From Automatic Structures to Borel Structures

We study the classes of B¨uchi and Rabin automatic structures. For B¨uchi (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 B¨uchi (Rabin) automata. A B¨uchi (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 the well-known L¨owenheim-Skolem theorem in model theory. We provide automata-theoretic versions of L¨owenheim-Skolem theorem for Rabin and B¨uchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every B¨uchi automatic structure have an injective B¨uchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of B¨uchi structures without...

• Type: Conference
• Year: 2008
• Where: LICS
• Authors: Greg Hjorth, Bakhadyr Khoussainov, Antonio Montalbán, André Nies
• Keywords: Automatic Structures, B¨uchi Automatic Structures, Computer Science, LICS 2008, Rabin Automatic Structure