# Theories of orders on the set of words

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 40, Issue: 1, page 53-74
- ISSN: 0988-3754

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topKuske, Dietrich. "Theories of orders on the set of words." RAIRO - Theoretical Informatics and Applications 40.1 (2010): 53-74. <http://eudml.org/doc/92789>.

@article{Kuske2010,

abstract = {
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 $(\mathbb N,+,\cdot)$ [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.
},

author = {Kuske, Dietrich},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {decidability; undecidability; first-order theory; words; subword order; lexicographic path; homomorphism preorder; infix order},

language = {eng},

month = {3},

number = {1},

pages = {53-74},

publisher = {EDP Sciences},

title = {Theories of orders on the set of words},

url = {http://eudml.org/doc/92789},

volume = {40},

year = {2010},

}

TY - JOUR

AU - Kuske, Dietrich

TI - Theories of orders on the set of words

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 40

IS - 1

SP - 53

EP - 74

AB -
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 $(\mathbb N,+,\cdot)$ [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.

LA - eng

KW - decidability; undecidability; first-order theory; words; subword order; lexicographic path; homomorphism preorder; infix order

UR - http://eudml.org/doc/92789

ER -

## References

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