This paper describes a purely functional library for computing level-$p$-complexity of Boolean functions, and applies it to two-level iterated majority. Boolean functions are simply functions from $n$ bits to one bit, and they can describe digital circuits, voting systems, etc. An example of a Boolean function is majority, which returns the value that has majority among the $n$ input bits for odd $n$. The complexity of a Boolean function $f$ measures the cost of evaluating it: how many bits of the input are needed to be certain about the result of $f$. There are many competing complexity measures but we focus on level-$p$-complexity -- a function of the probability $p$ that a bit is 1. The level-$p$-complexity $D_p(f)$ is the minimum expected cost when the input bits are independent and identically distributed with Bernoulli($p$) distribution. We specify the problem as choosing the minimum expected cost of all possible decision trees -- which directly translates to a clearly correct, but very inefficient implementation. The library uses thinning and memoization for efficiency and type classes for separation of concerns. The complexity is represented using polynomials, and the order relation used for thinning is implemented using polynomial factorisation and root-counting. Finally we compute the complexity for two-level iterated majority and improve on an earlier result by J.~Jansson.
翻译:使用瘦化,记忆化和多项式计算布尔函数的级别-p-复杂度
翻译摘要:
这篇论文介绍了一种纯函数库,用于计算布尔函数的级别-p-复杂度,并将其应用于两级迭代大多数。布尔函数仅是从n位到一个比特的函数,它们可以描述数字电路、投票系统等。布尔函数的一个例子是多数,对于奇数n,它返回在n个输入比特中具有多数的值。 布尔函数的复杂度度量其计算的成本:需要多少输入比特才能确定$f$的结果。有许多竞争的复杂度度量,但我们专注于级别-p-复杂度——一个与比特$p$的概率相关的函数。级别-p-复杂度$D_p(f)$是当输入比特独立且带有Bernoulli分布($p$)时,期望成本的最小值。我们将问题规定为选择所有可能决策树中的最小预期成本——这直接转化为一个明确正确但效率极低的实现。该库使用了瘦化和记忆化来提高效率,使用类型类来分离关注点。复杂度使用多项式表示,用于瘦化的顺序关系使用多项式分解和根计数实现。最后,我们计算了两级迭代大多数的复杂度,并改进了J.~Jansson的早期结果。