The complexity of nested counterfactuals and iterated knowledge base revisions

Abstract

We consider the computational complexity of evaluating nested counterfactuals over a propositional knowledge base. A counterfactual p > q is a conditional query with the meaning "If p would be true in the knowledge base, would it then hold that also q is true," which is different from material implication p ⇒ q. A nested counterfactual is a counterfactual statement where the premise p or the conclusion q is a counterfactual. Statements of the form P1 > (p2 ⋯ (pn>q)⋯ ) intuitively correspond to conditional queries involving a sequence of revisions. We show that evaluating such statements is Π2p-complete and that this task becomes PSPACE-complete if negation is allowed in a nesting of this form. We also consider nesting a counterfactual in the premise, i.e., (p>q) >r, and show that evaluating such statements is /7 4 -complete, thus most likely much harder than evaluating p > (q>r). Finally, we also address iterated nestings in the premise and the mix of nestings in the premise and the conclusion. © 1996 Academic Press, Inc.