This paper studies the hazard-free formula complexity of Boolean functions. As our main result, we prove that unate functions are the only Boolean functions for which the monotone formula complexity of the hazard-derivative equals the hazard-free formula complexity of the function itself. Consequently, every non-unate function breaks the so-called monotone barrier, as introduced and discussed by Ikenmeyer, Komarath, and Saurabh (ITCS 2023). Our second main result shows that the hazard-free formula complexity of random Boolean functions is at most $2^{(1+o(1))n}$. Prior to this, no better upper bound than $O(3^n)$ was known. Notably, unlike in the general case of Boolean circuits and formulas, where the typical complexity matches that of the multiplexer function, the hazard-free formula complexity is smaller than the optimal hazard-free formula for the multiplexer by an exponential factor in $n$. Additionally, we explore the hazard-free formula complexity of block composition of Boolean functions and obtain a result in the hazard-free setting that is analogous to a result of Karchmer, Raz, and Wigderson (Computational Complexity, 1995) in the monotone setting. We demonstrate that our result implies a lower bound on the hazard-free formula depth of the block composition of the set covering function with the multiplexer function, which breaks the monotone barrier.

翻译：本文研究了布尔函数的无险象公式复杂性。作为主要结果，我们证明了单边函数是唯一一类满足以下性质的布尔函数：其险象导数的单调公式复杂性等于函数本身的无险象公式复杂性。因此，每个非单边函数都会打破所谓的单调屏障，该概念由Ikenmeyer、Komarath和Saurabh（ITCS 2023）引入并讨论。我们的第二个主要结果表明，随机布尔函数的无险象公式复杂性至多为$2^{(1+o(1))n}$。在此之前，已知的最佳上界仅为$O(3^n)$。值得注意的是，与布尔电路和公式的一般情况（其典型复杂性与多路复用器函数相匹配）不同，无险象公式复杂性比多路复用器的最优无险象公式小一个关于$n$的指数因子。此外，我们探讨了布尔函数块组合的无险象公式复杂性，并在无险象设定下获得了一个结果，该结果类似于Karchmer、Raz和Wigderson（Computational Complexity, 1995）在单调设定中的结果。我们证明了我们的结果意味着集合覆盖函数与多路复用器函数块组合的无险象公式深度下界，该下界打破了单调屏障。