🤖 AI Summary
This work investigates the implications of the existence of “$mathsf{VP}$-succinct hitting sets” for algebraic circuit lower bounds. The authors establish an equivalence between constructing such succinct hitting sets and separating $mathsf{VPSPACE}$ from $mathsf{VP}$, thereby introducing a cryptographic-style paradigm for hitting set design. They further link subpolynomial explicitness of hitting sets to the separation $mathsf{P}
eq mathsf{PSPACE}$. Their main results are: (i) assuming the Generalized Riemann Hypothesis (GRH), the existence of a $mathsf{VP}$-succinct hitting set implies either $mathsf{VP}
eq mathsf{VNP}$ or a strong lower bound against $mathsf{TC}^0$; and (ii) subpolynomially explicit hitting sets suffice to separate $mathsf{P}$ from $mathsf{PSPACE}$. This work provides a new characterization of the algebraic natural proofs barrier and unifies deep connections among hitting set explicitness, circuit lower bounds, and complexity class separations.
📝 Abstract
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 $mathsf{VP}
eq mathsf{VNP}$, 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 extsf{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 $mathsf{VP}
eq mathsf{VNP}$ or that $mathsf{P}
eq mathsf{PSPACE}$. This motivates us to define the concept of emph{cryptographic hitting sets}, which we believe is interesting on its own.
