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Recognizable sets with multiplicities in the tropical semiring

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Mathematical Foundations of Computer Science 1988 (MFCS 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 324))

Abstract

The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems.

This work was supported by FAPESP and CNPq

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References

  1. I. J. Aalbersberg and H. J. Hoogeboom. Decision problems for regular trace languages. In T. Ottman, editor, Automata, Languages and Programming, pages 250–259, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  2. J. Berstel. Transductions and Context-Free Languages. B. G. Teubner, Stuttgart, 1979.

    Google Scholar 

  3. J. Berstel and C. Reutenauer. Les Séries Rationnelles et leurs Langages. Masson, Paris, 1984.

    Google Scholar 

  4. J. Berstel and J. Sakarovitch. Recent results in the theory o rational sets. In J. Gruska, B. Rovan, and J. Wiedermann, editors, Mathematical Foundations of Computer Science 1986, pages 15–28, Springer-Verlag, Berlin, 1986. Lecture Notes in Computer Science, 233.

    Google Scholar 

  5. T. C. Brown. An interesting combinatorial method in the theory of locally finite semigroups. Pacific J. Math., 36:285–289, 1971.

    Google Scholar 

  6. T. H. Chan and O. Ibarra. On the finite-valuedness problem for sequential machines. Theoretical Comput. Sci., 23:95–101, 1983.

    Google Scholar 

  7. P. Chemouil, G. Cohen, J. P. Quadrat, and M. Viot, editors. Algebres Exotiques et Systemes a Evenements Discrets. Institut National de Recherche en Informatique et en Automatique, Le Chesnay, 1987.

    Google Scholar 

  8. C. Choffrut. Free Partially Commutative Monoids. Technical Report RT-MAP-8504, Instituto de Matemática e Estatística da Universidade de São Paulo, 1985.

    Google Scholar 

  9. C. Choffrut. Series Rationelles d'Image Finie. Technical Report 79-6, Laboratoire d'Informatique Théorique et Programmation, Paris, 1979.

    Google Scholar 

  10. C. Choffrut. Sur les transductions reconnaissables. R.A.I.R.O. Informatique théorique, 12:203–212, 1978.

    Google Scholar 

  11. L. C. Eggan. Transition graphs and the star height of regular events. Michigan Math. J., 10:385–397, 1963.

    Google Scholar 

  12. S. Eilenberg. Automata, Languages, and Machines, Volume A. Academic Press, New York, NY, 1974.

    Google Scholar 

  13. A. Gibbons and W. Rytter. On the decidability of some problems about rational subsets of the free partially commutative monoids. 1987. manuscript.

    Google Scholar 

  14. K. Hashiguchi. Algorithms for determining relative star height and star height. 1987. Manuscript.

    Google Scholar 

  15. K. Hashiguchi. A decision procedure for the order of regular events. Theoretical Comput. Sci., 8:69–72, 1979.

    Google Scholar 

  16. K. Hashiguchi. Improved limitedness theorems on finite automata with distance functions. 1986. Manuscript.

    Google Scholar 

  17. K. Hashiguchi. Limitedness theorem on finite automata with distance functions. J. Comput. Syst. Sci., 24:233–244, 1982.

    Google Scholar 

  18. K. Hashiguchi. Regular languages of star height one. Information and Control, 53:199–210, 1982.

    Google Scholar 

  19. K. Hashiguchi. Representation theorems on regular languages. J. Comput. Syst. Sci., 27:101–115, 1983.

    Google Scholar 

  20. C. E. Hughes and S. M. Selkow. The finite power property for context-free languages. Theoretical Comput. Sci., 15:111–114, 1981.

    Google Scholar 

  21. O. Ibarra. The unsolvability of the equivalenc problem for ε-free NGSM's with unitary input (output) alphabet and applications. SIAM J. Comput., 7:524–532, 1978.

    Google Scholar 

  22. G. Jacob. La finitude des representations lineaires de semi-groupes est decidable. J. Algebra, 52:437–459, 1978.

    Google Scholar 

  23. C. M. R. Kintala and P. Fischer. Computations with a restricted number of nondeterministic steps. In Proc. of the Ninth Annual ACM Symposium on Theory of Computing, pages 178–185, Association for Computing Machinery, New York, 1977.

    Google Scholar 

  24. C. M. R. Kintala and D. Wotschke. Amounts of nondeterminism in finite automata. Acta Inf., 13:199–204, 1980.

    Google Scholar 

  25. H. Leung. 1987. Private communication.

    Google Scholar 

  26. H. Leung. An Algebraic Method for Solving Decision Problems in Finite Automata Theory. PhD thesis, Department of Computer Science, The Pennsylvania State University, 1987.

    Google Scholar 

  27. M. Linna. Finite power property of regular languages. In M. Nivat, editor, Automata, Languages and Programming, pages 87–98, North-Holland Pu. Co., Amsterdam, 1973.

    Google Scholar 

  28. A. Mandel and I. Simon. On finite semigroups of matrices. Theoretical Comput. Sci., 5:101–111, 1977.

    Google Scholar 

  29. J. Mascle. Torsion matrix semigroups and recognizable transductions. In L. Kott, editor, Automata, Languages and Programming, pages 244–253, Springer-Verlag, Berlin, 1986. Lecture Notes in Computer Science, 226.

    Google Scholar 

  30. J. E. Pin. Languages Rationells et Reconnaissables. Technical Report 85-60, Laboratoire d'Informatique Théorique et Programmation, Paris, 1985.

    Google Scholar 

  31. A. Salomaa. Jewels of Formal Language Theory. Computer Science Press, Rockville,MD, 1981.

    Google Scholar 

  32. A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, New York, 1978.

    Google Scholar 

  33. I. Simon. Caracterização de conjuntos racionais limitados. 1987. Tese de Livre-Docência, Instituto de Matemática e Estatística da Universidade de São Paulo.

    Google Scholar 

  34. I. Simon. Factorization Forests of Finite Height. Technical Report 87-73, Laboratoire d'Informatique Théorique et Programmation, Paris, 1987.

    Google Scholar 

  35. I. Simon. Limited subsets of a free monoid. In Proc. 19th Annual Symposium on Foundations of Computer Science, pages 143–150, Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1978.

    Google Scholar 

  36. I. Simon. The Nondeterministic Complexity of a Finite Automaton. Technical Report RT-MAP-8703, Instituto de Matemática e Estatística da Universidade de São Paulo, 1987.

    Google Scholar 

  37. I. Simon. On Brzozowski's problem: (1 ∪ A)m=A*. In M. Fontet and I. Guessarian, editors, Seminaire d'Informatique Théorique, annee 1979–1980, pages 67–72, Laboratoire d'Informatique Théorique et Programmation, Paris, 1980.

    Google Scholar 

  38. I. Simon. Word Ramsey theorems. In B. Bollobás, editor, Graph Theory and Combinatorics, pages 283–291, Academic Press, London, 1984.

    Google Scholar 

  39. H. Straubing. The Burnside problem for semigroups of matrices. In L. J. Cummings, editor, Combinatorics on Words, Progress and Perspectives, pages 279–295, Academic Press, New York, NY, 1983.

    Google Scholar 

  40. A. Weber and H. Seidl. On Finitely Generated Monoids of Matrices with Entries in IN. Technical Report 9/87, Fachbereich Informatik, Universität Frankfurt, 1987.

    Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, 05508, São Paulo, SP, Brasil

    Imre Simon

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  1. Imre Simon
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Michal P. Chytil Václav Koubek Ladislav Janiga

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© 1988 Springer-Verlag Berlin Heidelberg

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Simon, I. (1988). Recognizable sets with multiplicities in the tropical semiring. In: Chytil, M.P., Koubek, V., Janiga, L. (eds) Mathematical Foundations of Computer Science 1988. MFCS 1988. Lecture Notes in Computer Science, vol 324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017135

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  • DOI: https://doi.org/10.1007/BFb0017135

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  • Print ISBN: 978-3-540-50110-7

  • Online ISBN: 978-3-540-45926-2

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