A theory of game trees, based on solution trees

In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied

Game tree algorithms and solution trees

In this paper, a theory of game tree algorithms is presented, entirely based upon the concept of solution tree. Two types of solution trees are distinguished: max and min trees. Every game tree algorithm tries to prune nodes as many as possible from the game tree. A cut-off criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without affecting the result. Further, we show that any algorithm actually constructs a superposition of a max and a min solution tree. Finally, we will see, how solution trees and the related cutoff criterion are applied in major game tree algorithms, like alpha-beta and MTD

Trends in game tree search

This paper deals with algorithms searching trees generated by two-person, zero-sum games with perfect information. The standard algorithm in this field is alpha-beta. We will discuss this algorithm as well as extensions, like transposition tables, iterative deepening and NegaScout. Special attention is devoted to domain knowledge pertaining to game trees, more specifically to solution trees. The above mentioned algorithms implement depth first search. The alternative is best first search. The best known algorithm in this area is Stockman's SSS*. We treat a variant equivalent to SSS* called SSS-2. These algorithms are provably better than alpha-beta, but it needs a lot of tweaking to show this in practice. A variant of SSS-2, cast in alpha-beta terms, will be discussed which does realize this potential. This algorithm is however still worse than NegaScout. On the other hand, applying a similar idea as the one behind NegaScout to this last SSS version yields the best (sequential) game tree searcher known up till now: MTD(f)

Control Strategies for Two Player Games

Computer games have been around for almost as long as computers. Most of these games, however, have been designed in a rather ad hoc manner because many of their basic components have never been adequately defined. In this paper some deficiencies in the standard model of computer games, the minimax model, are pointed out and the issues that a general theory must address are outlined. Most of the discussion is done in the context of control strategies, or sets of criteria for move selection. A survey of control strategies brings together results from two fields: implementations of real games and theoretical predictions derived on simplified game-trees. The interplay between these results suggests a series of open problems that have arisen during the course of both analytic experimentation and practical experience as the basis for a formal theory

A Theory of Game Trees, Based on Solution Trees

In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respctively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied. Keywords: Game tree search, Minimax search, Solution trees, Alpha-beta, SSS*, MTD, (Nega)Scout. 1 Introduction A game tree models the behavior of a two-player game. The nodes in such a tree represent positions of a game, whereas edges represent moves. Given a payoff in the leaves of a game tree, the best move in each position can be determined by means of the so-called minimax function. Ov..

A Theory of Game Trees, Based on Solution Trees

In this paper, a theory of game tree algorithms is presented, entirely based upon the concept of solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that a game tree algorithm actually constructs a superposition of a max and a min solution tree. For algorithms searching a game tree, a cutoff criterion is suitable. A cut-off criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without affecting the result. Finally, we will see, how solution trees and the related cutoff criterion are applied in the major game tree algorithms