Automatic Structures: Richness and Limitations

TL;DR

This paper investigates which algebraic structures can be automatically presented using regular sets and automata, characterizes some classes, and proves the isomorphism problem's high complexity.

Contribution

It provides a characterization of automatic Boolean algebras, shows many structures lack automatic presentations, and establishes the isomorphism problem as -complete.

Findings

Boolean algebras are automatically presentable

Certain structures like free Abelian groups and random graphs are not automatic

Isomorphism problem for automatic structures is -complete

Abstract

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.