A central open problem in complexity theory concerns the question of
whether all efficient randomized algorithms can be simulated by
efficient deterministic algorithms. The celebrated ``hardness
v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84),
Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness
assumptions under which $\prBPP = \prP$, but these hardness ... more >>>

We prove two results about randomised query complexity $\mathrm{R}(f)$. First, we introduce a linearised complexity measure $\mathrm{LR}$ and show that it satisfies an inner-optimal composition theorem: $\mathrm{R}(f\circ g) \geq \Omega(\mathrm{R}(f) \mathrm{LR}(g))$ for all partial $f$ and $g$, and moreover, $\mathrm{LR}$ is the largest possible measure with this property. In particular, ... more >>>

We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times n$ matrix multiplication, we construct a hitting set generator with seed length $O(\sqrt{n} \cdot ... more >>>