# 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 [17,12]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation [20] the equivalence problem is in EPNP, 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.

## Keywords

Boolean Function Equivalence Problem Binary Decision Diagram Polynomial Hierarchy Order Binary Decision Diagram## Preview

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## References

- [1]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 - [2]E. Allender and R. Rubinstein. P-printable sets.
*SIAM Journal on Computing*, 17(6):1193–1202, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - [3]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 - [4]R. Beigel. Relativized counting classes: Relations among thresholds, parity, and mods.
*Journal of Computer and System Sciences*, 42(1):76–96, 1991.zbMATHCrossRefMathSciNetGoogle Scholar - [5]R. Beigel, R. Chang, and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory, 26(3):293–310, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
- [6]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 - [7]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 - [8]A. Blass and Y. Gurevich. On the unique satisfiability problem.
*Information and Control*, 55:80–88, 1982.zbMATHCrossRefMathSciNetGoogle Scholar - [9]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.
- [10]B. Borchert, D. Ranjan, and F. Stephan. On the computational complexity of some classical equivalence relations on boolean functions.
*Theory of Computing Systems*, 31:679–693, 1998.zbMATHCrossRefMathSciNetGoogle Scholar - [11]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 - [12]R. Bryant. Symbolic boolean manipulation with ordered binary decision diagrams.
*ACM Computing Surveys*, 24(3):293–318, 1992.CrossRefGoogle Scholar - [13]J. Cai and L. Hemachandra. On the power of parity polynomial time.
*Mathematical Systems Theory*, 23(2):95–106, 1990.zbMATHCrossRefMathSciNetGoogle Scholar - [14]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 - [15]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 - [16]S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes.
*Journal of Computer and System Sciences*, 48(1):116–148, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - [17]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 - [18]L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions.
*Theoretical Computer Science*, 43:43–58, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [19]J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems.
*SIAM Journal on Computing*, 17(2):309–335, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - [20]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 - [21]J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities.
*Theoretical Computer Science*, 34:17–32, 1984.zbMATHCrossRefMathSciNetGoogle Scholar - [22]E. Hemaspaandra and L. Hemaspaandra. Quasi-injective reductions.
*Theoretical Computer Science*, 123(2):407–413, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - [23]L. Hemaspaandra, S. Jain, and N. Vereshchagin. Banishing robust Turing completeness.
*International Journal of Foundations of Computer Science*, 4(3):245–265, 1993.zbMATHCrossRefMathSciNetGoogle Scholar - [24]L. Hemaspaandra and J. Rothe. Unambiguous computation: Boolean hierarchies and sparse Turing-complete sets.
*SIAM Journal on Computing*, 26(3):634–653, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - [25]J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP.
*Journal of Computer and System Sciences*, 44(2):272–286, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - [26]M. Ogiwara and L. Hemachandra. A complexity theory for closure properties.
*Journal of Computer and System Sciences*, 46(3):295–325, 1993.zbMATHCrossRefMathSciNetGoogle Scholar - [27]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 - [28]R. Rao, J. Rothe, and O. Watanabe. Upward separation for FewP and related classes.
*Information Processing Letters*, 52(4):175–180, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - [29]U. Schöning. Probabilistic complexity classes and lowness.
*Journal of Computer and System Sciences*, 39(1):84–100, 1989.zbMATHCrossRefMathSciNetGoogle Scholar - [30]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 - [31]S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy.
*SIAM Journal on Computing*, 21(2):316–328, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - [32]L. Valiant. The relative complexity of checking and evaluating.
*Information Processing Letters*, 5(1):20–23, 1976.zbMATHCrossRefMathSciNetGoogle Scholar - [33]N. Vereshchagin. Relativizable and nonrelativizable theorems in the polynomial theory of algorithms.
*Russian Academy of Sciences-Izvestiya-Mathematics*, 42(2):261–298, 1994.CrossRefMathSciNetGoogle Scholar - [34]K. Wagner. The complexity of combinatorial problems with succinct input representations.
*Acta Informatica*, 23:325–356, 1986zbMATHCrossRefMathSciNetGoogle Scholar