A hierarchy of classes Hk of propositional formulas has been introduced by Yamasaki and Doshita (1983) and Gallo and Scutella (1988). The classes Hk have polynomially solvable satisfiability problem, and the basic class is the set of Horn formulas. We show that the k-resolution is complete and sound for Hk, where k-resolution is a restriction of ordinary resolution, in that at least one of the parent clauses has at most k literals. Furthermore, we discuss to what extent the class of unsatisfiable formulas in Hk and the class of unsatisfiable formulas for which a k-resolution refutation exists coincide.