Skip to main content
SpringerLink
Log in

Inverse Monoids of Higher-Dimensional Strings

  • Conference paper
  • First Online:
Theoretical Aspects of Computing - ICTAC 2015 (ICTAC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9399))

Included in the following conference series:

Abstract

Halfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph morphism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Finally, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    unambiguity can be generalized to hypergraphs by viewing every binary relation of the form \(\exists \mathbf {z_1} \mathbf {z_2} \mathbf {z_3}\, a(\mathbf {z_1},x,\mathbf {z_2},y,\mathbf {z_3})\) with tuples of FO-variables \(\mathbf {z_1}\), \(\mathbf {z_2}\) and \(\mathbf {z_3}\) of adequate lengths as a primitive binary relation.

References

  1. Abramsky, S.: A structural approach to reversible computation. Theor. Comp. Sci. 347(3), 441–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abrial, J.-R.: Modeling in Event-B - System and Software Engineering. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  3. 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 

  4. Blume, C., Bruggink, H.J.S., Friedrich, M., König, B.: Treewidth, pathwidth and cospan decompositions with applications to graph-accepting tree automata. J. Visual Lang. Comput. 24(3), 192–206 (2013)

    Article  Google Scholar 

  5. Blumensath, A., Janin, D.: A syntactic congruence for languages of birooted trees. Semigroup Forum 91(161), 1–24 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Bruggink, H.J.S., König, B.: On the recognizability of arrow and graph languages. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 336–350. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  8. Courcelle, B.: The monadic second-order logic of graphs V: on closing the gap between definability and recognizability. Theor. Comp. Sci. 80(2), 153–202 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-order Logic, A Language Theoretic Approach, of Encyclopedia of Mathematics and its Applications, vol. 138. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Diekert, V., Lohrey, M., Miller, A.: Partially commutative inverse monoids. Semigroup Forum 77(2), 196–226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fountain, J., Gomes, G., Gould, V.: The free ample monoid. Int. J. Algebra Comput. 19, 527–554 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gadducci, F., Heckel, R.: An inductive view of graph transformation. In: Recent Trends in Algebraic Development Techniques, 12th International Workshop, WADT 1997, Selected Papers, pp. 223–237 (1997)

    Google Scholar 

  14. Hudak, P., Janin, D.: Tiled polymorphic temporal media. In: Work on Functional Art, Music, Modeling and Design (FARM), pp. 49–60. ACM Press (2014)

    Google Scholar 

  15. Janin, D.: Algebras, automata and logic for languages of labeled birooted trees. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 312–323. Springer, Heidelberg (2013)

    Google Scholar 

  16. Janin, D.: Overlapping tile automata. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 431–443. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  17. Janin, D.: On languages of labeled birooted trees: algebras, automata and logic. Inf. Comput. 243, 222–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Janin, D.: Towards a higher-dimensional string theory for the modeling of computerized systems. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 7–20. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  19. Janin, D.: Free inverse monoids up to rewriting. Research report, LaBRI, Université de Bordeaux (2015)

    Google Scholar 

  20. Janin, D.: Walking automata in the free inverse monoid. Research report, LaBRI, Université de Bordeaux (2015)

    Google Scholar 

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

    Google Scholar 

  22. Janin, D., Berthaut, F., DeSainte-Catherine, M., Orlarey, Y., Salvati, S.: The T-calculus: towards a structured programming of (musical) time and space. In: Work on Functional Art, Music, Modeling and Design (FARM), pp. 23–34. ACM Press (2013)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Kellendonk, J., Lawson, M.V.: Universal groups for point-sets and tilings. J. Algebra 276, 462–492 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, River Edge (1998)

    Book  MATH  Google Scholar 

  27. Lewis, H.R.: A new decidable problem, with applications (extended abstract). In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 62–73. IEEE Press (1977)

    Google Scholar 

  28. Margolis, S.W., Meakin, J.C.: E-unitary inverse monoids and the Cayley graph of a group presentation. J. Pure Appl. Algebra 58, 46–76 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Meakin, J.: Groups and semigroups: connections and contrasts. In: Groups St Andrews 2005, London Mathematical Society, Lecture Note Series 340, vol. 2. Cambridge University Press (2007)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Thomas, W.: Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 118–143. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  33. Thomas, W.: Logic for computer science: the engineering challenge. In: Wilhelm, R. (ed.) Informatics. LNCS, vol. 2000, pp. 257–267. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

Download references

Acknowledgements

The idea of developing a notion of higher dimensional strings has been suggested to the author by Mark V. Lawson in 2012. Their presentations have also benefited from numerous and helpful comments from anonymous referees of serval versions of this paper.

Author information

Authors and Affiliations

  1. LaBRI CNRS UMR 5800, Bordeaux INP, Université de Bordeaux, INRIA Bordeaux Sud-Ouest, 33405, Talence, France

    David Janin

Authors
  1. David Janin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Janin .

Editor information

Editors and Affiliations

  1. University of Lübeck, Lübeck, Schleswig-Holstein, Germany

    Martin Leucker

  2. Pontificia Universidad Javeriana-Cali, Cali, Colombia

    Camilo Rueda

  3. Pontificia Universidad Javeriana-Cali, CNRS LIX École Polytechnique de Paris, Palaiseau Cedex, France

    Frank D. Valencia

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Janin, D. (2015). Inverse Monoids of Higher-Dimensional Strings. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25150-9_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25149-3

  • Online ISBN: 978-3-319-25150-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation