PTIME Queries Revisited

The existence of a language expressing precisely the PTIME queries on arbitrary structures remains the central open problem in the theory of database query languages. As it turns out, two variants of this question have been formulated. Surprisingly, despite the importance of the problem, the relationship between these variants has not been systematically explored. A first contribution of the present paper is to revisit the basic definitions and clarify the connection between these two variants. We then investigate two relaxations to the original problem that appear as tempting alternatives in the absence of a language for the PTIME queries. The first consists in trying to express the PTIME queries using a richer language that can also express queries beyond PTIME, but for which there exists a query processor evaluating all PTIME queries in PTIME. The second approach, studied by many researchers, is to focus on PTIME properties on restricted sets of graphs. Our results are mostly negative, and point out limitations to both approaches. Finally, we turn to a natural class of languages that we call finitely generated, whose syntax is obtained by applying a fixed set of constructors to a given set of building blocks. We identify a broad class of such languages that cannot express all the PTIME queries.