Restrictive Acceptance Suffices for Equivalence Problems

Abstract

One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach-weaker in strength of evidence but more broadly applicable-to suggesting that concrete NP problems are not NP-complete. In particular we introduce the class EP, the subclass of NP consisting of those languages accepted by NP machines that when they accept always have a number of accepting paths that is a power of two. Since if any NP-complete set is in EP then all NP sets are in EP, it follows-with whatever degree of strength one believes that EP differs from NP-that membership in EP can be viewed as evidence that a problem is not NP-complete.We show that the negation equivalence problem for OBDDs (ordered binary decision diagrams [22,13]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation [25] the equivalence problem is in EP NP , thus tightening the existing NP NP upper bound. We show that FewP [2], bounded ambiguity polynomial time, is contained in EP, a result that is not known to follow from the previous SPP upper bound. For the three problems and classes just mentioned with regard to EP, no proof of membership/containment in UP is known, and for the problem just mentioned with regard to EP NP , no proof of membership in UP NP is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.

Untitled

“…It is at most as hard as "threshold counting," since C = P ⊆ PP, and it is not substantially easier, since PP ⊆ NP C = P . The class C = P has been given several names and characterizations; it equals the classes 2 coNQP [18] and ES [11].…”