We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}\phi$. The paper's two unconditional contributions constrain possible characterizations. First, for ... more >>>

We prove a lower bound of $\Omega\left(n^{1.5}\right)$ for the number of product gates in non-commutative arithmetic circuits for an explicit $n$-variate degree-$n$ polynomial $f_{n}$ (over every field).

We observe that this implies that over certain non-commutative rings $R$, any arithmetic circuit that computes the induced polynomial function $f_{n}: R^n \rightarrow ... more >>>

Streaming algorithms in adversarial settings have attracted considerable attention recently. We show that, in the white-box adversarial streaming model [ABJ+22], the fundamental problem of estimating the $F_p$ moment to within any constant factor requires $\Omega(n)$ memory. In this model, the internal state of the (randomized) streaming algorithm is visible to ... more >>>