DFA with a Bounded Activity Level

  • Marius Konitzer
  • Hans Ulrich Simon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)


Lookahead DFA are used during parsing for sake of resolving conflicts (as described in more detail in the introduction). The parsing of an input string w may require many DFA-explorations starting from different letter positions. This raises the question how many of these explorations can be active at the same time. If there is a bound on this number depending on the given DFA M only (i.e., the bound is valid for all input strings w), we say that M has a bounded activity level. The main results in this paper are as follows. We define an easy-to-check property of DFA named prefix-cyclicity and show that precisely the non prefix-cyclic DFA have a bounded activity level. Moreover, the largest possible number ℓ M of mutually overlapping explorations of a given non prefix-cyclic DFA M with t + 1 states, the so-called maximum activity level of M, is bounded from above by 2 t  − 1, and this bound is tight. We show furthermore that the maximum activity levels of equivalent DFA coincide so as to form an invariant of the underlying regular language, which leads us to a characterization of prefix-cyclicity in terms of the Nerode relation. We finally establish some complexity results. For instance, the problem of computing ℓ M for a given non prefix-cyclic DFA M is shown to be PSPACE-complete.


parsing lookahead DFA computational complexity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bermudez, M.E., Schimpf, K.M.: Practical arbitrary lookahead LR parsing. Journal of Computer and System Sciences 41(2), 230–250 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Boullier, P.: Contribution à la construction automatique d’analyseurs lexicographiques et syntaxiques. Ph.D. thesis, Université d’Orléans (1984)Google Scholar
  3. 3.
    Chung, F.R.K., Grinstead, C.M.: A survey of bounds for classical Ramsey numbers. Journal of Graph Theory 7(1), 25–37 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Farré, J., Fortes Gálvez, J.: A bounded graph-connect construction for LR-regular parsers. In: Wilhelm, R. (ed.) CC 2001. LNCS, vol. 2027, pp. 244–258. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Fredricksen, H.: Schur numbers and the Ramsey numbers N(3,3,.,3;2). Journal of Combinatorial Theory, Series A 27(3), 376–377 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gosling, J., Joy, B., Steele, G.: The Java\(^{\mbox{{\tiny TM}}}\) Language Specification. Addison-Wesley (1996)Google Scholar
  7. 7.
    Knuth, D.E.: On the translation of languages from left to right. Information and Control 8(6), 607–639 (1965)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Konitzer, M.: Laufzeitanalyse und Optimierung von Parsern für LR-reguläre Grammatiken. Ph.D. thesis, Ruhr-University Bochum (2013)Google Scholar
  9. 9.
    Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Symposium on Foundations of Computer Science, pp. 254–266 (1977)Google Scholar
  10. 10.
    Schmitz, S.: Approximating Context-Free Grammars for Parsing and Verification. Ph.D. thesis, Université de Nice-Sophia Antipolis (2007)Google Scholar
  11. 11.
    Čulik, K., Cohen, R.: LR-regular grammars - an extension of LR(k) grammars. Journal of Computer and System Sciences 7(1), 66–96 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wan, H.: Upper bounds for Ramsey numbers R(3,3,...,3) and Schur numbers. Journal of Graph Theory 26(3), 119–122 (1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Marius Konitzer
    • 1
  • Hans Ulrich Simon
    • 1
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

Personalised recommendations