  Tree-adjoining grammars are a generalization of context-free grammars that
are well suited to model human languages and are thus popular in computational
linguistics. In the tree-adjoining grammar recognition problem, given a grammar
$\Gamma$ and a string $s$ of length $n$, the task is to decide whether $s$ can
be obtained from $\Gamma$. Rajasekaran and Yooseph's parser (JCSS'98) solves
this problem in time $O(n^{2\omega})$, where $\omega < 2.373$ is the matrix
multiplication exponent. The best algorithms avoiding fast matrix
multiplication take time $O(n^6)$.
  The first evidence for hardness was given by Satta (J. Comp. Linguist.'94):
For a more general parsing problem, any algorithm that avoids fast matrix
multiplication and is significantly faster than $O(|\Gamma| n^6)$ in the case
of $|\Gamma| = \Theta(n^{12})$ would imply a breakthrough for Boolean matrix
multiplication.
  Following an approach by Abboud et al. (FOCS'15) for context-free grammar
recognition, in this paper we resolve many of the disadvantages of the previous
lower bound. We show that, even on constant-size grammars, any improvement on
Rajasekaran and Yooseph's parser would imply a breakthrough for the $k$-Clique
problem. This establishes tree-adjoining grammar parsing as a practically
relevant problem with the unusual running time of $n^{2\omega}$, up to lower
order factors.
