Logic and {$p$}-recognizable sets of integers

Definable relations and first-order query languages over strings

“…An alternative approach to definability of strings, based on classical infinite model theory rather than finite model theory, dates back to the 1960s [Büchi 1960;Bruyère et al 1994]. One considers an infinite structure M consisting of * , , where is a set of functions, predicates and constants on * .…”

“…Let S len be the model * , reg (we will explain the notation later). Then, subsets of * definable in S len are precisely the regular languages [Büchi 1960;Bruyère et al 1994;Blumensath and Grädel 2000]; moreover, this implies decidability of the first-order theory of S len [Hodgson 1983;Blumensath and Grädel 2000]. …”

“…In the case of the model S len above, this algebra is not new: in fact, the definable n-ary relations are exactly the ones recognizable under a natural notion of automaton running over n-tuples [Bruyère et al 1994;Elgot and Mezei 1965]. We refer to these automata-definable relations as the regular relations: the formal definition is given in Section 3.1.1.…”

“…As mentioned above, we know that there exists a regular string algebra [Büchi 1960;Bruyère et al 1994;Blumensath and Grädel 2000], that is, an algebra that exactly captures the regular sets when restricted to unary relations. An obvious question to ask, then, is whether new algebras of string relations arise through the model-theoretic approach.…”

“…It is known [Bruyère et al 1994;Blumensath and Grädel 2000] that a structure is automatic iff it can be interpreted in the structure S len ; hence, S len is, in some sense, the universal automatic structure. The first part of this article can be seen as a study of subclasses of automatic structures definable within S len that are significantly more restrictive, and that might have stronger model-theoretic or computational properties than a rich structure like S len .…”