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与Sakarovitch关于$\varphi$-表示自动机的经典定理，并展示了以另一种更具计算直接性的方式推导该定理的方法。基于此，我们能够以统一且简洁的方式，为文献中关于这些表示的现有结果提供简单且无需归纳的证明。特别地，我们可轻松、"自动地"复现Dekking与Van Loon近期论文中的多项成果，同时获得了关于$\varphi$-表示的若干新结论。