Failure of Feasible Disjunction Property for $k$-DNF Resolution and NP-hardness of Automating It

We show that for every integer $k\ge 2$, the Res($k$) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a recent result of Atserias and M"uller [FOCS, 2019] to Res($k$). We show that if NP is not included in P (resp. QP, SUBEXP) then for every integer $k\ge 1$, Res($k$) is not automatable in polynomial (resp. quasi-polynomial, subexponential) time.