Additive AND/OR graphs

Additive AND/OR graphs are defined as AND/ /ORgraphs without circuits, which can be considered as folded AND/OR trees; i.e. the cost of a common subproblem is added to the cost as many times as the subproblem occurs, but it is computed only once. Additive AND/OR graphs are naturally obtained by reinterpreting the dynamic programming method in the light of the problem-reduction approach. An example of this reduction is given. A top-down and a bottom-up method are proposed for searching additive AND/OR graphs. These methods are, respectively, extensions of the "arrow" method proposed by Nilsson for searching AND/OR trees and Dijkstra's algorithm for finding the shortest path. A proof is given that the two methods find an optimal solution whenever a solution exists. 1) introduction In the literature on artificial intelligence, AND/OR trees have proved to be a good formalism for representing the problem-reduction approach to problem solving. Usually, the search is for any solution tree, but in a paper by Nilsson the problem is presented of finding the best solution tree, where arcs have a given cost, and the cost of a tree is simply the sum of the costs of the arcs. Nilsson gives there an algorithm which assumes available, for each node, an estimate of the cost of the "optimal solution tree rooted at that node.