Temporal logic with overlap temporal relations generated by time states themselves

We study a temporal logic with non–standard temporal accessibility relations. This logic is generated by semantic underground models, and any such a model has a base formed by a frame with temporal relations generated by temporal states themselves; potentially, any state possesses its own temporal accessibility relation, and it is possible that all of them can be different. We consider this to be the most plausible modelling, because any time state has, in principle, its own view on what is past (or future). Time relations may have non–empty overlaps and they can be totally intransitive. Thus, this approach may be suitable for analysis of the most general cases of reasoning about computation, information flows, reliability, and other areas of AI and CS. The main mathematical question under consideration here is the existence of algorithms for solving satisfiability problems. Here we solve this problem and find the required algorithms. In the final part of our paper we formulate some interesting open problems.