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 (sets of) 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：本文描述了一个纯函数式库，用于计算布尔函数的层级p复杂度，并将其应用于两层级联多数函数。布尔函数即从n个比特到单比特的映射，可描述数字电路、投票系统等。多数函数是布尔函数的一个实例——当n为奇数时，它返回n个输入比特中出现频率最高的值。布尔函数f的复杂度衡量其求值代价：为确定f的结果所需检查的输入比特数。虽然存在多种竞争性复杂度度量，但我们聚焦于层级p复杂度——该指标是比特取值为1的概率p的函数。层级p复杂度D_p(f)定义为输入比特独立同分布于Bernoulli(p)分布时，最小期望代价。我们将问题表述为在所有可能决策树中选择最小期望代价——这直接对应一个逻辑正确但效率极低的实现。本库采用削减与缓存技术提升效率，并通过类型类实现关注点分离。复杂度以多项式（集合）表示，用于削减的序关系通过多项式因式分解与根计数实现。最后我们计算了两层级联多数函数的复杂度，改进了J.Jansson此前的研究结果。