Translation of Counter-free automata into Linear Temporal Logic
As mentioned in the comments, the translation is shown in:
Diekert, Volker, and Paul Gastin. &quot;First-order definable languages.&quot; (2008)
http://www.lsv.fr/Publis/PAPERS/PDF/DG-WT08.pdf
And it goes via a characterization of $LTL$ as $FO[&lt;]$.