Jacobo Torán
- 1 ANGLUIN, D. On counting problems and the polynomial-time hierarchy. Theoret. Comput. Sci. 12 (1980), 161-173.Google Scholar
- 2 BAKER, T. P, GILL, J., AND SOLOVAY, R.M. Retativizations of the P = 9NP question. SIAM J. Comput. 4 (1975), 431-442.Google Scholar
- 3 BALC~ZAR, J. L., DIAZ, J., AND GABARR(), J. Structural Complexity, vol. I. Springer-Verlag, 1987. Google Scholar
- 4 BENNET, C. H., AND GILL, J. Relative to a random oracle A p A #= NpA ~: co_NP.4 with probability 1. SIAM J. Comput. IO (1981), 96-112.Google Scholar
- 5 CHANDRA, A. K., KOZEN, D. C., AND STOCKMEYER, L. J. ALTERNATION J. A CM 28 (1981), 114-133. Google Scholar
- 6 FURST, M., SAXE, J. B., AND SIPSER, M. Parity circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17 (1984), 13-27.Google Scholar
- 7 GILL, J. Computational complexity of probabllistic Turing machines. SIAM J. Comput. 6 (1977), 675-695.Google Scholar
- 8 HARTMANIS, J. The structural complexity column: Sparse complete sets for NP and the optimal collapse of the polynomial hierarchy. Bull. EA TCS 32 (1987), 73- 81.Google Scholar
- 9 H~STAD, J. Computational limitations for small depth circuits. Ph.D. dissertation, M.I.T., Cambridge, Mass., (1986).Google Scholar
- 10 Ko, K. Relativized polynommI time hierarchies having exactly K levels. In Proceedings of the 3rd Structure in Complexity Theory Conference. IEEE, New York, 1988, pp. 251-252.Google Scholar
- 11 PAPADIMITRIOU, C. H. Games against nature In Proceedings of the 24th Foundations of Computer Science. IEEE, New York, 1983, pp. 446-450.Google Scholar
- 12 PAPADIMITRIOU, C. H., AND ZACHOS, S. Two remarks on the power of counting. In Proceedings of the 6th GI Conference on Theoretical Computer Science. Lecture Notes in Computer Science, vol. 145. Springer-Verlag, New York, 1983, pp. 269-276. Google Scholar
- 13 Russo, D.A. Structural propemes of complexity classes. Ph.D. dissertation. Univ. California, Santa Barbara, Santa Barabara, Calif., 1985. Google Scholar
- 14 SCH6MNG, U. Complexity and structure. In Lecture Notes in Computer Smence, vol. 211. Springer-Verlag, New York, 1985.Google Scholar
- 15 SIMON, J On some central problems in computauonat complexW. Ph.D. dissertation. Cornell Univ., 1975. Google Scholar
- 16 STOCKMEYER, L.J. The polynomial time hierarchy. Theoret. Comput. Sci. 3 (i977), 1-22.Google Scholar
- 17 TODA, S. On the computational power of PP and ~ P. In Proceedings of the 30th Foundations of Computer Science. IEEE, New York, 1989, 514-519.Google Scholar
- 18 TOR~N, J. Structural properties of the counting hierarchms, Ph.D. dissertation Facultat d'Informatica de Barcelona, Barcelona, Spain, 1988.Google Scholar
- 19 TOR~N, J. An oracle characterization of the counting hierarchy. In Proceedings of the 3rd Structure in Complexity Theory Conference. IEEE, New York, 1988, pp. 213-223Google Scholar
- 20 VALIANT, L.G. The complexW of computing the permanent Theoret. Comput. Sci. 8 (1979), 189-201.Google Scholar
- 21 WAGNER, K. The complexity of combinatorial problems with succinct input representation. Acta Inf. 23 (1986), 325-356. Google Scholar
- 22 WAGNER, K. AND WECHSUNG, G. Computational complexity. Reidel (1986).Google Scholar
- 23 WRATHALL, C. Complete sets and the polynomial time hmrarchy. Theoret. Comput. Sci. 3 (1977), 23-33.Google Scholar
- 24 YAO, A. Separating the polynomial time hierarchy with oracles. In Proceedings of the 26th Annual Foundations of Computer Science. IEEE, New York, 1985. pp. 1-10. Google Scholar
Index Terms
- Complexity classes defined by counting quantifiers
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Reviews
Marius Zimand
The counting quantifier C p f , where <__?__Pub Fmt italic>p<__?__Pub Fmt /italic> is a polynomial and <__?__Pub Fmt italic>f<__?__Pub Fmt /italic> is an arbitrary function from strings to integers, is defined as follows: C p f y Q x,y means that there are at least <__?__Pub Fmt italic>f<__?__Pub Fmt /italic>(<__?__Pub Fmt italic>x<__?__Pub Fmt /italic>) strings <__?__Pub Fmt italic>y<__?__Pub Fmt /italic> of length <__?__Pub Fmt italic>p<__?__Pub Fmt /italic>(|<__?__Pub Fmt italic>x<__?__Pub Fmt /italic>|) satisfying the predicate <__?__Pub Fmt italic>Q<__?__Pub Fmt /italic>(<__?__Pub Fmt italic>x,y<__?__Pub Fmt /italic>). The counting hierarchy (CH) arises by combining the counting quantifier with the existential and universal quantifiers. This hierarchy is important because it expresses the complexity of many natural problems, but little was known about the structural properties of CH before the publication of this paper. The author addresses four main topics: the investigation of the Boolean properties of the classes in CH; the characterization of CH in terms of nondeterministic and probabilistic machines with access to oracles; relativized separations of counting classes—<__?__Pub Caret>NP from G (exact counting), NP from ? P, and ? P from PP; and the absolute separation of log-time counting complexity classes.
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