We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the variables of $x$, each of which is associated with a cost of testing and an independent probability of being true. If a strategy for solving the problem is adaptive in the sense that its next test can depend on the outcomes of previous tests, it has lower expected cost but may take up to exponential space to store. In contrast, a non-adaptive strategy may have higher expected cost but can be stored in linear space and benefit from parallel resources. The adaptivity gap, the ratio between the expected cost of the optimal non-adaptive and adaptive strategies, is a measure of the benefit of adaptivity. We present lower bounds on the adaptivity gap for the SBFE problem for popular classes of Boolean functions, including read-once DNF formulas, read-once formulas, and general DNFs. Our bounds range from $\Omega(\log n)$ to $\Omega(n/\log n)$, contrasting with recent $O(1)$ gaps shown for symmetric functions and linear threshold functions.
翻译:我们考虑Stochastic Boulean 函数评价问题,在这个问题上,我们的任务是对已知的布林函数在未知的比特字符字符串上有效评估美元美元(美元)xx美元(美元)长度。我们通过顺序测试变量x美元(美元)确定美元(xx美元),每个变量都与测试成本和独立真实可能性相关。如果解决问题的战略是适应性的,因为其下一个测试可能取决于先前测试的结果,那么其预期成本较低,但可能达到存储的指数空间。相反,非适应性战略可能具有更高的预期成本,但可以储存在线性空间,并受益于平行资源。适应性差距,即最佳非适应性和适应性战略的预期成本比率是适应性效益的尺度。我们对于波林功能的流行型SBFE问题(包括读取 DNF 公式、读取式公式和普通 DNFS ) 的适应性差幅较小,我们从最近\ Om\ 美元(美元) 和 美元(美元) 直线函数显示的比值为 ngega/ 美元/ 美元/ 美元/ 美元 美元/ 美元 美元/ 美元 美元/ 美元 美元 的比值。