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Undecidability of infinite post correspondence problem for instances of Size 9

Published online by Cambridge University Press:  08 November 2006

Vesa Halava
Affiliation:
Department of Mathematics and TUCS – Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland; vesa.halava@utu.fi; harju@utu.fi
Tero Harju
Affiliation:
Department of Mathematics and TUCS – Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland; vesa.halava@utu.fi; harju@utu.fi
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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.

Type
Research Article
Copyright
© EDP Sciences, 2006

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References

Blondel, V.D. and Canterini, V., Undecidable problems for probabilistic automata of fixed dimension. Theory Comput. Syst. 36 (2003) 231245. CrossRef
Claus, V., Some remarks on PCP(k) and related problems. Bull. EATCS 12 (1980) 5461.
Ehrenfeucht, A., Karhumäki, J. and Rozenberg, G., The (generalized) Post Correspondence Problem with lists consisting of two words is decidable. Theoret. Comput. Sci. 21 (1982) 119144. CrossRef
Halava, V. and Harju, T., 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) 5768. CrossRef
Halava, V., Harju, T. and Hirvensalo, M., Binary (generalized) Post Correspondence Problem. Theoret. Comput. Sci. 276 (2002) 183204. CrossRef
Halava, V., Harju, T. and Karhumäki, J., Decidability of the binary infinite Post Correspondence Problem. Discrete Appl. Math. 130 (2003) 521526. CrossRef
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.
Harju, T., Karhumäki, J. and Krob, D., Remarks on generalized Post correspondence problem, in STACS'96, edited by C. Puech and R. Reischuk. Lect. Notes Comput. Sci. 1046 (1996) 3948.
Matiyasevich, Y. and Sénizergues, G., Decision problems for semi–Thue systems with a few rules. Theor. Comput. Sci. 330 (2005) 145169. CrossRef
Post, E., A variant of a recursively unsolvable problem. Bull. Amer. Math. Soc. 52 (1946) 264268. CrossRef
K. Ruohonen, Reversible machines and Post's correspondence problem for biprefix morphisms. J. Inform. Process. Cybernet. EIK 21 (1985) 579–595.

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