Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
