A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.
"Automata presenting structures: A survey of the finite string case." Bull. Symbolic Logic 14 (2) 169 - 209, June 2008. https://doi.org/10.2178/bsl/1208442827