Undecidability of infinite post correspondence problem for instances of Size 9

Vesa Halava; Tero Harju

RAIRO - Theoretical Informatics and Applications (2006)

  • Volume: 40, Issue: 4, page 551-557
  • ISSN: 0988-3754

Abstract

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In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem was shown to be undecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36 (2003) 231–245) showed that this problem is undecidable for domain alphabets of size 105. Here we give a proof that the infinite Post Correspondence Problem is undecidable for instances where the morphisms have domains of 9 letters. The proof uses a recent result of Matiyasevich and Sénizergues and a modification of a result of Claus.

How to cite

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Halava, Vesa, and Harju, Tero. "Undecidability of infinite post correspondence problem for instances of Size 9." RAIRO - Theoretical Informatics and Applications 40.4 (2006): 551-557. <http://eudml.org/doc/249719>.

@article{Halava2006,
abstract = { In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem was shown to be undecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36 (2003) 231–245) showed that this problem is undecidable for domain alphabets of size 105. Here we give a proof that the infinite Post Correspondence Problem is undecidable for instances where the morphisms have domains of 9 letters. The proof uses a recent result of Matiyasevich and Sénizergues and a modification of a result of Claus. },
author = {Halava, Vesa, Harju, Tero},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Infinite Post Correspondence Problem; undecidability; word problem; semi–Thue system.; Post correspondence problem},
language = {eng},
month = {11},
number = {4},
pages = {551-557},
publisher = {EDP Sciences},
title = {Undecidability of infinite post correspondence problem for instances of Size 9},
url = {http://eudml.org/doc/249719},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Halava, Vesa
AU - Harju, Tero
TI - Undecidability of infinite post correspondence problem for instances of Size 9
JO - RAIRO - Theoretical Informatics and Applications
DA - 2006/11//
PB - EDP Sciences
VL - 40
IS - 4
SP - 551
EP - 557
AB - In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem was shown to be undecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36 (2003) 231–245) showed that this problem is undecidable for domain alphabets of size 105. Here we give a proof that the infinite Post Correspondence Problem is undecidable for instances where the morphisms have domains of 9 letters. The proof uses a recent result of Matiyasevich and Sénizergues and a modification of a result of Claus.
LA - eng
KW - Infinite Post Correspondence Problem; undecidability; word problem; semi–Thue system.; Post correspondence problem
UR - http://eudml.org/doc/249719
ER -

References

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  1. V.D. Blondel and V. Canterini, Undecidable problems for probabilistic automata of fixed dimension. Theory Comput. Syst.36 (2003) 231–245.  Zbl1039.68061
  2. V. Claus, Some remarks on PCP(k) and related problems. Bull. EATCS12 (1980) 54–61.  
  3. A. Ehrenfeucht, J. Karhumäki and G. Rozenberg, The (generalized) Post Correspondence Problem with lists consisting of two words is decidable. Theoret. Comput. Sci.21 (1982) 119–144.  Zbl0493.68076
  4. V. Halava and T. Harju, Infinite solutions of the marked Post Correspondence Problem, in Formal and Natural Computing, edited by J. Karhumäki, W. Brauer, H. Ehrig and A. Salomaa. Lecture Notes in Comput. Sci.2300 (2002) 57–68.  Zbl1060.03068
  5. V. Halava, T. Harju and M. Hirvensalo, Binary (generalized) Post Correspondence Problem. Theoret. Comput. Sci.276 (2002) 183–204.  Zbl1023.03038
  6. V. Halava, T. Harju and J. Karhumäki, Decidability of the binary infinite Post Correspondence Problem. Discrete Appl. Math.130 (2003) 521–526.  Zbl1023.03039
  7. T. Harju and J. Karhumäki, Morphisms, in Handbook of Formal Languages, volume 1, edited by G. Rozenberg and A. Salomaa, Springer-Verlag (1997) 439–510.  
  8. T. Harju, J. Karhumäki and D. Krob, Remarks on generalized Post correspondence problem, in STACS'96, edited by C. Puech and R. Reischuk. Lect. Notes Comput. Sci.1046 (1996) 39–48.  
  9. Y. Matiyasevich and G. Sénizergues, Decision problems for semi–Thue systems with a few rules. Theor. Comput. Sci.330 (2005) 145–169.  Zbl1078.03033
  10. E. Post, A variant of a recursively unsolvable problem. Bull. Amer. Math. Soc.52 (1946) 264–268.  Zbl0063.06329
  11. K. Ruohonen, Reversible machines and Post's correspondence problem for biprefix morphisms. J. Inform. Process. Cybernet. EIK 21 (1985) 579–595.  Zbl0604.68057

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