  We study word structures of the form $(D,<,P)$ where $D$ is either
$\mathbb{N}$ or $\mathbb{Z}$, $<$ is the natural linear ordering on $D$ and
$P\subseteq D$ is a predicate on $D$. In particular we show:
  (a) The set of recursive $\omega$-words with decidable monadic second order
theories is $\Sigma_3$-complete.
  (b) Known characterisations of the $\omega$-words with decidable monadic
second order theories are transfered to the corresponding question for
bi-infinite words.
  (c) We show that such "tame" predicates $P$ exist in every Turing degree.
  (d) We determine, for $P\subseteq\mathbb{Z}$, the number of predicates
$Q\subseteq\mathbb{Z}$ such that $(\mathbb{Z},\le,P)$ and $(\mathbb{Z},\le,Q)$
are indistinguishable.
  Through these results we demonstrate similarities and differences between
logical properties of infinite and bi-infinite words.
