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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.CCC.2023.6
URN: urn:nbn:de:0030-drops-182764
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18276/
Hirahara, Shuichi ;
Lu, Zhenjian ;
Ren, Hanlin
Bounded Relativization
Abstract
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ℭ, we say that a statement is ℭ-relativizing if the statement holds relative to every oracle ? ∈ ℭ. It is easy to see that every result that relativizes also ℭ-relativizes for every complexity class ℭ. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing.
First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0, BPE^{MCSP}/2^{εn} ⊈ SIZE[2ⁿ/n].
We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021).
Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ≠ L. For example:
- Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ≠ BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ≠ L.
- Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ≠ L.
- Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ≠ L.
In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ≠ EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible.
BibTeX - Entry
@InProceedings{hirahara_et_al:LIPIcs.CCC.2023.6,
author = {Hirahara, Shuichi and Lu, Zhenjian and Ren, Hanlin},
title = {{Bounded Relativization}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {6:1--6:45},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18276},
URN = {urn:nbn:de:0030-drops-182764},
doi = {10.4230/LIPIcs.CCC.2023.6},
annote = {Keywords: relativization, circuit lower bound, derandomization, explicit construction, pseudodeterministic algorithms, interactive proofs}
}
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Keywords: |
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relativization, circuit lower bound, derandomization, explicit construction, pseudodeterministic algorithms, interactive proofs |
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Collection: |
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38th Computational Complexity Conference (CCC 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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10.07.2023 |
