 recognition, we'll define this recursively.</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a>First, let's define a path as a sequence of symbol-state pairs, where each pair represents a transition in the automaton. For a path $p$, we denote the last state in the path as $\text{last-state}(p)$.</span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-25