  The compact directed acyclic word graph (CDAWG) is the minimal compact
automaton that recognizes all the suffixes of a string. Classically the CDAWG
has been implemented as an index of the string it recognizes, requiring $o(n)$
space for a copy of the string $T$ being indexed, where $n=|T|$. In this work,
we propose using the CDAWG as an index for grammar-compressed strings. While
this enables all analyses supported by the CDAWG on any grammar-compressed
string, in this work we specifically consider pattern matching. Using the CDAWG
index, pattern matching can be performed on any grammar-compressed string in
$\mathcal{O}(\text{ra}(m)+\text{occ})$ time while requiring only
$\mathcal{O}(\text{er}(T))$ additional space, where $m$ is the length of the
pattern, $\text{ra}(m)$ is the grammar random access time, $\text{occ}$ is the
number of occurrences of the pattern in $T$, and $\text{er}(T)$ is the number
of right-extensions of the maximal repeats in $T$. Our experiments show that
even when using a na\"ive random access algorithm, the CDAWG index achieves
state of the art run-time performance for pattern matching on
grammar-compressed strings. Additionally, we find that all of the grammars
computed for our experiments are smaller than the number of right-extensions in
the string they produce and, thus, their CDAWGs are within the best known
$\mathcal{O}(\text{er}(T))$ space asymptotic bound.
