The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF and DNF formulas of polynomial size with approximate degree $\Omega(n^{1-\delta}),$ essentially matching the trivial upper bound of $n.$ This ... more >>>

The main motivation for studying linear data structures and circuits is the intuition that non-linear advice cannot help in computing a linear operator. Jukna and Schnitger formalized this as a conjecture which states that all circuits computing a linear operator can be ``linearized," with only a constant size blow-up. We ... more >>>

Is randomness ever necessary for space-efficient computation? It is commonly conjectured that L = BPL, meaning that halting decision algorithms can always be derandomized without increasing their space complexity by more than a constant factor. In the past few years (say, from 2017 to 2022), there has been some exciting ... more >>>