Theories of orders on the set of words

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 Σ 1 -theory of (N, +,.) [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.