Rich ω-words and monadic second-order arithmetic

Abstract

Rich ω-words are one-sided infinite strings which have every finite word as a subword (infix). Infix-regular w-words are one-sided infinite strings for which the infix set of a suffix is a regular language. We show that for a regular ω-language F (a set of predicates definable in Büchi's restricted monadic second order arithmetic) the following conditions are equivalent:

• 1. F contains a rich ω-word.

• 2. F is of second Baire category in the Cantor space of ω-words.

• 3. F is a non-nullset for a class of measures (including the natural Lebesgue measure on Cantor space).

• 4. F has maximum Hausdorff dimension.

This shows that, although we cannot fully translate Compton's result (Theorem 1 below) on rich ℤ-words (in the MSO theory of the integers) 2 to MSO arithmetic on naturals, a set definable in MSO arithmetic and containing a rich w-word is large in several respects simultaneously. Moreover, we show under the assumption of an exchanging property for ‘distinguishing’ prefixes that two regular w-words not necessarily being rich but having the same sets of infixes occurring infinitely often are indistinguishable by MSO formulas or, equivalently, by finite automata.