Automatic presentations of groups and semigroups
Effectively deciding the satisfiability of logical sentences over structures is an area well-studied in the case of finite structures. There has been growing work towards considering this question for infinite structures. In particular the theory of automatic structures, considered here, investigates structures representable by finite automata. The closure properties of finite automata lead naturally to algorithms for deciding satisfiability for some logics. The use of finite automata to investigate infinite structures has been inspired by the interplay between the theory of finite automata and the theory of semigroups. This inspiration has come in particular from the theory of automatic groups and semigroups, which considers (semi)groups with regular sets of normal forms over their generators such that generator-composition is also regular. The work presented here is a contribution to the foundational problem for automatic structures: given a class of structures, classify those members that have an automatic presentation. The classes considered here are various interesting subclasses of the class of finitely generated semigroups, as well as the class of Cayley Graphs of groups. Although similar, the theories of automatic (semi)groups and automatic presentation differ in their construction. A classification for finitely generated groups allows a direct comparison of the theory of automatic presentations with the theory of automatic groups.