Khoussainov, Bakhadyr and Nies, Andre and Rubin, Sasha and Stephan, Frank - Automatic Structures: Richness and Limitations

lmcs:2219 - Logical Methods in Computer Science, April 26, 2007, Volume 3, Issue 2
Automatic Structures: Richness and Limitations

Authors: Khoussainov, Bakhadyr and Nies, Andre and Rubin, Sasha and Stephan, Frank

We study the existence of automatic presentations for various algebraic structures. An automatic presentation of a structure is a description of the universe of the structure by a regular set of words, and the interpretation of the relations by synchronised automata. Our first topic concerns characterising classes of automatic structures. We supply a characterisation of the automatic Boolean algebras, and it is proven that the free Abelian group of infinite rank, as well as certain Fraisse limits, do not have automatic presentations. In particular, the countably infinite random graph and the random partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. Our second topic is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is \Sigma_1^1-complete.


Source : oai:arXiv.org:cs/0703064
DOI : 10.2168/LMCS-3(2:2)2007
Volume: Volume 3, Issue 2
Published on: April 26, 2007
Submitted on: September 15, 2006
Keywords: Computer Science - Discrete Mathematics,Computer Science - Logic in Computer Science,F.1.1,F.4.3


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