Short $\mathsf{Res}^*(\mathsf{polylog})$ refutations if and only if narrow $\mathsf{Res}$ refutations

In this note we show that any k-CNF which can be refuted by a quasi-polynomial Res(polylog) refutation has a “narrow” refutation in Res (i.e., of poly-logarithmic width). Notice that while Res(polylog) is a complete proof system, this is not the case for Res if we ask for a narrow refutation. In particular is not even possible to express all CNFs with narrow clauses. But even for constant width CNF the former system is complete and the latter is not (see for example [BG01]). We are going to show that the formulas “left out” are the ones which require large Res(polylog) refutations. We also show the converse implication: a narrow Resolution refutation can be simulated by a short Res(polylog) refutation.