On relativized exponential and probabilistic complexity classes

Q4 Mathematics
Hans Heller
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引用次数: 79

Abstract

An oracle X is constructed such that the exponential complexity class ΔEP,X2 equals the probabilistic class R(R(X)). This shows that it will be difficult to prove that ΔEP2 is different from R(R), although it seems very unlikely that these two classes are equal. The result subsumes several known results about relativized computations:

  • (i)

    the existence of relativized polynomial hierarchies extending two levels (Long, T., 1978, Dissertation, Purdue Univ., Lafayette, Ind.; Heller, H., 1984(a), SIAM J. Comput. 13, 717–725; Heller, H., 1984(b), Math. Systems Theory 17, 71–84);

  • (ii)

    the existence of an oracle X such that BPP(X) ⊄ ΔP,X2 (Stockmeyer, L., 1983, “Proc. 15th STOC” pp. 118–126),

  • (iii)

    the existence of an oracle X such that NP(X) is polynomially Turing reducible to a sparse set (Wilson, C., 1983, “Proc. 24th FOCS,”, pp. 329–334; Immerman, N., and Mahaney, S., 1983, “Conference on Computational Complexity Theory,” Santa Barbara, March 21–25).

The result shows possible inclusion relations for nonrelativized complexity classes and points out that certain results about probabilistic complexity classes and about polynomial size circuits cannot be improved unless methods are applied which do not relativize.

关于相对指数和概率复杂度类
构造一个oracle X,使指数复杂度类ΔEP,X2等于概率类R(R(X))。这表明很难证明ΔEP2不同于R(R),尽管这两个类似乎不太可能相等。该结果包含了几个关于相对计算的已知结果:(i)扩展两个层次的相对多项式层次的存在性(Long, T., 1978, Dissertation, Purdue university, Lafayette, Ind.);刘海涛,1984(a),中国科学院学报(自然科学版)13 (3):717-725;海勒,H., 1984(b),数学。(ii)使得BPP(X)≤ΔP,X2的一个神谕X的存在性(Stockmeyer, L., 1983,“Proc. 15 STOC”pp. 118-126),(iii)使得NP(X)多项式图灵可约为一个稀疏集的一个神谕X的存在性(Wilson, C., 1983,“Proc. 24 FOCS”,pp. 329-334;Immerman, N.和Mahaney, S., 1983,“计算复杂性理论会议”,圣巴巴拉,3月21-25日)。结果显示了非相对化复杂度类可能存在的包含关系,并指出除非采用非相对化的方法,否则无法改进关于概率复杂度类和多项式大小电路的某些结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
信息与控制
信息与控制 Mathematics-Control and Optimization
CiteScore
1.50
自引率
0.00%
发文量
4623
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