Advice from Nonadaptive Queries to NP.pdf

[CGH + 88] J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wag- ner, and G. Wechsung. The boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6):1232{1252, 1988. [HU79] J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979. [KL82] R. Karp and R. Lipton. Turing machines that take advice. LEnseignement Mathematique, 28:191{209, 1982. [KSW87] J. Kobler, U. Schoning, and K. Wagner. The dierence and truth-table hierar- chies of NP. R.A.I.R.O. Informatique theorique et Applications, 21(4):419{435, 1987. [KT90] J. Kobler and T. Thierauf. Complexity classes with advice. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 305{315. IEEE Computer Society Press, July 1990. To appear in SIAM Journal on Computing. [LLS75] R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time re- ducibilities. Theoretical Computer Science, 1(2):103{124, 1975. [Wec85] G. Wechsung. On the boolean closure of NP. In Proceedings of the 5th Confer- ence on Fundamentals of Computation Theory, pages 485{493. Springer-Verlag Lecture Notes in Computer Science #199 , 1985. (An unpublished precursor of this paper was coauthored by K. Wagner). 14 m, Mod NP[k] m functions are less powerful as advice to P evaluators than Mod NP[k] 2 functions, unless the boolean hierarchy collapses. Theorem 4.6 Let m 2 be odd and k  0. Then P==Mod NP[k] m = P==Mod NP[kbk=mc] 2 . Proof. Given Theorems 4.4 and 3.1, it suces to prove P==Mod NP[k] m  P==# NP[kbk=mc] . Let L 2 P==Mod NP[k] m via a function f 2 Mod NP[k] m and a set B 2 P. Let f 1 ; . . . ; f k be FP functions such that for all x 2   , f(x) = SAT(f 1 (x)) +


+ SAT(f k (x)) (mod m). For i = 1; . . . ; k, we dene the NP sets A i = f x j at least i of f 1 (x); . . . ; f k (x) are in SAT g and let h i be a many-one reduction from A i to SAT. Then f(x) = SAT(h 1 (x)) +


+ SAT(h k (x)) (mod m). Since the sets A i form