Inherent Complexity of Recursive Queries

We give lower bounds on the complexity of certain Datalog queries. Our notion of complexity applies to compile-time optimization techniques for Datalog; thus, our results indicate limitations of these techniques. The main new tool is linear first-order formulas, whose depth (respectively, number of variables) matches the sequential (respectively, parallel) complexity of Datalog programs. We define a combinatorial game (a variant of Ehrenfeucht–Fraı̈ssé games) that can be used to prove nonexpressibility by linear formulas. We thus obtain lower bounds for the sequential and parallel complexity of Datalog queries. We prove syntactically tight versions of our results, by exploiting uniformity and invariance properties of Datalog queries.