… as integers, set-variables are interpreted as set of integers and … the set of formulas is restricted
to first-order, that is, when set-… in this way are precisely the star-free languages, that is, all …

… In this small section, we show that the p-star-free sets are exactly the sets of integers whose
p-adic representations made up a star-free language. Capital Greek letters will represent …

… some integer. Indeed, the class of languages definable in this logic is exactly the star-free …
by Abelian groups (this follows from a generic correspondence theorem between star-free …

… Let n be a fixed integer; we extend the theorem of Schützenberger, McNaughton, and … on
star-free sets of finite and ω-words to sets of words of length less than ωn for an integer n. In …

… Now we say that a word u satisfies a sentence ~ if ~ is true when variables are interpreted
as integers, set-varlables as sets of integers, < as the usual relation "smaller than" and R x is …

… to the recognizability of sets of integers. Section 4 contains … star-free sets of integers. In the
next section we study the dependence of the star-free property in the special case of integer …

… a position modulo a certain integer, and more generally predicates that count the number
of occurrences of a particular letter modulo an integer. Similarly, star-free expressions can be …

… star-free event R there exists a star-free expression of proper dot-depth. It is easy to verify that
any quotient of a star-free event is again star-free… Then for any two nonnegative integers m, …

… as the aperiodic or star-free family [4, 51, and is identical to the family of noncounting regular
languages [2, 43. It was shown by Schiitzenberger [5] that 9 CA* is star-free iff its syntactic …

… set of states, unless that permutation is the identity mapping. Another point of view is that
these automata are counter-free, since they cannot count modulo any integer … case for star-free …