Skip to main content
SpringerLink
Log in

Where Automatic Structures Benefit from Weighted Automata

  • Chapter
Algebraic Foundations in Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7020))

Abstract

In this paper, we report on applications of weighted automata in the theory of automatic structures. All (except one) result were known before, but their proof using weighted automata is novel. More precisely, we prove that the extension of first-order logic by the infinity ∃  ∞ , the modulo ∃ (p,q), and the (new) boundedness quantifier is decidable. The first two quantifiers are handled using closure properties of the class of recognizable formal power series and the fact that the preimage of a value under a recognizable formal power series is regular if the semiring is finite. Our reasoning regarding the boundedness quantifier uses Weber’s decidability result of finite-valued rational transductions. We also show that the isomorphism problem of automatic structures is undecidable using an undecidability result on recognizable formal power series due to Honkala.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bárány, V.: Invariants of automatic presentations and semi-synchronous transductions. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 289–300. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Bárány, V., Grädel, E., Rubin, S.: Automata-based presentations of infinite structures. In: Finite and Algorithmic Model Theory, pp. 1–76. Cambridge University Press, Cambridge (2011)

    Chapter  Google Scholar 

  3. Bárány, V., Kaiser, Ł., Rubin, S.: Cardinality and counting quantifiers on omega-automatic structures. In: STACS 2008, pp. 385–396. IFIB Schloss Dagstuhl (2008)

    Google Scholar 

  4. Blumensath, A.: Automatic structures. Tech. rep., RWTH Aachen (1999)

    Google Scholar 

  5. Blumensath, A., Grädel, E.: Automatic Structures. In: LICS 2000, pp. 51–62. IEEE Computer Society Press, Los Alamitos (2000)

    Google Scholar 

  6. Braun, G., Strüngmann, L.: Breaking up finite automata presentable torsion-free abelian groups. International Journal of Algebra and Computation (accepted, 2011)

    Google Scholar 

  7. Campbell, C., Robertson, E., Ruškuc, N., Thomas, R.: Automatic semigroups. Theoretical Computer Science 250, 365–391 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Corran, R., Hoffmann, M., Kuske, D., Thomas, R.: Singular Artin monoids of finite type are automatic. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 250–261. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  9. Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. C. R. Acad. Sci. Paris, Ser. I 339, 5–10 (2004)

    Article  MathSciNet  Google Scholar 

  10. Delhommé, C., Goranko, V., Knapik, T.: Automatic linear orderings (2003) (manuscript)

    Google Scholar 

  11. Droste, M., Kuich, W.: Semirings and formal power series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 3–28. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  12. Elgot, C.: Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 98, 21–51 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  13. Epstein, D., Cannon, J., Holt, D., Levy, S., Paterson, M., Thurston, W.: Word Processing In Groups. Jones and Bartlett Publishers, Boston (1992)

    MATH  Google Scholar 

  14. Ershov, Y., Goncharov, S., Nerode, A., Remmel, J. (eds.): Handbook of recursive mathematics, vol. 1, 2. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  15. Fohry, E., Kuske, D.: On graph products of automatic and biautomatic monoids. Semigroup Forum 72, 337–352 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  17. Hodgson, B.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Honkala, J.: On the problem whether the image of an ℕ-rational series equals ℕ. Fundamenta Informaticae 73(1-2), 127–132 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  20. Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic structures: richness and limitations. Log. Methods in Comput. Sci. 3(2) (2007)

    Google Scholar 

  21. Khoussainov, B., Rubin, S., Stephan, F.: Definability and regularity in automatic structures. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 440–451. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  22. Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Transactions on Computational Logic 6(4), 675–700 (2005)

    Article  MathSciNet  Google Scholar 

  23. Kuske, D.: Is Cantor’s theorem automatic? In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 332–345. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  24. Kuske, D., Liu, J., Lohrey, M.: The isomorphism problem on classes of automatic structures. In: LICS 2010, pp. 160–169. IEEE Computer Society Press, Los Alamitos (2010)

    Google Scholar 

  25. Kuske, D., Lohrey, M.: First-order and counting theories of ω-automatic structures. Journal of Symbolic Logic 73, 129–150 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuske, D., Lohrey, M.: Some natural problems in automatic graphs. Journal of Symbolic Logic 75(2), 678–710 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nies, A.: Describing groups. Bulletin of Symbolic Logic 13(3), 305–339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oliver, G., Thomas, R.: Automatic presentations for finitely generated groups. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 693–704. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  29. Rubin, S.: Automata presenting structures: A survey of the finite string case. Bulletin of Symbolic Logic 14, 169–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sakarovitch, J.: Easy multiplications. I. The realm of Kleene’s Theorem. Information and Computation 74, 173–197 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sakarovitch, J.: Rational and recognizable series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 405–453. Springer, Heidelberg (2009)

    Google Scholar 

  32. Tsankov, T.: The additive group of the rationals does not have an automatic presentation (2009), http://arxiv.org/abs/0905.1505

  33. Weber, A.: On the valuedness of finite transducers. Acta Informatica 27, 749–780 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Informatik, Technische Universität Ilmenau, Germany

    Dietrich Kuske

Authors
  1. Dietrich Kuske
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria

    Werner Kuich

  2. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    George Rahonis

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kuske, D. (2011). Where Automatic Structures Benefit from Weighted Automata. In: Kuich, W., Rahonis, G. (eds) Algebraic Foundations in Computer Science. Lecture Notes in Computer Science, vol 7020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24897-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24897-9_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24896-2

  • Online ISBN: 978-3-642-24897-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation