We address the well-known satisfiability problem (SAT), i.e., the problem of checking whether a given propositional formula is satisfiable. Although the general satisfiability problem is NP-complete, some particular cases of SAT are known to be easy. The most important of those cases is HORN-SAT, i.e., the satisfiability problem in the case of Horn clauses: actually, an instance of HORN-SAT can be solved in linear time. Yamasaki and Doshita (1983) have introduced a new subclass of SAT, S₀, which is polynomially solvable and which strictly includes HORN-SAT. Here we introduce a family of subclasses of SAT, Γ₀, Γ₁,…,Γ_k,…, such that HORN-SAT = Γ₀, S₀ = Γ₁, Γ_k Γ_k-1, k = 1,2,…, and Γ_k approaches to SAT as k increases. For each k, Γ_k is solvable in O(n^*n^k) time, where n is the number of propositional letters, m is the number of clauses, and n^* = O(nm) is the size of the input. An algorithm to check whether a given instance of SAT belongs to Γ_k, for any k, which runs in O(n^*n^k time, is also described.