自动结构：丰富性与局限性
Automatic Structures: Richness and Limitations
摘要：
我们研究了各种代数结构的自动表示的存在性。结构的自动表示是通过一组规则的单词来描述结构的宇宙，并通过同步自动机来解释关系。我们的第一个主题是关于自动结构的分类。本文给出了自动布尔代数的一个刻划，证明了无限秩的自由阿贝尔群以及一定的Fraisse极限不具有自动表示。特别地，可数无限随机图和随机偏序没有自动表示。此外，没有无限积分域是自动的。我们的第二个主题是同构问题。我们证明了所有自动结构类的同构问题的复杂性是1^1^1-完全的。
英文摘要：
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 Σ_1^1-complete.
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