Give a complete (resource free) characterisation of rewrite systems with polynomial derivational complexity.

The source page states the original problem together with its recorded qualifications and progress updates as follows.

It is well-known that well-founded monotone algebras form a complete characterisation for termination while such a result is currently unknown for polynomial derivational complexity. The notion of resource freeness is borrowed from implicit computational complexity theory. Here it refers to characterisations devoid of direct references to polynomial derivational complexity.

Currently suitably restricted matrix interpretations (see~[Moser and Zankl, 2010]) form the method for proving polynomial upper bounds on the derivational complexity. Thus it is perhaps important to emphasise that matrix interpretations as studied in~[Moser and Zankl, 2010] are not sufficient as a starting point to solve the problem. Consider the one-rule TRS $\mathsf{g}(x,x) \to \mathsf{g}(\mathsf{a},\mathsf{b})$. This TRS has linear derivational complexity, but no compatible matrix interpretation can exist.