随机开关电路的综合与分析
The Synthesis and Analysis of Stochastic Switching Circuits
摘要：
随机开关电路是由称为pswitch的随机开关组成的继电器电路。随机开关电路的研究在计算机科学、神经科学和生物化学的许多领域有着广泛的应用。本文讨论了随机开关电路的几个性质，包括鲁棒性、可表达性和概率近似性。首先，我们研究了鲁棒性，即在随机电路中引入大小为ε的误差对每个p开关的影响。我们分析了两种结构，并证明了简单的串并联电路是强大的小误差扰动，而一般的串并联电路没有。具体地说，在简单的串并联电路中，由小于ε的扰动引入的总误差由ε的常数倍限制，与电路的大小无关。接下来，我们研究了随机开关电路的可表达性：给定一个整数q和一个p开关集S=\{ \frac{1}{q}，\frac{2}{q}，. \frac{q-1}{q} \}，我们能用一个简单的串并联随机开关电路合成任何分母为q^n（对于任意n）的有理概率吗？我们推广了以前的结果，并证明了当q是2或3的倍数时，答案是肯定的。我们还证明了当q是大于3的素数时，答案是否定的。本文研究了任意p开关集S=\{s_1，s_2，. s_{|S|} \}。在这种情况下，我们提出了一种基于局部优化的算法来逼近任何所需的概率。分析表明，开关电路的逼近误差随电路规模的增加呈指数下降。
英文摘要：
Stochastic switching circuits are relay circuits that consist of stochastic switches called pswitches. The study of stochastic switching circuits has widespread applications in many fields of computer science, neuroscience, and biochemistry. In this paper, we discuss several properties of stochastic switching circuits, including robustness, expressibility, and probability approximation. First, we study the robustness, namely, the effect caused by introducing an error of size εto each pswitch in a stochastic circuit. We analyze two constructions and prove that simple series-parallel circuits are robust to small error perturbations, while general series-parallel circuits are not. Specifically, the total error introduced by perturbations of size less than εis bounded by a constant multiple of εin a simple series-parallel circuit, independent of the size of the circuit. Next, we study the expressibility of stochastic switching circuits: Given an integer q and a pswitch set S=\{\frac{1}{q},\frac{2}{q},...,\frac{q-1}{q}\}, can we synthesize any rational probability with denominator q^n (for arbitrary n) with a simple series-parallel stochastic switching circuit? We generalize previous results and prove that when q is a multiple of 2 or 3, the answer is yes. We also show that when q is a prime number larger than 3, the answer is no. Probability approximation is studied for a general case of an arbitrary pswitch set S=\{s_1,s_2,...,s_{|S|}\}. In this case, we propose an algorithm based on local optimization to approximate any desired probability. The analysis reveals that the approximation error of a switching circuit decreases exponentially with an increasing circuit size.
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