, 33405, Talence, France

    David Janin

Authors
  1. Achim Blumensath
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  2. David Janin
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Correspondence to Achim Blumensath.

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Communicated by Mark V. Lawson.

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Blumensath, A., Janin, D. A syntactic congruence for languages of birooted trees. Semigroup Forum 91, 675–698 (2015). https://doi.org/10.1007/s00233-014-9677-x

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