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

## TL;DR

This paper presents a decidable, graph-based method to determine the infiniteness and equivalence of certain word languages defined by equal subword counts, providing a self-contained proof without relying on constrained automata theory.

## Contribution

It introduces a new, self-contained graph-structural characterization for deciding infiniteness and equivalence of specific word languages, simplifying previous complex automata-based approaches.

## Key findings

- Decidable graph-structural characterization of language infiniteness
- Decidable equivalence criteria for the languages
- Self-contained proof avoiding automata theory

## Abstract

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.