Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata

TL;DR

This paper establishes lower bounds and hardness magnification results for shrinking cellular automata, linking their computational power to streaming algorithms and implications for circuit complexity.

Contribution

It introduces a hardness magnification result for shrinking cellular automata, paralleling recent results for the minimum circuit size problem, and compares their computational capabilities to streaming algorithms.

Findings

Every language accepted by an SCA can be accepted by a similar complexity streaming algorithm.

SCAs are more restricted than streaming algorithms in two key aspects.

There exists a language not accepted by any SCA in o(n / log n) time, despite being efficiently handled by streaming algorithms.

Abstract

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.