Skip to main content
SpringerLink
Log in

Unambiguity in Automata Theory

  • Conference paper
  • First Online:
Descriptional Complexity of Formal Systems (DCFS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9118))

Included in the following conference series:

Abstract

Determinism of devices is a key aspect throughout all of computer science, simply because of considerations of efficiency of the implementation. One possible way (among others) to relax this notion is to consider unambiguous machines: non-deterministic machines that have at most one accepting run on each input.

In this paper, we will investigate the nature of unambiguity in automata theory, presenting the cases of standard finite words up to infinite trees, as well as data-words and tropical automata. Our goal is to show how this notion of unambiguity is so far not well understood, and how embarrassing open questions remain open.

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

Notes

  1. 1.

    An automaton is polynomially ambiguous if the number of accepting runs over an input is bounded by a polynomial in the length of the input.

  2. 2.

    Other choices are possible, but these distinctions do not make any difference here.

  3. 3.

    In the context of trees, two forms of determinism for automata are possible: top-down determinism, i.e., from root to leaves, and bottom-up determinism, i.e., from leaves to root. The former (considered here) is known to be strictly weaker than general automata, even over finite trees. The later does not make real sense over infinite trees, since there may be no leaves.

  4. 4.

    A language of infinite trees is bi-unambiguous if it is accepted by a unambiguous infinite tree automaton as well as its complement.

  5. 5.

    Strictly speaking, Conjecture 6 is wrong, but has a corrected version.

References

  1. Almagor, S., Boker, U., Kupferman, O.: What’s decidable about weighted automata? In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 482–491. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  2. Berstel, J.: Transductions and Context-Free Languages. Leitfäden der Angewandten Mathematik und Mechanik [Guides to Applied Mathematics and Mechanics], vol. 38. B.G. Teubner, Stuttgart (1979)

    Google Scholar 

  3. Berstel, J., Sakarovitch, J.: Recent results in the theory of rational sets. In: Gruska, J., Rovan, B., Wiedermann, J. (eds.) Mathematical Foundations of Computer Science (Bratislava 1986). LNCS, vol. 233, pp. 15–28. Springer, Berlin (1986)

    Chapter  Google Scholar 

  4. Bilkowski, M., Skrzypczak, M.: Unambiguity and uniformization problems on infinite trees. In: CSL, volume of LIPIcs, pp. 81–100 (2013)

    Google Scholar 

  5. Bojańczyk, M., Lasota, S.: Fraenkel-mostowski sets with non-homogeneous atoms. In: Finkel, A., Leroux, J., Potapov, I. (eds.) RP 2012. LNCS, vol. 7550, pp. 1–5. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Cadilhac, M., Finkel, A., McKenzie, P.: Unambiguous constrained automata. Int. J. Found. Comput. Sci. 24(7), 1099–1116 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Carayol, A., Löding, C.: MSO on the infinite binary tree: choice and order. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 161–176. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  8. Carayol, A., Löding, C., Niwiński, D., Walukiewicz, I.: Choice functions and well-orderings over the infinite binary tree. Cent. Eur. J. Math. 8(4), 662–682 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carton, O., Michel, M.: Unambiguous Büchi automata. Theor. Comput. Sci. 297(1–3), 37–81 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Colcombet, T.: Forms of determinism for automata. In: Dürr, C., Wilke, T. (eds.) STACS 2012: 29th International Symposium on Theoretical Aspects of Computer Science, Volume 14 of LIPIcs, pp. 1–23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

    Google Scholar 

  11. Colcombet, T., Puppis, G., Skrypczak, M.: Unambiguous register automata. Unpublished

    Google Scholar 

  12. Coste, F., Fredouille, D.C.: Unambiguous automata inference by means of state-merging methods. In: Lavrač, N., Gamberger, D., Todorovski, L., Blockeel, H. (eds.) ECML 2003. LNCS (LNAI), vol. 2837, pp. 60–71. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  13. Cristau, J.: Automata and temporal logic over arbitrary linear time (2011). CoRR, abs/1101.1731

  14. Droste, M., Kuske, D.: Weighted automata. To appear in Handbook AutoMathA (2013)

    Google Scholar 

  15. Goldstine, J., Kappes, M., Kintala, C.M.R., Leug, H., Malcher, A., Wotschke, D.: Descriptional complexity of machines with limited resources. J. Univ. Comput. Sci. 8, 193–234 (2002)

    MATH  Google Scholar 

  16. Gurevich, Y., Shelah, S.: Rabin’s uniformization problem. J. Symb. Log. 48(4), 1105–1119 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Holzer, M., Kutrib, M.: Descriptional complexity of (un)ambiguous finite state machines and pushdown automata. In: Kučera, A., Potapov, I. (eds.) RP 2010. LNCS, vol. 6227, pp. 1–23. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  18. Kaminski, M., Francez, N.: Finite-memory automata. Theor. Comput. Sci. 134(2), 329–363 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kirsten, D., Lombardy, S.: Deciding unambiguity and sequentiality of polynomially ambiguous min-plus automata. In: 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, 26–28 February 2009, Freiburg, Germany, Proceedings, pp. 589–600 (2009)

    Google Scholar 

  20. Klimann, I., Lombardy, S., Mairesse, J., Prieur, C.: Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton. Theor. Comput. Sci. 327(3), 349–373 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Krob, D.: The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Int. J. Algebra Comput. 4(3), 405–425 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Leiss, E.L.: Succint representation of regular languages by boolean automata. Theor. Comput. Sci. 13, 323–330 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27(4), 1073–1082 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Leung, H.: Descriptional complexity of NFA of different ambiguity. Int. J. Found. Comput. Sci. 16(5), 975–984 (2005)

    Article  MATH  Google Scholar 

  25. Leung, H.: Structurally unambiguous finite automata. In: Ibarra, O.H., Yen, H.-C. (eds.) CIAA 2006. LNCS, vol. 4094, pp. 198–207. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  26. Lombardy, S., Mairesse, J.: Series which are both max-plus and min-plus are unambiguous. RAIRO - Theor. Inf. Appl. 40(1), 1–14 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lupanov, O.B.: A comparison of two types of finite sources. Problemy Kybernetiki 9, 321–326 (1963)

    Google Scholar 

  28. Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: Symposium on Switching and Automata Theory, pp. 188–191. IEEE (1971)

    Google Scholar 

  29. Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Trans. Comput. 20(10), 1211–1214 (1971)

    Article  MATH  Google Scholar 

  30. Niwiński, D., Walukiewicz, I.: Ambiguity problem for automata on infinite trees (1996). unpublished

    Google Scholar 

  31. Okhotin, A., Salomaa, K.: Descriptional complexity of unambiguous input-driven pushdown automata. Theor. Comput. Sci. 566, 1–11 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  32. Pin, J.É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Syst. 30(4), 383–422 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)

    MATH  MathSciNet  Google Scholar 

  34. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3, 114–125 (1959)

    Article  MathSciNet  Google Scholar 

  35. Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM J. Comput. 29(4), 1118–1131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  37. Schmidt, E.M.: Succinctness of descriptions of context-free, regular and finite languages. Ph.D. thesis, Cornell University (1977)

    Google Scholar 

  38. Schützenberger, M.-P.: On the definition of a family of automata. Inf. Control 4, 245–270 (1961)

    Article  MATH  Google Scholar 

  39. Schützenberger, M.-P.: Sur les relations fonctionnelles. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 209–213. Springer, Heidelberg (1975)

    Google Scholar 

  40. Seidl, H.: Deciding equivalence of finite tree automata. SIAM J. Comput. 19(3), 424–437 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  41. Stearns, R.E., Hunt III, H.B.: On the equivalence and containment problems for unambiguous regular expressions, grammars, and automata. In: 22nd Annual Symposium on Foundations of Computer Science, Nashville, Tennessee, USA, 28–30 October 1981, pp. 74–81 (1981)

    Google Scholar 

  42. Thomas, W.: Languages, automata and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 389–455. Springer, New York (1997). Chap. 7

    Chapter  Google Scholar 

  43. Weber, A.: Decomposing a k-valued transducer into k unambiguous ones. ITA 30(5), 379–413 (1996)

    MATH  Google Scholar 

Download references

Acknowledgment

I am really grateful to Jean-Éric Pin, Gabriele Puppis and Michał Skrypczak for their precious help and their discussions on the topic.

Author information

Authors and Affiliations

  1. CNRS, Université Paris 7 – Paris Diderot, Paris, France

    Thomas Colcombet

Authors
  1. Thomas Colcombet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Colcombet .

Editor information

Editors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Jeffrey Shallit

  2. Department of Mathematics, University of Turku, Turku, Finland

    Alexander Okhotin

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Colcombet, T. (2015). Unambiguity in Automata Theory. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19225-3_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19224-6

  • Online ISBN: 978-3-319-19225-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation