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

by   Augusto Modanese, et al.

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) β‰₯ 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 π—‰π—ˆπ—…π—’(s(n))-space streaming algorithm with π—‰π—ˆπ—…π—’(s(n)) update time for 𝖬𝖒𝖲𝖯[s], then 𝖯≠𝖭𝖯. We prove an equivalent 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, where n is the input length.



There are no comments yet.


page 1

page 2

page 3

page 4

βˆ™ 03/02/2020

Hardness of Sparse Sets and Minimal Circuit Size Problem

We develop a polynomial method on finite fields to amplify the hardness ...
βˆ™ 04/11/2021

Graph Streaming Lower Bounds for Parameter Estimation and Property Testing via a Streaming XOR Lemma

We study space-pass tradeoffs in graph streaming algorithms for paramete...
βˆ™ 11/12/2018

Streaming Hardness of Unique Games

We study the problem of approximating the value of a Unique Game instanc...
βˆ™ 07/12/2018

Sliding window order statistics in sublinear space

We extend the multi-pass streaming model to sliding window problems, and...
βˆ™ 02/13/2018

Lengths of Words Accepted by Nondeterministic Finite Automata

We consider two natural problems about nondeterministic finite automata....
βˆ™ 02/15/2019

Descriptive complexity for minimal time of cellular automata

Descriptive complexity may be useful to design programs in a natural dec...
βˆ™ 09/12/2019

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

After an apparent hiatus of roughly 30 years, we revisit a seemingly neg...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.