Deterministic State Space Planning with BDDs

This short paper (cf. [5]) proposes a planner that uses BDDs to compactly represent sets of propositionally represented states. Using this representation, accurate reachability analysis and backward chaining can apparently be carried out without necessarily encountering exponential representation explosion. The main objectives are the interest in optimal solutions, the generality and the conciseness of the approach. The algorithms are tested against the AIPS'98 planning competition problems and lead to substantial improvements to existing solutions. The projectMips abbreviates intelligent model checking and planning system and encapsulates algorithms and data structures for model checking and planning with binary decision diagrams (BDDs) [2]. BDDs have the expressive power to e ciently encode and manipulate large sets of bit-strings. Therefore, BDDs are capable of symbolically representing both the states within the planning space and the operators applied. The planning algorithm assumes a binary encoding of the planning states inferred by a domain-independent pre-compiler [3]. Let O be the set of all operators. Then the transition relation T for the planning problem given by the disjunction T (x; x) = W o2O o(x; x) evaluates to true if x is a successor of x. To build the sub-relations o(x; x) we use enumeration of all possible values of the parameters. As an operator usually depends only on a small subset of the state space information this is not a limitation. The same transition relation can be used to perform several di erent strategies. Either a simple breadthrst-search can be applied, or a bidirectional search simultaneously starting from the initial and the goal state. But also more elaborated algorithms like BDDA [4], a BDD-based version of A* are available. The BDD representation of the state space and the operators allows to reduce the planning problem to model checking [6] which usually performs a breadthrst-search. An iterative calculation of boolean expressions has to be performed to verify the formula EF(goal) of temporal logic: Is it possible to reach a goal state from the initial state within a nite number of transitions? If the answer to this question is yes the planning problem is solvable and the computation of a (minimal) witness delivers a plan. Figure 1 shows a typical pro le of the iteration process of symbolic (bidirectional) breadthrst-search to re ect the major advantage of BDDs: Even though the number of represented states in the search grows exponentially the BDD-sizes do not necessarily do the same.