Logic in Computer Science, Symposium on (2004)
July 13, 2004 to July 17, 2004
Bakhadyr Khoussainov , University of Auckland, New Zealand
Andre Nies , University of Auckland, New Zealand
Sasha Rubin , University of Auckland, New Zealand
Frank Stephan , National ICT Australia
This paper studies the existence of automatic presentations for various algebraic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Fra?ss? limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is \sum\nolimits_1^1-complete.
B. Khoussainov, S. Rubin, A. Nies and F. Stephan, "Automatic Structures: Richness and Limitations," Logic in Computer Science, Symposium on(LICS), Turku, Finland, 2004, pp. 44-53.