Descriptional Complexity of Error Detection

Part of the Emergence, Complexity and Computation book series (ECC, volume 24)


The neighbourhood of a language L consists of all strings that are within a given distance from a string of L. For example, additive distances or the prefix-distance are regularity preserving in the sense that the neighbourhood of a regular language is always regular. For error detection and error correction applications an important question is to determine the size of the minimal deterministic finite automaton (DFA) needed to recognize the neighbourhood of a language recognized by an n state DFA. This paper surveys recent work on the state complexity of neighbourhoods of regularity preserving distances.


State Complexity Hausdorff Distance Edit Distance Regular Language Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Benedikt, M., Puppis, G., Riveros, C.: Bounded repairability of word languages. J. Comput. Syst. Sci. 79, 1302–1321 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benedikt, M., Puppis, G., Riveros, C.: The per-character cost of repairing word languages. Theoret. Comput. Sci. 539, 38–67 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brzozowski, J., Jirásková, G., Li, B.: Quotient complexity of ideal languages. In: Latin American Theoretical Informatics Symposium, pp. 208–221 (2010)Google Scholar
  4. 4.
    Calude, C.S., Salomaa, K., Yu, S.: Additive distances and quasi-distances between words. J. Univers. Comput. Sci. 8, 141–152 (2002)MathSciNetMATHGoogle Scholar
  5. 5.
    Chatterjee, K., Henzinger, T.A., Ibsen-Jensen, R., Otop, J.: Edit distance for pushdown automata. In: 42nd ICALP, Proceedings. Part II. Lecture Notes in Computer Science, vol. 9135, pp. 121–133 (2015)Google Scholar
  6. 6.
    Choffrut, C., Pighizzini, G.: Distances between languages and reflexivity of relations. Theoret. Comput. Sci. 286, 117–138 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering Codes. Elsevier North-Holland Mathematical Library, vol. 54 (1997)Google Scholar
  8. 8.
    Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer, Berlin Heidelberg (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Dudzinski, K., Konstantinidis, S.: Formal descriptions of code properties: decidability, complexity, implementation. Int. J. Found. Comput. Sci. 23, 67–85 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    El-Mabrouk, N.: On the size of minimal automata for approximate string matching. Technical report, Institut Gaspard Monge, Université de Marne la Vallée, Paris (1997)Google Scholar
  11. 11.
    Gao, Y., Moreira, N., Reis, R., Yu, S.: A survey on operational state complexity. arXiv:1509.03254v1 [cs.FL], Sept 2015. (To appear in Computer Science Review.)
  12. 12.
    Han, Y.-S., Ko, S.-K., Salomaa, K.: The edit-distance between a regular language and a context-free language. Int. J. Found. Comput. Sci. 24, 1067–1082 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kari, L., Konstantinidis, S.: Descriptional complexity of error/edit systems. J. Autom. Lang. Comb. 9(2/3), 293–309 (2004)Google Scholar
  14. 14.
    Kari, L., Konstantinidis, S., Kopecki, S., Yang, M.: An efficient algorithm for computing the edit distance of a regular language via input-altering transducers (2014)Google Scholar
  15. 15.
    Konstantinidis, S.: An algebra of discrete channels that involve combinations of three basic error types. Inform. Comput. 167, 120–131 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Konstantinidis, S.: Transducers and the properties of error-detection, error-correction, and finite-delay decodability. J. Univers. Comput. Sci. 8, 278–291 (2002)MathSciNetMATHGoogle Scholar
  17. 17.
    Konstantinidis, S.: Computing the edit distance of a regular language. Inform. Comput. 205, 1307–1316 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Konstantinidis, S., Silva, P.: Maximal error-detecting capabilities of formal languages. J. Autom. Lang. Comb. 13(1), 55–71 (2008)Google Scholar
  19. 19.
    Konstantinidis, S., Silva, P.V.: Computing maximal error-detecting capabilities and distances of regular languages. Fund. Inform. 101, 257–270 (2010)MathSciNetMATHGoogle Scholar
  20. 20.
    Kutrib, M., Meckel, K., Wendlandt, M.: Parameterized prefix distance between regular languages. In: SOFSEM 2014: Theory and Practice of Computer Science, pp. 419–430 (2014)Google Scholar
  21. 21.
    Leung, H., Podolskiy, V.: The limitedness problem on distance automata: Hashiguchi’s method revisited. Theoret. Comput. Sci. 310, 147–158 (2004)Google Scholar
  22. 22.
    Levenshtein, V.I.: Binary codes capable of correcting deletions, insertions, and reversals. Sov. Phys. Dokl. 10(8), 707–710 (1966)MathSciNetMATHGoogle Scholar
  23. 23.
    Lothaire, M.: Applied combinatorics on words, Chapter 1 algorithms on words. In: Encyclopedia of Mathematics and It’s Applications, vol. 105. Cambridge University Press, New York (2005)Google Scholar
  24. 24.
    Mohri, M.: Edit-distance of weighted automata: general definitions and algorithms. Int. J. Found. Comput. Sci. 14(6), 957–982 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Morelos-Zaragoza, R.H.: The Art of Error Correcting Coding. John Wiley & Sons, Chichester, England (2006)CrossRefGoogle Scholar
  26. 26.
    Ng, T., Rappaport, D., Salomaa, K.: Quasi-distances and weighted finite automata. In: Proceedings of DCFS 2015, Lecture Notes Computer Science, vol. 9118, pp. 209–219 (2015)Google Scholar
  27. 27.
    Ng, T., Rappaport, D., Salomaa, K.: State complexity of neighbourhoods and approximate pattern matching. In: Potapov I., (ed.) Proceedings of DLT 2015. Lecture Notes Computer Science, vol. 9168, pp. 389–400 (2015)Google Scholar
  28. 28.
    Ng, T., Rappaport, D., Salomaa, K.: State complexity of prefix distance. In: Proceedings of CIAA 2015. Lecture Notes Computer Science, vol. 9223, pp. 238–249 (2015)Google Scholar
  29. 29.
    Pighizzini, G.: How hard is computing the edit distance? Inform. Comput. 165(1), 1–13 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Povarov, G.: Descriptive complexity of the Hamming neighborhood of a regular language. In: Language and Automata Theory and Applications, pp. 509–520 (2007). (An updated version available at:
  31. 31.
    Povarov, G.: Finite transducers and nondeterministic state complexity of regular languages. Russ. Math. (Iz. VUZ) 54(6), 19–25 (2010)Google Scholar
  32. 32.
    Salomaa, K., Schofield, P.: State complexity of additive weighted finite automata. Int. J. Found. Comput. Sci. 18(6), 1407–1416 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Shamkin, S.: Descriptional complexity of Hamming neighbourhoods of finite languages (in Russian). M.Sc. Thesis, Ural Federal University, Ekaterinburg, Russia (2011)Google Scholar
  34. 34.
    Schulz, K.U., Mihov, S.: Fast string correction with Levenshtein automata. Int. J. Doc. Anal. Recogn. 5, 67–85 (2002)CrossRefMATHGoogle Scholar
  35. 35.
    Shallit, J.: A Second Course in Formal Languages and Automata Theory, Cambridge University Press (2009)Google Scholar
  36. 36.
    van Lint, J.H.: Introduction to Coding Theory. Springer, Graduate Texts in Mathematics (1999)CrossRefMATHGoogle Scholar
  37. 37.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A., (eds.) Handbook of Formal Languages, pp. 41–110. Springer, Berlin, Heidelberg (1997)Google Scholar

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© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.School of ComputingQueen’s UniversityKingstonCanada

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