Two Techniques in the Area of the Star Problem

  • Daniel Kirsten
  • Jerzy Marcinkowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1644)


Decidability of the Star Problem, the problem whether the language ℙ* is recognizable for a recognizable language ℙ, remains open. We slightly generalize the problem and show that then its decidability status depends strongly on the assumptions considering the trace monoid and finiteness of ℙ. More precisely, we show that for finite set ℙ ⊂ {A, B}* ×{C}* and recognizable ℝ it is decidable whether ℙ* ∩ℝ is recognizable, but the problem becomes undecidable if we consider recognizable (infinite) ℙ or finite ℙ ⊂ {A, B}* × {C, D}*.


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  1. 1.
    J. Berstel. Transductions and Context-Free Languages. B.G. Teubner, 1979.Google Scholar
  2. 2.
    P. Cartier, D. Foata. Problèmes combinatoires de commutation et réarrangements, volume 85of LNCS. Springer-Verlag, Berlin, 1969.zbMATHGoogle Scholar
  3. 3.
    M. Clerbout, M. Latteux. Semi-commutations. Information and Computation, 73:59–74, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. Diekert, Y. Métivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Vol. 3, Beyond Words, pages 457–534. Springer-Verlag, Berlin, 1997.Google Scholar
  5. 5.
    V. Diekert, G. Rozenberg, editors. The Book of Traces. World Scientific, 1995.Google Scholar
  6. 6.
    M. Fliess. Matrices de hankel. J. Math. Pures et Appl., 53:197–224, 1974.MathSciNetzbMATHGoogle Scholar
  7. 7.
    P. Gastin, E. Ochmański, A. Petit, B. Rozoy. Decidability of the star problem in A* × {b}*. Information Processing Letters, 44(2):65–71, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    K. Hashiguchi. Limitedness theorem on finite automata with distance functions. Journal of Computer and System Sciences, 24:233–244, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Kirsten. Some undecidability results related to the star problem in trace monoids. In C. Meinel and S. Tison, editors, STACS’99 Proceedings, volume 1563 of LNCS, pages 227–236. Springer-Verlag, Berlin, 1999.Google Scholar
  10. 10.
    A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, 1977.Google Scholar
  11. 11.
    Y. Métivier. Une condition suffisante de reconnaissabilité dans un monoïde partiellement commutative. R.A.I.R.O.-Informatique Théorique et Applications, 20:121–127, 1986.zbMATHGoogle Scholar
  12. 12.
    Y. Métivier, G. Richomme. On the star operation and the finite power property in free partially commutative monoids. Proceedings of STACS 94, volume 775 of LNCS, pages 341–352. Springer-Verlag, Berlin, 1994.Google Scholar
  13. 13.
    Y. Métivier, G. Richomme. New results on the star problem in trace monoids. Information and Computation, 119(2):240–251, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    E. Ochmański. Regular Trace Languages (in Polish). PhD thesis, Warszawa, 1984.Google Scholar
  15. 15.
    G. Richomme. Some trace monoids where both the star problem and the finite power property problem are decidable. In I. Privara et al., editors, MFCS’94 Proceedings, volume 841 of LNCS, pages 577–586. Springer-Verlag, Berlin, 1994.Google Scholar
  16. 16.
    J. Sakarovitch. The “last” decision problem for rational trace languages. In I. Simon, editor, LATIN’92 Proceedings, LNCS 583, pages 460–473. Springer, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Daniel Kirsten
    • 1
  • Jerzy Marcinkowski
    • 2
  1. 1.Department of Computer ScienceDresden University of TechnologyDresdenGermany
  2. 2.Institute of Computer ScienceUniversity of Wroc lawWroc lawPoland

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