Recall the definition of a cluster algebra, which can be seen as a (possibly infinite) graph, where each vertex is a tuple of a quiver and Laurent expressions at some of the vertices of the quiver. The edges of this graph is given by mutations, and every vertex has the same degree.
Whenever the quiver is a Dynkin diagram, the graph is finite, and the Laurent polynomials have nice interpretations.
However, for a general seed quiver, the graph is infinite. Given a seed, and a Laurent polynomial, is there an algorithm to determine if the given Laurent polynomial appear as a cluster variable in the cluster algebra? Or might this be undecidable?
The reason it might be undecidable, is that a priori, the graph is infinite, so there is no upper bound on how many vertices we need to visit (by successive mutations), before concluding that the expression cannot be constructed, and the problem has "the same feel" as say the Collatz problem, or the problem of deciding if a set of matrices can produce the zero matrix by multiplication (the latter problem is undecidable).