In this paper we study multi-agents’ non-linear temporal logic TS4_{K_n} based on arbitrary (in particular, non-linear, finite or infinite) frames with reflexive and transitive accessibility relations, and individual symmetric accessibility relations R_i for agents. Our framework uses conception of interacting agents implemented via arbitrary finite paths of transitions by agents accessibility relations. Main problems we are dealing here are decidability and satisfiability problems for this logic. We prove that TS4_Kn is decidable and that the satisfiability problem for it is also decidable. We suggest an algorithm for checking satisfiability based on computation truth values of special inference rues in finite models modeling web connections by transitive Kripke frames.