A Solution to the GHI Problem for Best-First Search

In a search graph the node's value is dependent on the path leading to it. Diierent paths may lead to diierent values. Hence, it is diicult to determine the value of any node unambiguously. The problem is known as the graph-history-interaction (GHI) problem. This report provides a solution for best-rst search. First, we give a precise formulation of the problem. Then we review all important earlier proposals to overcome it, also for other searches than best rst. Next, our solution is given in outline. The solution is implemented in a best-rst search algorithm. Experimental results in the eld of computer chess connrm the claim that the GHI problem has been solved for best-rst search. Search algorithms are used in many domains, ranging from theorem proving to computer games. The algorithms are searching in a search space containing problem states (positions), often represented as nodes. A move which transforms a position into a new position is represented as an edge connecting the two nodes. In a search tree, it may happen that identical nodes are encountered at diierent places. If these so-called transpositions are not recognized, the search algorithm unnecessarily expands identical subtrees. Therefore, it is prootable to recognize transpositions and to ensure that for each set of identical nodes, only one subtree is expanded. In computer-chess programs using a depth-rst search algorithm, this idea is realized by storing the result of a node's investigation in a transposition table (e.g., Hyatt et al. (1984); Marsland (1986)). If an identical node is encountered in the search process, the result is retrieved from the transposition table and used without further investigation. If a (selective) best-rst search algorithm (which stores the whole search tree in memory) is used, the search tree is converted into a search graph, by joining identical nodes into one node, thereby merging the subtrees. These common ways of dealing with transpositions contain an important aw: determining whether nodes are identical is not the same as determining whether the search states represented by the nodes are identical. For two reasons, the path leading to a node cannot be ignored. First, the history of a node may partly determine the legitimacy of a move. For instance, in chess, castling rights are not only determined by the position of the pieces on the board, but also by the knowledge that in the position under investigation the King and Rook have not moved previously.