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.

• 03D05 - Automata and formal grammars in connection with logical questions
• 11A63 - Radix representation; digital problems
• 11A67 - Other number representations
• 11B85 - Automata sequences
• 68Q45 - Formal languages and automata
• 68R15 - Combinatorics on words