The complexity of set constraints has been extensively studied over the last years and was often found quite high. At the lower end of expressiveness, there are atomic set constraints which are conjunctions of inclusions t1 $subseteq$ t2 between first-order terms without set operators. It is well-known that satisfiability of atomic set constraints can be tested in cubic time. Also, entailment of atomic set constraints has been claimed decidable in polynomial time. We refute this claim. We show that entailment between atomic set constraints can express validity of quantified boolean formulas and is thus PSPACE hard. For infinite signatures, we also present a PSPACE-algorithm for solving atomic set constraints with negation. This proves that entailment of atomic set constraints is PSPACE-complete for infinite signatures. In case of finite signatures, this problem is even DEXPTIME-hard.
Download PDF Show BibTeX
Login to edit