Towards a Theory of Recursive Structures

In computer science, one is interested mainly in nite objects. Insofar as innnite objects are of interest, they must be computable, i.e., recursive, thus admitting an eeective nite representation. This leads to the notion of a recursive graph, or, more generally, a recursive structure , model or data base. In this paper we summarize our recent work on recursive structures and data bases, including (i) the high undecidability of many problems on recursive graphs and structures, (ii) a method for deducing results on the descriptive complexity of nitary NP optimization problems from results on the computational complexity (i.e., the degree of undecidability) of their innnitary analogues, (iii) completeness results for query languages on recursive data bases, (iv) correspondences between descriptive and computational complexity over recursive structures , and (v) zero-one laws for recursive structures.