eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-07-10
6:1
6:45
10.4230/LIPIcs.CCC.2023.6
article
Bounded Relativization
Hirahara, Shuichi
1
Lu, Zhenjian
2
Ren, Hanlin
2
https://orcid.org/0000-0002-7632-7574
National Institute of Informatics, Tokyo, Japan
University of Oxford, UK
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 𝒪 ∈ 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol264-ccc2023/LIPIcs.CCC.2023.6/LIPIcs.CCC.2023.6.pdf
relativization
circuit lower bound
derandomization
explicit construction
pseudodeterministic algorithms
interactive proofs