Complexity Classes Characterized by Semi-Random Sources

数理解析研究所講究録 943 巻 1996 年 1-14

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Complexity Classes Characterized by

Semi-Random Sources 上原隆平 (Ryuhei Uehara) Center for Information Science, Tokyo Woman’s Christian University, Zcmpukuji, Suginami-Ku, Tokyo 167, Japan, [email protected]

acterization for PSPACE improves a series of the research for Interactive Proof System.

Abstract The complexity classes characterized by semi-random sources were investigated. . Vazirani and V.V Vazirani [VV85] . Vazirani , and showed that where, BPP, [Vaz86] showed that -BPP is a set class , the class for the condition satisfy the of all languages which for by using any -random source. First, we show that

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$\mathrm{U}.\mathrm{V}$

$\forall_{\delta- \mathrm{R}}\mathrm{p}=\mathrm{R}\mathrm{P}$

$=$

$\forall_{\delta}$

$C$

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\ln$

$C$

$\forall_{\delta- C}$

$\delta$

$\forall_{\delta-\mathrm{p}}\mathrm{P}=\mathrm{B}\mathrm{P}\mathrm{P}$

$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{t}\cdot 1_{1}$

,

means that the class PP is weakened some semi-random source unless

by , whereas RP and BPP don’t BPP change by using any semi-random source. The characterization above of the complexby using semi-random source is ity defined by using any -random source. We introduce the dual characterization, which is defined by using some -random source. In other words, for the random class , the class is defined by the existence of a 6-random for . source which satisfies the Secondly, for these classes, we show that $=\mathrm{p}\mathrm{p}$

$\mathrm{c}\cdot \mathrm{l}\mathrm{a}\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{S}$

$\delta$

$\mathrm{B}1n\mathrm{m}[\mathrm{B}\mathrm{l}\mathrm{u}86]$

$\delta$

$C$


$\exists_{\delta- C}$

$\mathrm{c}\cdot \mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$

,

and

$C$

$\exists_{\delta- \mathrm{B}}\mathrm{p}\mathrm{p}=\exists_{\delta- \mathrm{P}\mathrm{P}=\mathrm{P}\mathrm{s}}\mathrm{p}\mathrm{A}\mathrm{c}\mathrm{E}$

These equations give the new characterization of NP and PSPACE, especially, the char-

The existence of a fair coin has been extensively assumed for applications such as randomizing algorithms, cryptographic protocols, and stochastic simulation experiments. However, it beset with a difficulty; the available sources of randomness, such as Zener diodes, and Geiger counters are imperfect. They don’t output unbiased, independent prorandom bits. J. von posed a simple algorithm to extract unbiased flips from an imperfect source, which is the simplest model of an imperfect source of randomness being a coin whose bias is unknown, considered when but fixed. M. the imperfect random source is a deterministic finite state Markov process. M. San. Vazirani introduced, as an extha and tremely general model of an imperfect source of randomness, a “slightly random source” in ’ in [SV86]. [SV84], or “semi-random The model of this random source is also -model” in [CG88]. This called -random source” source is referred as a in this paper. A semi-random source is as$\mathrm{N}\mathrm{e}\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}[\mathrm{N}\mathrm{e}\mathrm{u}51]$

$\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{l}$

$\exists_{\delta- \mathrm{R}\mathrm{P}}=\mathrm{N}\mathrm{P}$

Introduction


$\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}’$

.

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\ln$

$‘ i\mathrm{S}\mathrm{V}$

$‘\prime sem\dot{i}$

2

sumed that the previous bits output by the source can condition the next bit in an arbitrarily bad way. Accordingly, the next bit is output by the flip of a coin whose bias is fixed an adversary who has complete knowledge of the history of the process. The adversary limited to choosing a bias in with positive some . More prenumber cisely: $1_{\mathrm{J}}\mathrm{y}$

$[\delta, 1-\delta]$

$\mathrm{i}‘(\backslash ^{\mathrm{t}}$

$0 \leq\delta\leq\frac{1}{2}$

Definition 1 Let be a number such that semi-random source with parameter 6 outputs bits , such , that for all and for all $([\mathrm{S}\mathrm{V}84])$

$0 \leq\delta\leq‘\frac{1}{\mathit{2}}.$

$\delta$

$A$

$X_{1}X_{2}\cdots$

$i$

$x_{1},$ $x_{2},$

$\cdots$

$\delta\leq \mathrm{p}\mathrm{r}[_{\mathit{1}\mathrm{Y}_{i}}=x_{i}|X_{1}=x_{1},$

$\cdots,$

$z\mathrm{Y}_{i-1}=$

Notice that the class , defined by , is defined by the definition J. by letting . In other words, since a -random source is a fair random source, defines the same class as . In the paper, they showed that RP with . The class -BPP, to BPP, was introduced by U. Vazirani (he referred as SBPP): $\mathrm{R}\mathrm{P}$

$\mathrm{G}\mathrm{i}\mathrm{l}\mathrm{l}[\mathrm{G}\mathrm{i}\mathrm{l}77]$

$\delta=\frac{1}{2}$

$\frac{1}{2}$

$\mathrm{R}\mathrm{P}$

$\forall.\frac{1}{2}- \mathrm{R}\mathrm{P}$

$\forall_{\delta- \mathrm{R}\mathrm{P}}=$

$0< \delta\leq\frac{1}{2}$

$\forall_{\delta}$

$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}1^{)}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$

Definition 3 A language is in -BPP if there exists a $PTMM$ such that; for $x\in L,$ $M$ accepts with the probability greater than , and for $x\not\in L,$ $M$ accepts with the probability less than for all 6random sources. $L$

$([\mathrm{V}\mathrm{a}\mathrm{z}86])$

$\forall_{\delta}$

$\frac{3}{4}$

$\frac{1}{4}$

$x_{?-1}.]\leq 1-\delta$

.

A semi-random source with parameter will be termed -random source.

$\delta$

$\delta$

Notice that -BPP defines the class as BPP. He showed that -BPP BPP with in the paper. The proof of the result also given by . itriou in [Pap94], and the result is generalized by B. Chor and O. Goldreich in [CG88], D. Zuckerman in [Zuc91], and A. Srinivasan and D. Zuckerman in [SZ94]. In the same manner as and -BPP, we introduce the class , corresponding to : $\forall\frac{1}{2}$

$8\mathrm{a}\mathrm{m}\mathrm{e}$

$\forall_{\delta}$

In the paper, they proved that there is no way to generate fair random bits from one semi-random source . showed how to generate random bits from two independent semi-random sources). A semi-random source is weak as a random source in a sense as mentioned above. J. defined the classes, such as , BPP and , by using a fair random source. The influence by using a semirandom source, instead of a fair rand.om source, over these classes has been investigated. (The terminology of the classes below are unified by the author, and it will be ’ clear what a symbol means in the next paragraph.) The class , corresponding , was introduced by to . Vazirani and . Vazirani (they referred as ): $(\mathrm{U}.\mathrm{V}$

$\mathrm{V}\mathrm{a}\mathrm{z}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{i}[\mathrm{v}_{\mathrm{a}}\mathrm{z}85]$

$\mathrm{G}\mathrm{i}\mathrm{l}\mathrm{l}[\mathrm{G}\mathrm{i}\mathrm{l}77]$


$=$

$0< \delta\leq\frac{1}{2}$

$\mathrm{C}.\mathrm{H}$

$1\mathrm{S}*$

$\forall_{\delta- \mathrm{R}\mathrm{P}}$

$\mathrm{P}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{n}1^{-}$

$\forall_{\delta}$

$\forall_{\delta-\mathrm{p}\mathrm{P}}$

$\mathrm{P}\mathrm{P}$

$\mathrm{p}\mathrm{p}$

Definition 4 A language is in if that; there exists a $PTMM$ such for $x\in L$ , $M$ accepts with the probability greater than , $M$ accepts with the probability and for less than for all -random sources. $L$

$\forall_{\delta-\mathrm{p}\mathrm{P}}$

$\frac{1}{2}$

$x\not\in L,$

$\delta$

$\frac{1}{2}$

$\mathrm{t}‘\forall$




$\mathrm{V}.\mathrm{V}$

$SR_{p}$

Definition 2 ( A language is in if there exists a probabilistic Turing $mach_{\dot{i}}ne(P\tau M)M$ such that; for $x\in L,$ $M$ accepts with the $probab\dot{i}l\dot{i}ty.qrCater$ bhan for $M$ alall -random sources, and for rejects. $L$

$[\mathrm{V}\mathrm{V}85]\rangle$

Notice that defines the same class as . The first theorem in this paper is the following: $\forall\frac{1}{2}- \mathrm{P}\mathrm{P}$

$\mathrm{p}\mathrm{p}$

Theorem 1 For


$\frac{1}{2}$

$‘$

$\delta$

$u\mathit{1}a\mathrm{t}/s$

$x\not\in L,$

$0< \delta