We revisit a classic theorem of Frougny and Sakarovitch concerning automata for $\varphi$-representations, and show how to obtain it in a different and more computationally direct way. Using it, we can find simple, induction-free proofs of existing results in the literature about these representations, in a uniform and straightforward manner. In particular, we can easily and "automatically'' recover many of the results of recent papers of Dekking and Van Loon. We also obtain a number of new results on $\varphi$-representations.

翻译：我们重新审视了Frougny和Sakarovitch关于$\varphi$-表示自动机的一个经典定理，并提出了一种不同且计算上更直接的方法来获得该定理。借助该方法，我们可以用统一且直接的方式为文献中关于这些表示的现有结果找到简单、无需归纳的证明。特别地，我们能够轻松且“自动地”重现Dekking和Van Loon近期论文中的众多结果。我们还获得了关于$\varphi$-表示的一系列新结论。