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

@inproceedings{Modanese2021LowerBA,
  title={Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata},
  author={Augusto Modanese},
  booktitle={CSR},
  year={2021}
}
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… Expand

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References

SHOWING 1-10 OF 38 REFERENCES
One-Tape Turing Machine and Branching Program Lower Bounds for MCSP
TLDR
These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bound for any explicit functions against these computational models. Expand
Weak lower bounds on resource-bounded compression imply strong separations of complexity classes
TLDR
It is shown that search-MCSP[s(n)] (where the authors must output a s( n)-size circuit when it exists) admits extremely efficient AC0 circuits and streaming algorithms using Σ3 SAT oracle gates of small fan-in (related to the size s(n) they want to test). Expand
Hardness Magnification for all Sparse NP Languages
TLDR
There is a hardness magnification phenomenon for all equally-sparse NP languages, and analogous theorems for De Morgan formulas, B_2-formulas, branching programs, AC^0[6] and TC^0 circuits are proved. Expand
NP-hardness of circuit minimization for multi-output functions
TLDR
This work establishes the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits, and shows that computing the minimum circuit size of a given multi-output Boolean function f is NP- hard under many-one polynomial-time randomized reductions. Expand
Beyond Natural Proofs: Hardness Magnification and Locality
TLDR
This work considers the following essential questions associated with the hardness magnification program: Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [RR97]. Expand
Hardness magnification near state-of-the-art lower bounds
TLDR
A natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U2-formulas that can compute parity functions at the leaves (instead of just literals) and lower bounds similar to (3)-(5) hold for various regimes of parameters. Expand
Hardness Magnification for Natural Problems
  • I. Oliveira, R. Santhanam
  • Computer Science, Mathematics
  • 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2018
TLDR
It is shown that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models, and magnification is explored as an avenue to proving strong lower bounds. Expand
Complexity-Theoretic Aspects of Expanding Cellular Automata
The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. The respective polynomial-time complexity class is shownExpand
Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question
We investigate relativized versions of the open question of whether every language accepted nondeterministically in polynomial time can be recognized deterministically in polynomial time. For any setExpand
Circuit minimization problem
TLDR
It is argued that proving this problem to be NP-complete (if it is indeed true) would imply proving strong circuit lower bounds for the class DTIME(2°('~)), which appears beyond the currently known techniques. Expand
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4
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