Conspiracy Numbers and Caching for Searching And/Or Trees and Theorem-Proving

1 I n t r o d u c t i o n This paper applies the idea of conspiracy numbers [McAllester, 1988] to derive two heuristic algorithms for searching and/or trees. The first algorithm turns out to be a member of the class of A O * best-first algo­ r i thms [Nilsson, 1980], but it conforms to the principle of sunk costs (a rule of economic rationality not respected by tradit ional and/or tree search algorithms) and hence the standard guarantees of termination and admissibility do not apply usefully to i t . The second algori thm works depth-first. It guides the search done by an iterative deepening SLD-resolution theorem prover that we have implemented. In addition to caching successes, the prover caches failures and uses the latter to avoid repeated effort also. The prover ex­ ploits the fact that a new goal matches a cached success or failure if it is a substitution instance of the cached goal, not just if the two are identical. Unlike many heuristics for guiding search and saving effort, the con­ spiracy number and caching ideas introduced in this pa­ per can be implemented efficiently. Our experimental results indicate that conspiracy numbers and especially the new caching scheme are effective in practice. Section 2 develops the best-first conspiracy numbers algor i thm, relates it to tradi t ional and/or tree search 'This research was supported in part by the United States Office of Naval Research through grant N0014-88-K-0123. algorithms, and then presents the depth-first conspir­ acy numbers algorithm. Section 3 describes our theorem prover and compares it to a similar PROLOG-technology theorem prover [Stickel, 1986]. The caching done by our prover is discussed in Section 4. Finally, our experimen­ tal results appear in Section 5, and Section 6 contains our conclusions. 2 Bestf i rs t and depthf i rs t conspiracy number a lgor i thms And/or trees are well-known [Nilsson, 1980; Pearl, 1984], and we shall describe them here only to the extent neces­ sary to make the terminology of this paper understand­ able. Briefly, an and/or tree is a tree where each node is an and-node or an or-node. The children of and-nodes are required to be or-nodes, and vice versa. An and/or tree may be completely or partial ly explored. Each node of a completely explored and/or tree is solved or failed. An internal and-node is solved if each of its children is solved and it is failed if at least one of its children is failed. Conversely, an internal or-node is solved if at least one of its children is solved, and failed if all its children are failed. A partial ly explored and/or tree also contains leaf nodes called unexpanded leaves whose children have not yet been discovered. The solved/failed status of an unexpanded leaf is unknown, as is the status of any in­ ternal node of a partial ly explored tree whose status is not fixed by the status of its children. A solution of an and/or tree is a subtree that demon­ strates that the root of the tree is solved. The aim of searching an and/or tree is to find a solution. Concretely, a solution is a subtree such that all its nodes are solved, all the children of each of its and-nodes belong to the subtree, and at least one of the children of each of its or-nodes also belongs to the subtree. The basic search­ ing operation on a partial ly explored and/or tree is to expand an unexpanded leaf of the tree. When such a leaf is expanded there are three possible outcomes: it can be discovered to be solved, it can be discovered to be failed, or it can be discovered to have children, which are new unexpanded leaves. Different searching algo­ r i thms choose in different ways which unexpanded leaf to expand next.