First-Order Logic and Star-Free Sets

formulas are built up in the usual way by means of the connectives -I, v , A and the quantifiers 3 and V bounding up both types of variables. Now, we say that a word w on the alphabet A satisfies such a sentence 4 if 6 is true when variables are interpreted as integers, set-variables are interpreted as set of integers and the formula X,X is interpreted as “the letter in position x in w is an a.” McNaughton [3] was the first to consider the case where the set of formulas is restricted to first-order, that is, when set-variables are ignored. He proved that the languages defined in this way are precisely the star-free languages, that is, all languages obtained from finite languages by boolean operations and concatenation product. Later on, star-free languages have been considerably studied. First, a fundamental result of Schiitzenberger shows that star-free languages are exactly the languages recognized by an aperiodic finite monoid (i.e., a monoid all of whose groups are trivial). Further on, a great number of subclasses of star-free languages have been studied [6]. Among the most famous, let us quote the locally testable languages studied by McNaughton and Brzozowski and Simon and the piecewise testable languages, introduced by Simon. Star-free languages are defined by two types of operations: boolean operations on one hand and concatenation product on the other hand. This naturally defines a hierarchy based on the alternative use of these operations. The hierarchy was originally introduced by Brzozowski who showed with Knast [ 11 that the inclusion was proper on each level. Furthermore the class of locally testable languages 393 0022~0000/86 $3.00