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$\varphi$-表示自动机的经典定理，并展示了一种不同且计算上更直接的方法来获得该定理。利用该方法，我们能够以统一且简洁的方式，找到文献中关于这些表示已有结果的简单无归纳证明。特别地，我们可以轻松且“自动地”重现Dekking和Van Loon近期论文中的众多结果。此外，我们还获得了若干关于$\varphi$-表示的新结论。