A Theory of Heuristic Information in Game-Tree Search

Searching is an important process in most AI systems, especially in those AI production systems consisting of a global database, a set of production rules, and a control system. Because of the intractability of uninformed search procedures, the use of heuristic information is necessary in most searching processes of AI systems. This important concept of heuristic informatioD is the central topic of this book. We first use the 8-puzzle and the game tic-tac-toe (noughts and crosses) as examples to help our discussion. The 8-puzzle consists of eight numbered movable tiles set in a 3 x 3 frame. One cell of the frame is empty so that it is possible to move an adjacent numbered tile into the empty cell. Given two tile configurations, initial and goal, an 8-puzzle problem consists of changing the initial configuration into the goal configuration, as illustrated in Fig. 1.1. A solution to this problem is a sequence of moves leading from the initial configuration to the goal configuration, and an optimal solution is a solution having the smallest number of moves. Not all problems have solutions; for example, in Fig. 1.1, Problem 1 has many solutions while Problem 2 has no solution at all.

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This book presents the use of imperfect information (called heuristic information) in game-tree search. Its purpose is to investigate the theoretical background of the use of heuristic information in game-tree search. Computer programs playing games usually search the game-tree to a reasonable depth with a static evaluation function and make decisions based upon backed-up values. Since the information in either the backed-up values or the values returned directly by the static evaluation function is often imperfect, decision making is usually not optimal. Also, pathological cases show why intuition about game-tree search is not always correct. This book introduces a mathematical formulation of heuristic information and a theoretical model of game-tree search. In this model, notions of game-tree search are formulated in mathematical terms and a sound mathematical theory of heuristic information is developed. The conventional pathological cases disappear in this theory. This book is accessible to the general AI community, for example first year graduate students who have completed an introductory AI course and have at least some background in probability. The book is also a foundation for further work on game-tree search as well as on heuristic information in general AI.

Contenu
1 Introduction.- 2 Games and Minimax Values.- 2.1 Finite Perfect Information Games and Game Trees.- 2.2 Zero-Sum Two-Person Perfect Information Games.- 2.3 Subgames and Game Graphs.- 2.4 Example 1: G1-Games.- 2.5 Example 2: P2-Games.- 2.6 Minimax Values, the Minimax Procedure and the Alpha-Beta.- Procedure.- 3 Heuristic Game-Tree Searches.- 3.1 The Conventional Heuristic Game-Tree Search.- 3.1.1 Static Evaluation Functions.- 3.1.2 The Back-Up Process.- 3.2 Heuristic Arguments and the Pathological Phenomenon.- 3.3 New Back-Up Processes.- 3.3.1 The Product-Propagation Procedure.- 3.3.2 The M & N Procedure.- 3.3.3 The *-MIN Procedure.- 3.3.4 Average Propagation.- 4 Probability Spaces and Martingales.- 4.1 Borel Fields and Partitions.- 4.2 Probability Spaces.- 4.3 Random Variables.- 4.4 Product Spaces.- 4.5 Conditional Probabilities and Martingales.- 5 Probabilistic Game Models and Games Values.- 5.1 Probabilistic Game Models.- 5.2 Strategies and Game Values.- 5.2.1 Non-randomized Strategies.- 5.2.2 Randomized Strategies.- 5.2.3 Minimax Values.- 5.3 Pb-Game Models.- 5.4 Gd-Game Models.- 6 Heuristic Information.- 6.1 Examples: P2- and G1-Game Models.- 6.1.1 P2-Game Models.- 6.1.2 G1-GameModels.- 6.2 Formulation of Heuristic Information.- 6.3 Heuristic Search.- 6.4 Improved Visibility of a Heuristic Search.- 7 Estimation and Decision Making.- 7.1 Random Variable Estimators.- 7.2 Comparison of Estimators.- 7.3 Decision Making.- 7.3.1 Decision Models.- 7.3.2 Decision Qualities.- 8 Independence and Product-Propagation Rules.- 8.1 Product Models.- 8.2 Product Heuristic Information and Product Heuristic Searches.- 8.3 Product-Propagation Rules.- 9 Estimation of Minimax Values in Pb-Game Models.- 9.1 More About Probabilities on Pb-Game Trees.- 9.2 The Conditional Probability of a Forced Win, p(h,l).- 9.3 An Approximation of p(h,l).- 10 Estimation of Minimax Values in Gd-Game Models.- 10.1 Estimation in G1-Game Models.- 10.2 Estimation in Gd-Game Models.- 11 Conclusions.- References.