Towards a theory of recursive structures

In computer science, one is interested mainly in finite objects. Insofar as infinite objects are of interest, they must be computable, i.e., recursive, thus admitting an effective finite representation. This leads to the notion of a recursive graph, or, more generally, a recursive structure, model or data base. We summarize our recent work on recursive structures and data bases, including (i) high undecidability of specific problems, (ii) connections between the descriptive complexity of finitary problems and the computational complexity of their infinitary analogues, (iii) completeness for query languages, (iv) descriptive and computational complexity, and (v) zero-one laws.