In this paper, we propose a new resolution proof procedure for the branching-time logic programming language Cactus. The particular strength of the new proof procedure, called CSLD-resolution, is that it can handle, in a more general way, open-ended queries, i.e. goal clauses that include atoms which do not refer to specific moments in time, without the need of enumerating all their canonical instances. We also prove soundness, completeness and independence of the computation rule for CSLD-resolution. The new proof procedure overcomes the limitations of a family of proof procedures for temporal logic programming languages, which were based on the notions of canonical program and goal clauses. Moreover, it applies directly to Chronolog programs and it can be easily extended to apply to multi-dimensional logic programs as well as to Chronolog(MC) programs.