Note on the Infiniteness and Equivalence Problems for Word-MIX Languages

In this note we provide a (decidable) graph-structural characterisation of the infiniteness of $L(w_1, ..., w_k)$, where $L(w_1, ..., w_k) = \{w \in A^* | |w|_{w_1} = \cdots = |w|_{w_k}\}$ is the set of all words that contain the same number of subword occurrences of parameter words $w_1, ..., w_k$. We also provide the decidable characterisation of the equivalence for those languages. Although those two decidability results are also obtained from more general known decidability results on unambiguous constrained automata, this note tries to give a self-contained (without the knowledge about constrained automata) proof of the decidability.