Planning is a very important AI problem, and it is also a very time-consuming AI problem. To get an idea of how complex dif- ferent planning problems are, it is useful to describe the computational complexity of different general planning problems. This complexity has been described for problems in which planning is based on the (complete or partial) information about the current state of the system. In real-life planning problems, we can often complement the incompleteness of our explicit knowledge about the current state by using the implicit knowl- edge about this state which is contained in the description of the system's past behavior. For example, the information about the system's past fail- ures is very important in planning diagnostic and repair. To describe planning which can use the information about the past, a special lan- guage L was developed in 1997 by C. Baral, M. Gelfond and A. Provetti. In this paper, we expand the known results about computational com- plexity of planning (including our own previous results) to this more general class of planning problems.