Let $S$ be a relational structure (for example, a graph).

$S$ has an automatic presentation if (for some alphabet $\Sigma$ ) there is a language $L\subseteq \Sigma^*$ and a surjective map $f$ from $L$ onto the domain (universe) of $S$ such that

• $L$ can be checked by a finite automaton (Membership)
• The language of all convolutions of $x,y\in L$ where $f(x)=f(y)$ can be checked by a finite automaton (Equality)
• For all $n$ -ary relations $R_i$ in $S$ , the language of all convolutions of $x_1,x_2,\ldots,x_n\in L$ where $R_i(f(x_1),f(x_2),\ldots,f(x_n))$ can be checked by a finite automaton (Relations)

The class of languages accepted by finite automata, i.e. regular languages, is closed under operations like union, intersection, complementation etc, and it is decidable whether or not a finite automaton accepts the empty language. This allows any first order sentence over the structure to be decided - using union for 'and', complementation for 'not' etc., and emptiness for dealing with 'there exists'. As such, the first order theory of any structure with an automatic presentation is decidable.

Note that wrt group theory this is inspired by, but not equivalent to, the definition of automatic groups.