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

TL;DR

This paper proves matching nΩ(dk1/d) lower bounds for monotone AC0 and SAC0 formulas for all k ≤ loglogn, and slightly weaker nΩ(dk1/2d) lower bounds for non-monotone AC0 and SAC0 formulas.

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 $n^{O(dk^{1/d})}$ size monotone formulas of two distinct types: (unbounded fan-in) $AC^0$ formulas of depth $d+1$ and (semi-unbounded fan-in) $SAC^0$ formulas of $\bigwedge$-depth $d$ and $\bigwedge$-fan-in $k^{1/d}$. The results of this paper give matching $n^{Ω(dk^{1/d})}$ lower bounds for monotone $AC^0$ and $SAC^0$ formulas for all $k \le \log\log n$, as well as slightly weaker $n^{Ω(dk^{1/2d})}$ lower bounds for non-monotone $AC^0$ and $SAC^0$ formulas. These size-depth tradeoffs converge at $d = \log k$ to tight $n^{Ω(\log k)}$ 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 $n^{k^{1/\exp(O(d))}}$ tradeoff for this problem [BIP98].

Theorems & Definitions (113)

• Theorem 1.1: rossman2015correlationrossman2018formulas
• Proposition 1.2
• Proposition 1.3
• Example 1.4: The case $d=1$ and $k=5$ of Proposition \ref{['prop:upper']}
• Example 1.5: The case $d=2$ and $k=25$ of Proposition \ref{['prop:upper']}
• Theorem 1.6: Size-depth tradeoffs for $\mathit{AC^0}$ and $\mathit{SAC^0}$ formulas computing $\textup{sub-pmm}_{n,k}$
• Definition 2.1: Parameters $\|G\|$, $\Delta(G)$, $\lambda(G)$
• Definition 2.2: The operation $G \ominus F$
• Definition 2.3
• Example 2.4