Res($\oplus$) is the simplest fragment of $\text{AC}^0[2]\text{-Frege}$ for which no super-polynomial lower bounds on the size of proofs are known. Bhattacharya and Chattopadhyay [BC25] recently proved lower bounds of the form $\exp(\tilde\Omega(N^{\varepsilon}))$ on the size of Res($\oplus$) proofs whose depth is upper bounded by $O(N^{2 - \varepsilon})$, where $N$ is ... more >>>

We prove that relative to a random oracle answering $O(\log n)$-bit queries, there exists a function computable in $O(n)$ time by a random-access machine (RAM) but requiring $n^2/polylog(n)$ time by any multitape Turing machine. This provides strong evidence that simulating RAMs on multitape Turing machines inherently incurs a nearly quadratic ... more >>>

We prove that $\mathrm{deg}(f) \leq 2 \, \mathrm{rdeg}(f)^4$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.

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