Skip to main content
SpringerLink
Log in

Descriptional Complexity of (Un)ambiguous Finite State Machines and Pushdown Automata

  • Conference paper
Reachability Problems (RP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6227))

Included in the following conference series:

Abstract

Unambiguity and its generalization to quantified ambiguity are important concepts in, e.g., automata and complexity theory. Basically, an unambiguous machine has at most one accepting computation path for each accepted word. While unambiguous pushdown automata induce a language family strictly in between the deterministic and general context-free languages, unambiguous finite automata capture the regular languages, that is, they are equally powerful as deterministic and nondeterministic finite automata. However, their descriptional capacity is significantly different. In the present paper, we summarize and discuss developments relevant to (un)ambiguous finite automata and pushdown automata problems from the descriptional complexity point of view. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

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. Birget, J.C.: Intersection and union of regular languages and state complexity. Inform. Process. Lett. 43, 185–190 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Björklund, H., Martens, W.: The tractability frontier for NFA minimization. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 27–38. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  3. Borchardt, I.: Nonrecursive tradeoffs between context-free grammars with different constant ambiguity. Diploma thesis, Universität Frankfurt (1992) (in German)

    Google Scholar 

  4. Cho, S., Huynh, D.T.: The parallel complexity of finite-state automata problems. Inform. Comput. 97, 1–22 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47, 149–158 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chrobak, M.: Errata to “finite automata and unary languages”. Theoret. Comput. Sci. 302, 497–498 (2003)

    Article  MathSciNet  Google Scholar 

  7. Fischer, P.C., Kintala, C.M.R.: Real-time computations with restricted nondeterminism. Math. Systems Theory 12, 219–231 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gill, A., Kou, L.T.: Multiple-entry finite automata. J. Comput. System Sci. 9, 1–19 (1974)

    MATH  MathSciNet  Google Scholar 

  9. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962)

    Article  MathSciNet  Google Scholar 

  10. Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59, 75–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gold, E.M.: Complexity of automaton identification from given data. Inform. Control 37, 302–320 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goldstine, J., Kintala, C.M.R., Wotschke, D.: On measuring nondeterminism in regular languages. Inform. Comput. 86, 179–194 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goldstine, J., Leung, H., Wotschke, D.: On the relation between ambiguity and nondeterminism in finite automata. Inform. Comput. 100, 261–270 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Goldstine, J., Leung, H., Wotschke, D.: Measuring nondeterminism in pushdown automata. J. Comput. System Sci. 71, 440–466 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Goldstine, J., Price, J.K., Wotschke, D.: On reducing the number of states in a PDA. Math. Systems Theory 15, 315–321 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  16. Goldstine, J., Price, J.K., Wotschke, D.: On reducing the number of stack symbols in a PDA. Math. Systems Theory 26, 313–326 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gruber, H., Holzer, M.: Finding lower bounds for nondeterministic state complexity is hard (extended abstract). In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 363–374. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Gruber, H., Holzer, M., Kutrib, M.: On measuring non-recursive trade-offs. In: Descriptional Complexity of Formal Systems (DCFS 2009), pp. 187–198. Otto-von-Guericke-Universität Magdeburg (2009)

    Google Scholar 

  19. Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Reading (1978)

    MATH  Google Scholar 

  20. Hartmanis, J.: Context-free languages and Turing machine computations. In: Proc. Symposia in Applied Mathematics, vol. 19, pp. 42–51 (1967)

    Google Scholar 

  21. Hartmanis, J.: On the succinctness of different representations of languages. SIAM J. Comput. 9, 114–120 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hartmanis, J.: On Gödel speed-up and succinctness of language representations. Theoret. Comput. Sci. 26, 335–342 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  23. Herzog, C.: Pushdown automata with bounded nondeterminism and bounded ambiguity. Theoret. Comput. Sci. 181, 141–157 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Holzer, M., Kutrib, M.: Descriptional and computational complexity of finite automata. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 23–42. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  25. Holzer, M., Kutrib, M.: Nondeterministic finite automata - Recent results on the descriptional and computational complexity. Int. J. Found. Comput. Sci. 20, 563–580 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Holzer, M., Kutrib, M.: Descriptional complexity – An introductory survey. In: Scientific Applications of Language Methods. Imperial College Press, London (to appear, 2010)

    Google Scholar 

  27. Holzer, M., Salomaa, K., Yu, S.: On the state complexity of k-entry deterministic finite automata. J. Autom., Lang. Comb. 6, 453–466 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Hopcroft, J.E.: An nlogn algorithm for minimizing the state in a finite automaton. In: The Theory of Machines and Computations, pp. 189–196. Academic Press, London (1971)

    Google Scholar 

  29. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  30. Hromkovič, J., Schnitger, G.: Ambiguity and communication. In: Theoretical Aspects of Computer Science (STACS 2009), Dagstuhl, Germany. LIPICS, vol. 3, pp. 107–118 (2009)

    Google Scholar 

  31. Hromkovič, J.: Communication Complexity and Parallel Computing. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  32. Hromkovič, J., Seibert, S., Karhumäki, J., Klauck, H., Schnitger, G.: Communication complexity method for measuring nondeterminism in finite automata. Inform. Comput. 172, 202–217 (2002)

    Article  MATH  Google Scholar 

  33. Ibarra, O.H., Ravikumar, B.: On sparseness, ambiguity and other decision problems for acceptors and transducers. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 1986. LNCS, vol. 210, pp. 171–179. Springer, Heidelberg (1986)

    Google Scholar 

  34. Jiang, T., McDowell, E., Ravikumar, B.: The structure and complexity of minimal NFA’s over a unary alphabet. Int. J. Found. Comput. Sci. 2, 163–182 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jiang, T., Ravikumar, B.: Minimal NFA problems are hard. SIAM J. Comput. 22, 1117–1141 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kappes, M.: Descriptional complexity of deterministic finite automata with multiple initial states. J. Autom., Lang. Comb. 5, 269–278 (2000)

    MATH  MathSciNet  Google Scholar 

  37. Kintala, C.M.R.: Computations with a Restricted Number of Nondeterministic Steps. PhD thesis, Pennsylvania State University (1977)

    Google Scholar 

  38. Kintala, C.M.R., Wotschke, D.: Amounts of nondeterminism in finite automata. Acta Inform. 13, 199–204 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kuich, W.: Finite automata and ambiguity. Report 253 of the IIG, Technische Universität Graz (1988)

    Google Scholar 

  40. Kutrib, M.: The phenomenon of non-recursive trade-offs. Int. J. Found. Comput. Sci. 16, 957–973 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  41. Kutrib, M., Malcher, A.: Context-dependent nondeterminism for pushdown automata. Theoret. Comput. Sci. 376, 101–111 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Kutrib, M., Malcher, A., Werlein, L.: Regulated nondeterminism in pushdown automata. Theoret. Comput. Sci. 410, 3447–3460 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  43. Landau, E.: Über die Maximalordnung der Permutationen gegebenen Grades. Archiv der Math. und Phys. 3, 92–103 (1903)

    Google Scholar 

  44. Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen. Teubner (1909)

    Google Scholar 

  45. Leiss, E.L.: Succinct representation of regular languages by Boolean automata. Theoret. Comput. Sci. 13, 323–330 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  46. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  49. Lupanov, O.B.: A comparison of two types of finite sources. Problemy Kybernetiki 9, 321–326 (1963) (in Russian); German translation: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 328–335 (1966)

    Google Scholar 

  50. Malcher, A.: Minimizing finite automata is computationally hard. Theoret. Comput. Sci. 327, 375–390 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Chapter  Google Scholar 

  52. 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, 1211–1214 (1971)

    Article  MATH  Google Scholar 

  53. Okhotin, A.: A study of unambiguous finite automata over a one-letter alphabet. TUCS Technical Report No 951, Turku Centre for Computer Science (2009)

    Google Scholar 

  54. Pighizzini, G.: Deterministic pushdown automata and unary languages. In: Ibarra, O.H., Ravikumar, B. (eds.) CIAA 2008. LNCS, vol. 5148, pp. 232–241. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  55. Pighizzini, G., Shallit, J., Wang, M.W.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. J. Comput. System Sci. 65, 393–414 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  57. Ravikumar, B., Ibarra, O.H.: Relating the type of ambiguity of finite automata to the succinctness of their representation. SIAM J. Comput. 18, 1263–1282 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  58. Salomaa, A., Soittola, M.: Automata-theoretic Aspects of Formal Power Series. Springer, Heidelberg (1978)

    MATH  Google Scholar 

  59. Salomaa, K., Yu, S.: Limited nondeterminism for pushdown automata. Bull. EATCS 50, 186–193 (1993)

    MATH  Google Scholar 

  60. Salomaa, K., Yu, S.: Measures of nondeterminism for pushdown automata. J. Comput. System Sci. 49, 362–374 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  61. Schmidt, E.M.: Succinctness of Dscriptions of Context-Free, Regular and Finite Languages. PhD thesis, Cornell University, Ithaca, NY (1978)

    Google Scholar 

  62. Schmidt, E.M., Szymanski, T.G.: Succinctness of descriptions of unambiguous context-free languages. SIAM J. Comput. 6, 547–553 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  63. Sipser, M.: Lower bounds on the size of sweeping automata. J. Comput. System Sci. 21, 195–202 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  64. Stearns, R.E.: A regularity test for pushdown machines. Inform. Control 11, 323–340 (1967)

    Article  MATH  Google Scholar 

  65. Stearns, R.E., Hunt III, H.B.: On the equivalence and containment problems for unambiguous regular expressions, regular grammars, and finite automata. SIAM J. Comput. 14, 598–611 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  66. Valiant, L.G.: Regularity and related problems for deterministic pushdown automata. J. ACM 22, 1–10 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  67. Valiant, L.G.: A note on the succinctness of descriptions of deterministic languages. Inform. Control 32, 139–145 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  68. Veloso, P.A.S., Gill, A.: Some remarks on multiple-entry finite automata. J. Comput. System Sci. 18, 304–306 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  69. Vermeir, D., Savitch, W.: On the amount of nondeterminism in pushdown automata. Fund. Inform. 4, 401–418 (1981)

    MATH  MathSciNet  Google Scholar 

  70. Weber, A.: Über die Mehrdeutigkeit und Wertigkeit von endlichen Automaten und Transducern. Dissertation, Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt am Main (1987) (in German)

    Google Scholar 

  71. Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theoret. Comput. Sci. 88, 325–349 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392, Giessen, Germany

    Markus Holzer & Martin Kutrib

Authors
  1. Markus Holzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Kutrib
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Informatics, Masaryk University, Botanická 68a, 60200, Brno, Czech Republic

    Antonín Kučera

  2. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, England

    Igor Potapov

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Holzer, M., Kutrib, M. (2010). Descriptional Complexity of (Un)ambiguous Finite State Machines and Pushdown Automata. In: Kučera, A., Potapov, I. (eds) Reachability Problems. RP 2010. Lecture Notes in Computer Science, vol 6227. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15349-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15349-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15348-8

  • Online ISBN: 978-3-642-15349-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation