Easiness amplification and uniform circuit lower bounds
Name
LIPIcs-CCC-2017-8.pdf
Description
Published version
Size
592.51 KB
Format
Adobe PDF
Checksum (MD5)
473fe0220c830749099425d1116fc1a5
Author(s)
Williams, Richard Ryan
Murray, Cody
Date Issued
2017
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Williams, Richard Ryan and Murray, Cody. 2017. "Easiness amplification and uniform circuit lower bounds." Leibniz International Proceedings in Informatics, LIPIcs, 79.
Version
Final published version
Abstract
© Cody D. Murray and R. Ryan Williams. We present new consequences of the assumption that time-bounded algorithms can be "compressed" with non-uniform circuits. Our main contribution is an "easiness amplification" lemma for circuits. One instantiation of the lemma says: if n1+ϵ-time, O (n)-space computations have n1+o(1) size (non-uniform) circuits for some ϵ > 0, then every problem solvable in polynomial time and O(n) space has n1+o(1) size (non-uniform) circuits as well. This amplification has several consequences: An easy problem without small LOGSPACE-uniform circuits. For all ϵ > 0, we give a natural decision problem General Circuit nϵ-Composition that is solvable in n1+ϵ time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n1+o(1)-size circuits for the problem. This shows that there are problems solvable in n1+ϵ time which are not in LOGSPACE-uniform n1+o(1) size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. Problems without low-depth LOGSPACE-uniform circuits. For all ϵ > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in O (n1+ϵ) time, or in O((log n)d) space (though not necessarily simultaneously) that we prove does not have SPACE[(log n)e]-uniform circuits of O(n) size and O((log n)e) depth. We also show SAT does not have circuits of O (n) size and log2-o(1) n depth that can be constructed in log2-o(1) n space. A strong circuit complexity amplification. For every ϵ > 0, we give a natural problem Circuit nϵ-Composition and show that if it has O(n)-size circuits (uniform or not), then every problem solvable in 2O(n) time and 2O( p n log n) space (simultaneously) has 2O( p n log n)- size circuits (uniform or not). We also show the same consequence holds assuming SAT has O (n)-size circuits. As a corollary, if n1.1 time computations (or O(n) nondeterministic time computations) have O (n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.CCC.2017.8
