Lower Bounds from Succinct Hitting Sets
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909608008744960 |
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| author | Chatterjee, Prerona Tengse, Anamay |
| author_facet | Chatterjee, Prerona Tengse, Anamay |
| contents | We investigate the consequences of the existence of ``efficiently describable'' hitting sets for polynomial sized algebraic circuit ($\mathsf{VP}$), in particular, \emph{$\mathsf{VP}$-succinct hitting sets}. Existence of such hitting sets is known to be equivalent to a ``natural-proofs-barrier'' towards algebraic circuit lower bounds, from the works that introduced this concept (Forbes \etal (2018), Grochow \etal (2017)). We show that the existence of $\mathsf{VP}$-succinct hitting sets for $\mathsf{VP}$ would either imply that $\mathsf{VP} \neq \mathsf{VNP}$, or yield a fairly strong lower bound against $\mathsf{TC}^0$ circuits, assuming the Generalized Riemann Hypothesis (GRH).
This result is a consequence of showing that designing efficiently describable ($\mathsf{VP}$-explicit) hitting set generators for a class $\mathcal{C}$, is essentially the same as proving a separation between $\mathcal{C}$ and $\mathsf{VPSPACE}$: the algebraic analogue of \textsf{PSPACE}. More formally, we prove an upper bound on \emph{equations} for polynomial sized algebraic circuits ($\mathsf{VP}$), in terms of $\mathsf{VPSPACE}$.
Using the same upper bound, we also show that even \emph{sub-polynomially explicit hitting sets} for $\mathsf{VP}$ -- much weaker than $\mathsf{VP}$-succinct hitting sets that are almost polylog-explicit -- would imply that either $\mathsf{VP} \neq \mathsf{VNP}$ or that $\mathsf{P} \neq \mathsf{PSPACE}$. This motivates us to define the concept of \emph{cryptographic hitting sets}, which we believe is interesting on its own. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07612 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower Bounds from Succinct Hitting Sets Chatterjee, Prerona Tengse, Anamay Computational Complexity We investigate the consequences of the existence of ``efficiently describable'' hitting sets for polynomial sized algebraic circuit ($\mathsf{VP}$), in particular, \emph{$\mathsf{VP}$-succinct hitting sets}. Existence of such hitting sets is known to be equivalent to a ``natural-proofs-barrier'' towards algebraic circuit lower bounds, from the works that introduced this concept (Forbes \etal (2018), Grochow \etal (2017)). We show that the existence of $\mathsf{VP}$-succinct hitting sets for $\mathsf{VP}$ would either imply that $\mathsf{VP} \neq \mathsf{VNP}$, or yield a fairly strong lower bound against $\mathsf{TC}^0$ circuits, assuming the Generalized Riemann Hypothesis (GRH). This result is a consequence of showing that designing efficiently describable ($\mathsf{VP}$-explicit) hitting set generators for a class $\mathcal{C}$, is essentially the same as proving a separation between $\mathcal{C}$ and $\mathsf{VPSPACE}$: the algebraic analogue of \textsf{PSPACE}. More formally, we prove an upper bound on \emph{equations} for polynomial sized algebraic circuits ($\mathsf{VP}$), in terms of $\mathsf{VPSPACE}$. Using the same upper bound, we also show that even \emph{sub-polynomially explicit hitting sets} for $\mathsf{VP}$ -- much weaker than $\mathsf{VP}$-succinct hitting sets that are almost polylog-explicit -- would imply that either $\mathsf{VP} \neq \mathsf{VNP}$ or that $\mathsf{P} \neq \mathsf{PSPACE}$. This motivates us to define the concept of \emph{cryptographic hitting sets}, which we believe is interesting on its own. |
| title | Lower Bounds from Succinct Hitting Sets |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2309.07612 |