Semirings and Formal Power Series

Abstract

This chapter presents basic foundations for the theory of weighted automata: semirings and formal power series. A fundamental question is how to extend the star operation (Kleene iteration) from languages to series. For this, we investigate ordered, complete and continuous semirings and the related concepts of star semirings and Conway semirings. We derive natural properties for the Kleene star of cycle-free series and also of matrices often used to analyze the behavior of weighted automata. Finally, we investigate cycle-free linear equations which provide a useful tool for proving identities for formal power series.

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References

  1. 1.
    J. Adámek, E. Nelson, and J. Reiterman. Tree constructions of free continuous algebras. Journal of Computer and System Sciences, 24:114–146, 1982. CrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Berstel, editor. Séries formelles en variables non commutatives et applications. Laboratoire d’Informatique Théorique et Programmation, Ecole Nationale Supérieure de Techniques Avancées, Paris, 1978. MATHGoogle Scholar
  3. 3.
    J. Berstel and C. Reutenauer. Les séries rationelles et leurs langages. Masson, Paris, 1984. English translation: Rational Series and Their Languages, volume 12 of Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 1988. Google Scholar
  4. 4.
    S.L. Bloom and Z. Ésik. Iteration Theories, Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 1993. MATHGoogle Scholar
  5. 5.
    G. Carré. Graphs and Networks. Clarendon, Oxford, 1979. MATHGoogle Scholar
  6. 6.
    J.H. Conway. Regular Algebra and Finite Machines. Chapman & Hall, London, 1971. MATHGoogle Scholar
  7. 7.
    M. Droste and P. Gastin. On aperiodic and star-free formal power series in partially commuting variables. Theory of Computing Systems, 42:608–631, 2008. Extended abstract in: D. Krob, A.A. Milchalev, and A.V. Milchalev, editors, Formal Power Series and Algebraic Combinatorics, 12th Int. Conf., Moscow, pages 158–169. Springer, Berlin, 2000. MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Eilenberg. Automata, Languages and Machines, volume A. Academic Press, San Diego, 1974. Google Scholar
  9. 9.
    Z. Ésik and W. Kuich. Modern automata theory. www.dmg.tuwien.ac.at/kuich.
  10. 10.
    Z. Ésik and W. Kuich. Locally closed semirings. Monatshefte für Mathematik, 137:21–29, 2002. MATHCrossRefGoogle Scholar
  11. 11.
    Z. Ésik and W. Kuich. Equational axioms for a theory of automata. In C. Martin-Vide, V. Mitrana, and G. Paun, editors, Formal Languages and Applications, volume 148 of Studies in Fuzziness and Soft Computing, pages 183–196. Springer, Berlin, 2004. Google Scholar
  12. 12.
    Z. Ésik and W. Kuich. Finite automata. In M. Droste, W. Kuich, and H. Vogler, editors, Handbook of Weighted Automata. Chapter 3. Springer, Berlin, 2009. Google Scholar
  13. 13.
    K. Głazek. A Guide to the Literature on Semirings and Their Applications in Mathematics and Information Science. Kluwer Academic, Dordrecht, 2002. Google Scholar
  14. 14.
    J.A. Goguen, J.W. Thatcher, E.G. Wagner, and J.B. Wright. Initial algebra semantics and continuous algebras. Journal of the Association for Computing Machinery, 24:68–95, 1977. MATHMathSciNetGoogle Scholar
  15. 15.
    J. Golan. Semirings and Their Applications. Kluwer Academic, Dordrecht, 1999. MATHGoogle Scholar
  16. 16.
    M. Goldstern. Vervollständigung von Halbringen. Diplomarbeit, Technische Universität Wien, 1985. Google Scholar
  17. 17.
    I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer, Berlin, 1981. MATHGoogle Scholar
  18. 18.
    P. Hájek. Metamathematics of Fuzzy Logic. Kluwer Academic, Dordrecht, 1998. MATHGoogle Scholar
  19. 19.
    U. Hebisch. The Kleene theorem in countably complete semirings. Bayreuther Mathematische Schriften, 31:55–66, 1990. MATHMathSciNetGoogle Scholar
  20. 20.
    U. Hebisch and H.J. Weinert. Halbringe—Algebraische Theorie und Anwendungen in der Informatik. Teubner, Leipzig, 1993. English translation: Semirings—Algebraic Theory and Applications in Computer Science. World Scientific, Singapore, 1998. MATHGoogle Scholar
  21. 21.
    B. Heidergott, G.J. Olsder, and J. van der Woude. Max Plus at Work. Princeton University Press, Princeton, 2006. MATHGoogle Scholar
  22. 22.
    G. Karner. On limits in complete semirings. Semigroup Forum, 45:148–165, 1992. MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    G. Karner. Continuous monoids and semirings. Theoretical Computer Science, 318:355–372, 2004. MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    D. Kozen. A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation, 110:366–390, 1994. MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    D. Krob. Monoides et semi-anneaux complets. Semigroup Forum, 36:323–339, 1987. MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    D. Krob. Monoides et semi-anneaux continus. Semigroup Forum, 37:59–78, 1988. MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    W. Kuich. The Kleene and the Parikh theorem in complete semirings. In ICALP’87, volume 267 of Lecture Notes in Computer Science, pages 212–225. Springer, Berlin, 1987. Google Scholar
  28. 28.
    W. Kuich. Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, Chapter 9, pages 609–677. Springer, Berlin, 1997. Google Scholar
  29. 29.
    W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 1986. MATHGoogle Scholar
  30. 30.
    D.J. Lehmann. Algebraic structures for transitive closure. Theoretical Computer Science, 4:59–76, 1977. MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Springer, Berlin, 1986. MATHGoogle Scholar
  32. 32.
    G. Markowsky. Chain-complete posets and directed sets with applications. Algebra Universalis, 6:53–68, 1976. MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    M. Mohri. Semiring frameworks and algorithms for shortest-distance problems. Journal of Automata, Languages and Combinatorics, 7:321–350, 2002. MATHMathSciNetGoogle Scholar
  34. 34.
    I. Petre and A. Salomaa. Algebraic systems and pushdown automata. In M. Droste, W. Kuich, and H. Vogler, editors, Handbook of Weighted Automata. Chapter 7. Springer, Berlin, 2009. Google Scholar
  35. 35.
    G. Rahonis. Fuzzy languages. In M. Droste, W. Kuich, and H. Vogler, editors, Handbook of Weighted Automata. Chapter 12. Springer, Berlin, 2009. Google Scholar
  36. 36.
    J. Sakarovitch. Kleene’s theorem revisited. In Trends, Techniques, and Problems in Theoretical Computer Science, 4th International Meeting of Young Computer Scientists, volume 281 of Lecture Notes in Computer Science, pages 39–50. Springer, Berlin, 1987. Google Scholar
  37. 37.
    J. Sakarovitch. Éléments de Théorie des Automates. Vuibert, Paris, 2003. Google Scholar
  38. 38.
    J. Sakarovitch. Rational and recognisable power series. In M. Droste, W. Kuich, and H. Vogler, editors, Handbook of Weighted Automata. Chapter 4. Springer, Berlin, 2009. Google Scholar
  39. 39.
    A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, Berlin, 1978. MATHGoogle Scholar
  40. 40.
    W. Wechler. The Concept of Fuzziness in Automata and Language Theory. Akademie Verlag, Berlin, 1978. MATHGoogle Scholar
  41. 41.
    X. Zhao. Locally closed semirings and iteration semirings. Monatshefte für Mathematik, 144:157–167, 2005. MATHCrossRefGoogle Scholar
  42. 42.
    U. Zimmermann. Linear and Combinatorial Optimization in Ordered Algebraic Structures, volume 10 of Annals of Discrete Mathematics. North-Holland, Amsterdam, 1981. MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany
  2. 2.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

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