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On the Number of Nonterminal Symbols in Unambiguous Conjunctive Grammars

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Descriptional Complexity of Formal Systems (DCFS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7386))

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Abstract

It is demonstrated that the family of languages generated by unambiguous conjunctive grammars with 1 nonterminal symbol is strictly included in the languages generated by 2-nonterminal grammars, which is in turn a proper subset of the family generated using 3 or more nonterminal symbols. This hierarchy is established by considering grammars over a one-letter alphabet, for which it is shown that 1-nonterminal grammars generate only regular languages, 2-nonterminal grammars generate some non-regular languages, but all of them have upper density zero, while 3-nonterminal grammars may generate some non-regular languages of non-zero density. It is also shown that the equivalence problem for 2-nonterminal grammars is undecidable.

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References

  1. Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003)

    Google Scholar 

  2. Baader, F., Okhotin, A.: Complexity of language equations with one-sided concatenation and all Boolean operations. In: 20th International Workshop on Unification, UNIF 2006, Seattle, USA, August 11, pp. 59–73 (2006)

    Google Scholar 

  3. Culik II, K., Gruska, J., Salomaa, A.: Systolic trellis automata, I and II. International Journal of Computer Mathematics 15, 195–212; 16, 3–22 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dyer, C.: One-way bounded cellular automata. Information and Control 44, 261–281 (1980)

    Google Scholar 

  5. Gruska, J.: Descriptional complexity of context-free languages. In: Mathematical Foundations of Computer Science, MFCS 1973, Strbské Pleso, High Tatras, Czechoslovakia, September 3-8, pp. 71–83 (1973)

    Google Scholar 

  6. Jeż, A.: Conjunctive grammars can generate non-regular unary languages. International Journal of Foundations of Computer Science 19(3), 597–615 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jeż, A., Okhotin, A.: Conjunctive grammars over a unary alphabet: undecidability and unbounded growth. Theory of Computing Systems 46(1), 27–58 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jeż, A., Okhotin, A.: Complexity of equations over sets of natural numbers. Theory of Computing Systems 48(2), 319–342 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jeż, A., Okhotin, A.: One-nonterminal conjunctive grammars over a unary alphabet. Theory of Computing Systems 49(2), 319–342 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jeż, A., Okhotin, A.: Unambiguous conjunctive grammars over a one-letter alphabet. TUCS Technical Report 1043, Turku Centre for Computer Science (submitted for publication, April 2012)

    Google Scholar 

  11. Okhotin, A.: Conjunctive grammars. Journal of Automata, Languages and Combinatorics 6(4), 519–535 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. Informatique Théorique et Applications 38(1), 69–88 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Okhotin, A.: On the number of nonterminals in linear conjunctive grammars. Theoretical Computer Science 320(2-3), 419–448 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Okhotin, A.: Unambiguous Boolean grammars. Information and Computation 206, 1234–1247 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Okhotin, A.: Fast Parsing for Boolean Grammars: A Generalization of Valiant’s Algorithm. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 340–351. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Okhotin, A., Reitwießner, C.: Parsing unary Boolean grammars using online convolution. In: Advances and Applications of Automata on Words and Trees (Dagstuhl seminar 10501), December 12-17 (2010)

    Google Scholar 

  17. Okhotin, A., Rondogiannis, P.: On the expressive power of univariate equations over sets of natural numbers. Information and Computation 212, 1–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, Poland

    Artur Jeż

  2. Department of Mathematics, University of Turku, Finland

    Alexander Okhotin

Authors
  1. Artur Jeż
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  2. Alexander Okhotin
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Editor information

Editors and Affiliations

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

    Martin Kutrib

  2. Departamento de Ciência de Computadores, Universidade do Porto, Faculdade de Ciências, Rua do Campo Alegre 1021-1055, 4169-007, Porto, Portugal

    Nelma Moreira  & Rogério Reis  & 

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Jeż, A., Okhotin, A. (2012). On the Number of Nonterminal Symbols in Unambiguous Conjunctive Grammars. In: Kutrib, M., Moreira, N., Reis, R. (eds) Descriptional Complexity of Formal Systems. DCFS 2012. Lecture Notes in Computer Science, vol 7386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31623-4_14

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  • DOI: https://doi.org/10.1007/978-3-642-31623-4_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31622-7

  • Online ISBN: 978-3-642-31623-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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