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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2023.34
URN: urn:nbn:de:0030-drops-175373
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17537/
Chen, Lijie
New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method
Abstract
In this paper, we obtain several new results on lower bounds and derandomization for ACC⁰ circuits (constant-depth circuits consisting of AND/OR/MOD_m gates for a fixed constant m, a frontier class in circuit complexity):
1) We prove that any polynomial-time Merlin-Arthur proof system with an ACC⁰ verifier (denoted by MA_{ACC⁰}) can be simulated by a nondeterministic proof system with quasi-polynomial running time and polynomial proof length, on infinitely many input lengths. This improves the previous simulation by [Chen, Lyu, and Williams, FOCS 2020], which requires both quasi-polynomial running time and proof length.
2) We show that MA_{ACC⁰} cannot be computed by fixed-polynomial-size ACC⁰ circuits, and our hard languages are hard on a sufficiently dense set of input lengths.
3) We show that NEXP (nondeterministic exponential-time) does not have ACC⁰ circuits of sub-half-exponential size, improving the previous sub-third-exponential size lower bound for NEXP against ACC⁰ by [Williams, J. ACM 2014].
Combining our first and second results gives a conceptually simpler and derandomization-centric proof of the recent breakthrough result NQP := NTIME[2^polylog(n)] ̸ ⊂ ACC⁰ by [Murray and Williams, SICOMP 2020]: Instead of going through an easy witness lemma as they did, we first prove an ACC⁰ lower bound for a subclass of MA, and then derandomize that subclass into NQP, while retaining its hardness against ACC⁰.
Moreover, since our derandomization of MA_{ACC⁰} achieves a polynomial proof length, we indeed prove that nondeterministic quasi-polynomial-time with n^ω(1) nondeterminism bits (denoted as NTIMEGUESS[2^polylog(n), n^ω(1)]) has no poly(n)-size ACC⁰ circuits, giving a new proof of a result by Vyas. Combining with a win-win argument based on randomized encodings from [Chen and Ren, STOC 2020], we also prove that NTIMEGUESS[2^polylog(n), n^ω(1)] cannot be 1/2+1/poly(n)-approximated by poly(n)-size ACC⁰ circuits, improving the recent strongly average-case lower bounds for NQP against ACC⁰ by [Chen and Ren, STOC 2020].
One interesting technical ingredient behind our second result is the construction of a PSPACE-complete language that is paddable, downward self-reducible, same-length checkable, and weakly error correctable. Moreover, all its reducibility properties have corresponding AC⁰[2] non-adaptive oracle circuits. Our construction builds and improves upon similar constructions from [Trevisan and Vadhan, Complexity 2007] and [Chen, FOCS 2019], which all require at least TC⁰ oracle circuits for implementing these properties.
BibTeX - Entry
@InProceedings{chen:LIPIcs.ITCS.2023.34,
author = {Chen, Lijie},
title = {{New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {34:1--34:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-263-1},
ISSN = {1868-8969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17537},
URN = {urn:nbn:de:0030-drops-175373},
doi = {10.4230/LIPIcs.ITCS.2023.34},
annote = {Keywords: Circuit Lower Bounds, Derandomization, Algorithmic Method, ACC}
}
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Keywords: |
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Circuit Lower Bounds, Derandomization, Algorithmic Method, ACC |
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Collection: |
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14th Innovations in Theoretical Computer Science Conference (ITCS 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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01.02.2023 |
