Formula Size-Depth Tradeoffs for Iterated Sub-Permutation Matrix Multiplication

Abstract: We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le \log k$, this problem is solvable by $n<sup>{O(dk<sup>{1/d})}$ size monotone formulas of two distinct types: (unbounded fan-in) $AC<sup>0$ formulas of depth $d+1$ and (semi-unbounded fan-in) $SAC<sup>0$ formulas of $\bigwedge$-depth $d$ and $\bigwedge$-fan-in $k<sup>{1/d}$. The results of this paper give matching $n<sup>{\Omega(dk<sup>{1/d})}$ lower bounds for monotone $AC<sup>0$ and $SAC<sup>0$ formulas for all $k \le \log\log n$, as well as slightly weaker $n<sup>{\Omega(dk<sup>{1/2d})}$ lower bounds for non-monotone $AC<sup>0$ and $SAC<sup>0$ formulas. These size-depth tradeoffs converge at $d = \log k$ to tight $n^{\Omega(\log k)}$ lower bounds for both unbounded-depth monotone formulas [Ros15] and bounded-depth non-monotone formulas [Ros18]. Our non-monotone lower bounds extend to the more restricted Iterated Permutation Matrix Multiplication problem, improving the previous $n<sup>{k<sup>{1/\exp(O(d))}}$ tradeoff for this problem [BIP98].