Restrictive acceptance suffices for equivalence problems
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 [17,12]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation  the equivalence problem is in EPNP, thus tightening the existing NPNP upper bound. We show that FewP , 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 EPNP, no proof of membership in UPNP is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.
KeywordsBoolean Function Equivalence Problem Binary Decision Diagram Polynomial Hierarchy Order Binary Decision Diagram
Unable to display preview. Download preview PDF.
- M. Agrawal and T. Thierauf. The boolean isomorphism problem. In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, pages 422–430. IEEE Computer Society Press, October 1996.Google Scholar
- R. Beigel. On the relativized power of additional accepting paths. In Proceedings of the 4th Structure in Complexity Theory Conference, pages 216–224. IEEE Computer Society Press, June 1989.Google Scholar
- R. Beigel, J. Gill, and U. Hertrampf. Counting classes: Thresholds, parity, mods, and fewness. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, pages 49–57. Springer-Verlag Lecture Notes in Computer Science #415, February 1990.Google Scholar
- A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 132–137. IEEE Computer Society Press, June 1992.Google Scholar
- B. Borchert, L. Hemaspaandra, and J. Rothe. Powers-of-two acceptance suffices for equivalence and bounded ambiguity problems. Technical Report TR96-045, Electronic Colloquium on Computational Complexity, http://www.eccc.uni-trier.de/eccc/, August 1996.
- B. Borchert and F. Stephan. Looking for an analogue of Rice’s Theorem in circuit complexity theory. In Proceedings of the 1997 Kurt Gödel Colloquium, pages 114–127. Springer-Verlag Lecture Notes in Computer Science #1289, 1997.Google Scholar
- J. Feigenbaum, S. Kannan, M. Vardi, and M. Viswanathan. Complexity of problems on graphs represented as OBDDs. In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, pages 216–226. Springer-Verlag Lecture Notes in Computer Science #1373, February 1998.Google Scholar
- M. Fellows and N. Koblitz. Self-witnessing polynomial-time complexity and prime factorization. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 107–110. IEEE Computer Society Press, June 1992.Google Scholar
- S. Fortune, J. Hopcroft, and E. Schmidt. The complexity of equivalence and containment for free single program schemes. In Proceedings of the 5th International Colloquium on Automata, Languages, and Programming, pages 227–240. Springer-Verlag Lecture Notes in Computer Science #62, 1978.Google Scholar
- M. Harrison. Counting theorems and their applications to classi cation of switching functions. In A. Mukhopadyay, editor, Recent Developments in Switching Theory, pages 4–22. Academic Press, 1971.Google Scholar
- C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings 6th GI Conference on Theoretical Computer Science, pages 269–276. Springer-Verlag Lecture Notes in Computer Science #145, 1983.Google Scholar
- Y. Takenaga, M. Nouzoe, and S. Yajima. Size and variable ordering of OBDDs representing threshold functions. In Proceedings of the 3rd Annual International Computing and Combinatorics Conference, pages 91–100. Springer-Verlag Lecture Notes in Computer Science #1276, August 1997.Google Scholar