Theories of orders on the set of words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 40 (2006) no. 1, pp. 53-74.

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 Σ 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.

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Kuske, Dietrich. Theories of orders on the set of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 40 (2006) no. 1, pp. 53-74. doi : 10.1051/ita:2005039. http://www.numdam.org/articles/10.1051/ita:2005039/

[1] F. Baader and T. Nipkow, Term Rewriting and All That. Cambridge University Press (1998). | MR | Zbl

[2] A. Björner, The Möbius function of subword order, in Invariant theory and tableaux, IMA Springer. Math. Appl. 19 (1990) 118-124. | Zbl

[3] A. Boudet and H. Comon, About the theory of tree embedding, in TAPSOFT'93, Springer. Lect. Notes Comp. Sci. 668 (1993) 376-390.

[4] J.R. Büchi, Transfinite automata recursions and weak second order theory of ordinals, in Logic, Methodology and Philos. Sci., North Holland, Amsterdam (1965) 3-23. | Zbl

[5] J.R. Büchi and S. Senger, Coding in the existential theory of concatentation. Archiv Math. Logik 26 (1986) 101-106. | Zbl

[6] H. Comon and R. Treinen, Ordering constraints on trees, in CAAP'94, Springer. Lect. Notes Comp. Sci. 787 (1994) 1-14. | Zbl

[7] H. Comon and R. Treinen, The first-order theory of lexicographic path orderings is undecidable. Theor. Comput. Sci. 176 (1997) 67-87. | Zbl

[8] D.E. Daykin, To find all “suitable” orders of 0,1-vectors. Congr. Numer. 113 (1996) 55-60. Festschrift for C. St. J. A. Nash-Williams. | Zbl

[9] N. Dershowitz, Orderings for term rewriting systems. Theor. Comput. Sci. 17 (1982) 279-301. | Zbl

[10] V.G. Durnev, Undecidability of the positive 3 -theory of a free semigroup. Sibirsky Matematicheskie Jurnal 36 (1995) 1067-1080. | Zbl

[11] F.D. Farmer, Homotopy spheres in formal languages. Stud. Appl. Math. 66 (1982) 171-179. | Zbl

[12] A. Finkel and Ph. Schnoebelen, Well-structured transition systems everywhere! Theor. Comput. Sci. 256 (2001) 63-92. | Zbl

[13] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38 (1931) 173-198. | JFM

[14] T. Harju and L. Illie, On quasi orders of words and the confluence property. Theor. Comput. Sci. 200 (1998) 205-224. | Zbl

[15] G. Higman, Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2 (1952) 326-336. | Zbl

[16] S. Kosub and K. Wagner, The boolean hierarchy of NP-partitions, in STACS 2000, Springer. Lect. Notes Comp. Sci. 1770 (2000) 157-168. | Zbl

[17] D. Kuske, Emptiness is decidable for asynchronous cellular machines, in CONCUR 2000. Springer. Lect. Notes Comp. Sci. 1877 (2000) 536-551. | Zbl

[18] U. Leck, Nonexistence of a Kruskal-Katona type theorem for subword orders. Technical Report 98-06, Universität Rostock, Fachbereich Mathematik (1998). | Zbl

[19] M. Lothaire, Combinatorics on Words. Addison-Wesley (1983). | MR | Zbl

[20] G.S. Makanin, The problem of solvability of equations in a free semigroup, Math. Sbornik, 103 (1977) 147-236. In Russian; English translation in: Math. USSR Sbornik 32 (1977). | Zbl

[21] Y. Matiyasevich, Hilbert's Tenth Problem. MIT Press (1993). | Zbl

[22] P. Narendran and M. Rusinowitch, The theory of total unary rpo is decidable, in Computational Logic 2000, Springer. Lect. Notes Artificial Intelligence 1861 (2000) 660-672. | Zbl

[23] M.O. Rabin, Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141 (1969) 1-35. | Zbl

[24] V.L. Selivanov, Boolean hierarchy of partitions over reducible bases. Algebra and Logic 43 (2004) 77-109. | Zbl

[25] R. Treinen, A new method for undecidability proofs of first order theories. J. Symbolic Comput. 14 (1992) 437-458. | Zbl

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