On the subword equivalence problem for infinite words

Abstract

Two infinite words x and y are said to be subword equivalent if they have the same set of finite subwords (factors). The subword equivalence problem is the question whether two infinite words are subword equivalent. We show that, under mild hypotheses, the decidability of the subword equivalence problem implies the decidability of the ω-sequence equivalence problem, a problem which has been shown to be decidable by Čulik and Harju for morphic words (i.e. words generated by iterating a morphism).

We prove that the subword equivalence problem is decidable for two morphic words, provided the morphisms are primitive and have bounded delays. We also prove that the subword equivalence problem is decidable for any pair of morphic words in the case of a binary alphabet. The subword equivalence problem is also shown to be decidable for two p-automatic words. We also prove a Cobham-like theorem: let p and q be two multiplicatively independent integers, and let x be a p-automatic word and let y be a q-automatic word, both with bounded gaps. If x and y are subword equivalent, then they are both ultimately periodic. Our results hold in fact for a stronger version, namely for the subword inclusion problem.