次线性时间收缩元胞自动机的下界与硬度放大
Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata
摘要：
最小电路尺寸问题（MCSP）是一个参数为$s$的字符串压缩问题，其中给定长度为$n$的输入上的布尔函数的真值表，必须回答它是否可以由尺寸至多为$s(n) \ge n$的布尔电路计算。最近，McKay，Murray和Williams（STOC，2019）证明了涉及（一次通过）流算法的MCSP的硬度放大结果：对于任何合理的$s$，如果没有$\mathsf{poly}(s(n))$ -空间流算法，$\mathsf{MCSP}[s]$的$\mathsf{poly}(s(n))$更新时间，则$\mathsf{P} \neq \mathsf{NP}$。我们证明了一个类似的结果（可证明）严格的收缩细胞自动机（SCA），这是细胞自动机的细胞可以自发地删除自己的能力较低的模型。我们发现，每一种语言接受的SCA也可以接受流算法的类似的复杂性，我们确定了两个不同的方面，SCA比流算法更受限制。我们还表明，有一种语言，不能接受任何SCA在$o(n / \log n)$的时间，即使它承认一个$O(\log n)$空间流算法与$O(\log n)$更新时间。
英文摘要：
The minimum circuit size problem (MCSP) is a string compression problem with a parameter $s$ in which, given the truth table of a Boolean function over inputs of length $n$, one must answer whether it can be computed by a Boolean circuit of size at most $s(n) \ge n$. Recently, McKay, Murray, and Williams (STOC, 2019) proved a hardness magnification result for MCSP involving (one-pass) streaming algorithms: For any reasonable $s$, if there is no $\mathsf{poly}(s(n))$-space streaming algorithm with $\mathsf{poly}(s(n))$ update time for $\mathsf{MCSP}[s]$, then $\mathsf{P} \neq \mathsf{NP}$. We prove an analogous result for the (provably) strictly less capable model of shrinking cellular automata (SCAs), which are cellular automata whose cells can spontaneously delete themselves. We show every language accepted by an SCA can also be accepted by a streaming algorithm of similar complexity, and we identify two different aspects in which SCAs are more restricted than streaming algorithms. We also show there is a language which cannot be accepted by any SCA in $o(n / \log n)$ time, even though it admits an $O(\log n)$-space streaming algorithm with $O(\log n)$ update time.
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