Pp-definability Is Co-nexptime-complete

PP-Def is the problem which takes as input a relation r and a finite set Γ of relations on the same finite domain A, and asks whether r is definable by a conjunctive query over (A,Γ), i.e., by a formula of the form ∃~yφ(~x, ~y) where φ(~x, ~y) is a conjunction of atomic formulas built using the relations in Γ∪ {=}, and where the variables range over A. (Such formulas ∃~yφ(~x, ~y) are called primitive positive formulas.) PP-Def is known to be in co-NEXPTIME , and has been shown to be tractable on the boolean domain. We show that there exists k > 2 such that PP-Def is co-NEXPTIME -complete on k-element domains, answering a question of Creignou, Kolaitis and Zanuttini. We also show that two related problems are NEXPTIME -complete.