Document Open Access Logo

Greibach Normal Form for omega-Algebraic Systems and Weighted Simple omega-Pushdown Automata

Authors Manfred Droste, Sven Dziadek, Werner Kuich

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2019.38.pdf
  • Filesize: 462 kB
  • 14 pages

Document Identifiers

Author Details

Manfred Droste
  • Institut für Informatik, Universität Leipzig, Germany
Sven Dziadek
  • Institut für Informatik, Universität Leipzig, Germany
Werner Kuich
  • Institut für Diskrete Mathematik und Geometrie, Technische Unversität Wien, Austria

Funding

  • Dziadek, Sven: supported by DFG Research Training Group 1763 (QuantLA).
  • Kuich, Werner: partially supported by Austrian Science Fund (FWF): grant no. I1661 N25.

Cite As Get BibTex

Manfred Droste, Sven Dziadek, and Werner Kuich. Greibach Normal Form for omega-Algebraic Systems and Weighted Simple omega-Pushdown Automata. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.FSTTCS.2019.38

Abstract

In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of omega-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted languages, or omega-algebraic series, can be represented as solutions of mixed omega-algebraic systems of equations and by weighted omega-pushdown automata.
In our first main result, we show that mixed omega-algebraic systems can be transformed into Greibach normal form. Our second main result proves that simple omega-reset pushdown automata recognize all omega-algebraic series that are a solution of an omega-algebraic system in Greibach normal form. Simple reset automata do not use epsilon-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Quantitative automata
  • Theory of computation → Grammars and context-free languages
Keywords
  • Weighted omega-Context-Free Grammars
  • Algebraic Systems
  • Greibach Normal Form
  • Weighted Automata
  • omega-Pushdown Automata

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. A. Blass and Y. Gurevich. A note on nested words. Microsoft Research, 2006. URL: https://www.microsoft.com/en-us/research/publication/180-a-note-on-nested-words/.
  2. S. L. Bloom and Z. Ésik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer, 1993. URL: https://doi.org/10.1007/978-3-642-78034-9.
  3. N. Chomsky and M. P. Schützenberger. The Algebraic Theory of Context-Free Languages. In P. Braffort and D. Hirschberg, editors, Computer Programming and Formal Systems, volume 35 of Studies in Logic and the Foundations of Mathematics, pages 118-161. Elsevier, 1963. URL: https://doi.org/10.1016/S0049-237X(08)72023-8.
  4. R. S. Cohen and A. Y. Gold. Theory of ω-Languages I: Characterizations of ω-Context-Free Languages. Journal of Computer and System Sciences, 15(2):169-184, 1977. URL: https://doi.org/10.1016/S0022-0000(77)80004-4.
  5. J. H. Conway. Regular Algebra and Finite Machines. Chapman & Hall, 1971. Google Scholar
  6. M. Droste, S. Dziadek, and W. Kuich. Logic for ω-Pushdown Automata. Information and Computation, 2019. Special issue on "Weighted Automata", Accepted for publication. Google Scholar
  7. M. Droste, S. Dziadek, and W. Kuich. Weighted Simple Reset Pushdown Automata. Theoretical Computer Science, 777:252-259, 2019. URL: https://doi.org/10.1016/j.tcs.2019.01.016.
  8. M. Droste, Z. Ésik, and W. Kuich. The Triple-Pair Construction for Weighted ω-Pushdown Automata. Electronic Proceedings in Theoretical Computer Science, 252:101-113, 2017. URL: https://doi.org/10.4204/EPTCS.252.12.
  9. M. Droste and W. Kuich. A Kleene Theorem for Weighted ω-Pushdown Automata. Acta Cybernetica, 23:43-55, 2017. URL: https://doi.org/10.14232/actacyb.23.1.2017.4.
  10. M. Droste, W. Kuich, and H. Vogler, editors. Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-01492-5.
  11. M. Droste and V. Perevoshchikov. A Logical Characterization of Timed Pushdown Languages. In 10th International Computer Science Symposium in Russia (CSR 2015), volume 9139 of LNCS, pages 189-203. Spinger, 2015. URL: https://doi.org/10.1007/978-3-319-20297-6_13.
  12. S. Eilenberg. Automata, Languages, and Machines, volume 59, Part A of Pure and Applied Mathematics. Elsevier, 1974. URL: https://doi.org/10.1016/S0079-8169(08)60880-6.
  13. C. C. Elgot. Matricial theories. Journal of Algebra, 42(2):391-421, 1976. URL: https://doi.org/10.1016/0021-8693(76)90106-X.
  14. Z. Ésik and W. Kuich. A Semiring-Semimodule Generalization of ω-Context-Free Languages. In J. Karhumäki, H. Maurer, G. Păun, and G. Rozenberg, editors, Theory Is Forever, volume 3113 of LNCS, pages 68-80. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27812-2_7.
  15. Z. Ésik and W. Kuich. A Semiring-Semimodule Generalization of ω-Regular Languages II. Journal of Automata, Languages and Combinatorics, 10(2-3):243-264, 2005. URL: https://doi.org/10.25596/jalc-2005-243.
  16. Z. Ésik and W. Kuich. Modern Automata Theory, 2007. URL: http://www.dmg.tuwien.ac.at/kuich.
  17. Z. Esik and W. Kuich. On Iteration Semiring-Semimodule Pairs. Semigroup Forum, 75(1):129-159, 2007. URL: https://doi.org/10.1007/s00233-007-0709-7.
  18. S. A. Greibach. A New Normal-Form Theorem for Context-Free Phrase Structure Grammars. J. ACM, 12(1):42-52, 1965. URL: https://doi.org/10.1145/321250.321254.
  19. W. Kuich. Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages: Volume 1 Word, Language, Grammar, chapter 9, pages 609-677. Springer, 1997. URL: https://doi.org/10.1007/978-3-642-59136-5_9.
  20. W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of EATCS Monographs on Theoretical Computer Science. Springer, 1986. URL: https://doi.org/10.1007/978-3-642-69959-7.
  21. A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science. Springer, 1978. URL: https://doi.org/10.1007/978-1-4612-6264-0.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact