Automatic Structures: Richness and Limitations

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. To this end, we supply a characterisation of the automatic Boolean algebras, and it is proven that the free Abelian group of infinite rank as well as many Fra¨ıss´e 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 Σ1 1-complete.

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Bakhadyr Khoussainov, André Nies, Sasha Rubin, Frank Stephan
• Keywords: Automatic Boolean Algebras, Automatic Presentations, Automatic Structures, CORR 2007, Education