  In this article we give an algorithm for determining the generators and
relations for the rings of semi-invariant functions on irreducible components
of representation spaces for gentle string algebras. These rings of
semi-invariants turn out to be semigroup rings to which we can associate a
so-called matching graph. Under this association, generators for the semigroup
can be seen by certain walks on this graph, and relations are given by certain
configurations in the graph. This allows us to determine degree bounds for the
generators and relations of these rings. We furthermore show that these bounds
also hold for acyclic string algebras in general.
