From Automatic Structures to Borel Structures

Greg Hjorth, Bakh Khoussainov, Antonio Montalbán, André Nies
2008 Logic in Computer Science  
We study the classes of Büchi and Rabin automatic structures. For Büchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Büchi (Rabin) automata. A Büchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automatatheoretic content of
more » ... the well-known Löwenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Löwenheim-Skolem theorem for Rabin and Büchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Büchi automatic structure have an injective Büchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Büchi structures without injective Büchi and Rabin presentations. To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Büchi automatic structures. We show that the isomorphism problem is not even a Σ 1 2 -set.
doi:10.1109/lics.2008.28 dblp:conf/lics/HjorthKMN08 fatcat:cko4fcoz3jczzkpfb4kyd462zq