Symmetric Formulas for Products of Permutations (2211.15520v1)

Abstract: We study the formula complexity of the word problem $\mathsf{Word}{S_n,k} : {0,1}^{kn^2} \to {0,1}$: given $n$-by-$n$ permutation matrices $M_1,\dots,M_k$, compute the $(1,1)$-entry of the matrix product $M_1\cdots M_k$. An important feature of this function is that it is invariant under action of $S_n^{k-1}$ given by [ (\pi_1,\dots,\pi{k-1})(M_1,\dots,M_k) = (M_1\pi_1^{-1},\pi_1M_2\pi_2^{-1},\dots,\pi_{k-2}M_{k-1}\pi_{k-1}^{-1},\pi_{k-1}M_k). ] This symmetry is also exhibited in the smallest known unbounded fan-in ${\mathsf{AND},\mathsf{OR},\mathsf{NOT}}$-formulas for $\mathsf{Word}{S_n,k}$, which have size $n^{O(\log k)}$. In this paper we prove a matching $n^{\Omega(\log k)}$ lower bound for $S_n^{k-1}$-invariant formulas computing $\mathsf{Word}{S_n,k}$. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes $\mathsf{NC}^1$ and $\mathsf{Logspace}$. Our more general main theorem gives a nearly tight $n^{d(k^{1/d}-1)}$ lower bound on the $G^{k-1}$-invariant depth-$d$ ${\mathsf{MAJ},\mathsf{AND},\mathsf{OR},\mathsf{NOT}}$-formula size of $\mathsf{Word}_{G,k}$ for any finite simple group $G$ whose minimum permutation representation has degree~$n$. We also give nearly tight lower bounds on the $G^{k-1}$-invariant depth-$d$ ${\mathsf{AND},\mathsf{OR},\mathsf{NOT}}$-formula size in the case where $G$ is an abelian group.