Automatic Structures: Richness and Limitations

Bakhadyr Khoussainov; Andre Nies; Sasha Rubin; Frank Stephan

doi:10.2168/LMCS-3(2:2)2007

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

lmcs:2219 - Logical Methods in Computer Science, April 26, 2007, Volume 3, Issue 2 - https://doi.org/10.2168/LMCS-3(2:2)2007

Automatic Structures: Richness and LimitationsArticle

Authors: Bakhadyr Khoussainov ; Andre Nies ; Sasha Rubin ; Frank Stephan

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.

https://doi.org/10.2168/LMCS-3(2:2)2007

Source: arXiv.org:cs/0703064

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

Licence: arXiv.org - Assumed non-exclusive license to distribute

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Mathematics Subject Classification 2020¹

Sources:

[1] zbMATH Open.

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