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.

Restrictive Acceptance Suffices for Equivalence Problems

“…4 From Theorem 3.4 it immediately follows that FewP ⊆ EP, since EP = RC {2 i |i∈IN} and {2 i | i ∈ IN} is clearly a P-printable, non-gappy set. The comments attached to our on-line technical report version [10] give some of the history of the proof of our results and of some valuable comments made by Richard Beigel, in particular that FewP is also contained in the EP analog based on any integer n (note that the acceptance sets for such classes are P-printable and non-gappy).…”

Lower Bounds and the Hardness of Counting Properties

“…We mention in passing that Theorem 9 seems neither to imply nor to be implied by a result of Borchert, Hemaspaandra, and Rothe that shows that certain "restricted counting classes" contain FewP [BHROO,Theorem 3.4]. On the one hand, the result of Borchert et al applies only to promise classes; but on the other hand, the result of Borchert et al (conditionally) concludes containment results rather than hardness results.…”

An Overview Of Some Semantic And Syntactic Complexity Classes

“…It was shown in [16] that NP ⊆ RP Unambiguous−SAT , where RP is the randomized polynomial time as defined in [10]. The semantic version of C = P was defined in [26] as Half P. Another notable semantic complexity class is EP, which was defined in [27]. An EP machine has an acceptance criterion of power of two and a rejection criterion of zero.…”

“…An EP machine has an acceptance criterion of power of two and a rejection criterion of zero. It was shown in [27] that the syntactic version of EP, called ES, equals C = P. Various other interesting semantic complexity classes were introduced in [20] and [25].…”

“…Definition 11. A language L is in syntactic complexity class ES, as defined in [27], if there exist a polynomial p and a polynomial-time predicate R such that, for each x,…”