Computational complexity of planning and approximate planning in the presence of incompleteness

In the last several years, there have been several studies about the computational complexity of classical planning assuming that the planner has complete knowledge about the initial situation. Recently, there have been proposals to use `sensing' actions to plan in the presence of incompleteness. In this paper we study the complexity of planning in such cases. In our study we use the action description language A proposed in 1991 by Gelfond and Lifschitz, and its extensions. It is known that if we consider only plans of tractable (polynomial) duration, planning in A - with complete information about the initial situation - is NP-complete: even checking whether a given objective is attainable from a given initial state is NP-complete. In this paper, we show that the planning problem in the presence of incompleteness is indeed harder: it belongs to the next level of the complexity hierarchy (in precise terms, it is Σ₂P-complete). To overcome the complexity of this problem, Baral and Son have proposed several approximations. We show that under certain conditions, one of these approximations - 0-approximation - makes the problem NP-complete (thus indeed reducing its complexity).