\)</span> extracts the output strings from a set of paths - <span class="math inline">\(P' = \{p \circ \langle (\sigma_1, \gamma, q') \rangle \circ \epsilon\text{-path} \mid p \in P, \text{last-state}(p) = q, (q', \gamma) \in \delta(q, \sigma_1), \epsilon\text{-path} \in \text{epsilon-paths}(q')\}\)</span></p>
<p>The function <span class="math inline">\(\text{last-state}(p)\)</span> returns the state at the end of path <span class="math inline">\(p\)</span>.</p>
<p>To initialize the transduction, we need to