Complexity of Planning with Partial Observability

We show that for conditional planning with partial observability the existence problem of plans with success probability 1.0 is 2-EXP-complete. This result completes the complexity picture for nonprobabilistic propositional planning. We also give new more direct and informative proofs for the EXP-hardness of conditional planning with full observability and the EXPSPACE-hardness of conditional planning without observability. The proofs demonstrate how lack of full observability allows the encoding of exponential space Turing machines in the planning problem, and how the necessity to have branching in plans corresponds to the move to a complexity class defined in terms of alternation from the corresponding deterministic complexity class. Lack of full observability necessitates the use of beliefs states, the number of which is exponential in the number of states, and alternation corresponds to the choices a branching plan can make.