Automatic Structures and Their Complexity Bakhadyr Khoussainov and Mia Minnes

In recent years there has been increasing interest in the study of structures that can be presented by automata. The underlying idea in this line of research consists of applying properties of automata and techniques of automata theory to decision problems that arise in logic and applications. A typical example of a decision problem is the model checking problem, stated as follows. For a structure A (e.g. a graph, a fragment of the arithmetic, the real numbers with addition) design an algorithm that, given a formula φ(x̄) in a formal logical system and a tuple ā from the structure, decides if φ(ā) is true in A. In particular, when the formal system is the first order predicate logic or the monadic second order logic, we would like to know if the theory of the structure, that is the collection of all sentences of the logic true in the structure, is decidable. Büchi used automata to prove that the monadic second order theory of the successor function on ω is decidable ([?], [?]). Rabin, in [?], extended this by proving decidability of the monadic second order theory of the binary tree. There have been numerous applications and extensions of these results in logic, algebra, verification, model checking, and databases ([?] is an algebra application; [?], [?] treat logics and verifications; [?], and [?] give applications to databases). Using simple closure properties and the decidability of the emptiness problem for finite and tree automata, one can easily prove that the first order (and monadic second order) theories of some wellknown structures are decidable. Examples of such structures are Presburger arithmetic and some of its extensions, the term algebra, real numbers under addition, finitely generated abelian groups, and the atomless Boolean algebra. Direct proofs of these results, without the use of automata, require non-trivial technical work. A structure A = (A;R0, . . . , Rm) is automatic if the domain A and all the relations R0, . . . , Rm of the structure are recognized by finite automata (precise definitions are in the next section). For instance, an automatic graph is one whose set of vertices and set of edges can be recognized by finite automata. There are several motivating results that are foundational for the development of the theory of automatic structures. Khoussainov and Nerode proved that for any given automatic structure there is an algorithm that solves the model checking problem in the first order logic (see [?]). In particular, the first order theory of the structure is decidable. This result is extended by adding the ∃ (there are infinitely many) and ∃ (there are m many mod n) quantifiers to the first order logic (see [?], [?]). Blumensath and Grädel proved a logical characterization theorem stating that automatic structures are exactly those definable in the fragment of arithmetic (ω; +, |2,≤, 0), where + and ≤ have their usual meanings and |2 is a weak divisibility predicate for which x|2y