| Issue |
RAIRO-Theor. Inf. Appl.
Volume 40, Number 1, January-March 2006
|
|
|---|---|---|
| Page(s) | 53 - 74 | |
| DOI | https://doi.org/10.1051/ita:2005039 | |
| Published online | 15 October 2005 | |
Theories of orders on the set of words
Universität Leipzig, Germany; kuske@informatik.uni-leipzig.de
Received:
26
May
2004
Accepted:
3
February
2005
It is shown that small fragments of the first-order theory of the
subword order, the (partial) lexicographic path ordering on words,
the homomorphism preorder, and the infix order are undecidable. This
is in contrast to the decidability of the monadic second-order
theory of the prefix order [M.O. Rabin, Trans. Amer. Math. Soc., 1969] and of the theory of the
total lexicographic path ordering [P. Narendran and M. Rusinowitch, Lect. Notes Artificial
Intelligence, 2000] and, in case of the
subword and the lexicographic path order, improves upon a result by
Comon & Treinen [H. Comon and R. Treinen, Lect. Notes Comp. Sci., 1994]. Our proofs rely on the
undecidability of the positive ∑1-theory of 
[Y. Matiyasevich, Hilbert's Tenth Problem, 1993] and on Treinen's technique [R. Treinen, J. Symbolic Comput., 1992] that allows to
reduce Post's correspondence problem to logical theories.
Mathematics Subject Classification: 03D35
© EDP Sciences, 2006
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