Showing papers on "Game tree published in 1986"

Quantitative deduction and its fixpoint theory

Abstract: Logic programming provides a model for rule-based reasoning in expert systems. The advantage of this formal model is that it makes available many results from the semantics and proof theory of first-ordet predicate logic. A disadvantage is that in expert systems one often wants to use, instead of the usual two truth values, an entire continuum of “uncertainties” in between. That is, instead of the usual “qualitative” deduction, a form of “quantitative” deduction is required. We present an approach to generalizing the Tarskian semantics of Horn clause rules to justify a form of quantitative deduction. Each clause receives a numerical attenuation factor. Herbrand interpretations, which are subsets of the Herbrand base, are generalized to subsets which are fuzzy in the sense of Zadeh. We show that as result the fixpoint method in the semantics of Horn clause rules can be developed in much the same way for the quantitative case. As for proof theory, the interesting phenomenon is that a proof should be viewed as a two-person game. The value of the game turns out to be the truth value of the atomic formula to be proved, evaluated in the minimal fixpoint of the rule set. The analog of the PROLOG interpreter for quantitative deduction becomes a search of the game tree ( = proof tree) using the alpha-beta heuristic well known in game theory.

Probabilistic Boolean decision trees and the complexity of evaluating game trees

Abstract: The Boolean Decision tree model is perhaps the simplest model that computes Boolean functions; it charges only for reading an input variable. We study the power of randomness (vs. both determinism and non-determinism) in this model, and prove separation results between the three complexity measures. These results are obtained via general and efficient methods for computing upper and lower bounds on the probabilistic complexity of evaluating Boolean formulae in which every variable appears exactly once (AND/OR tree with distinct leaves). These bounds are shown to be exactly tight for interesting families of such tree functions. We then apply our results to the complexity of evaluating game trees, which is a central problem in AI. These trees are similar to Boolean tree functions, except that input variables (leaves) may take values from a large set (of valuations to game positions) and the AND/OR nodes are replaced by MIN/MAX nodes. Here the cost is the number of positions (leaves) probed by the algorithm. The best known algorithm for this problem is the alpha-beta pruning method. As a deterministic algorithm, it will in the worst case have to examine all positions. Many papers studied the expected behavior of alpha-beta pruning (on uniform trees) under the unreasonable assumption that position values are drawn independently from some distribution. We analyze a randomized variant of alphabeta pruning, show that it is considerably faster than the deterministic one in worst case, and prove it optimal for uniform trees.

Values of graph-restricted games

Abstract: We consider the problem of modifying n-person games so as to take account of the difficulties imposed by lack of communications, and the opportunities this might accord to intermediaries.In this model, the members of a finite set are simultaneously players in a game and vertices of a graph. A combination of these two structures gives rise to a new, modified game in which the only effective coalitions are those corresponding to connected partial graphs. We study the relationship between the power indices of the original game and the restricted game; for the special case where the graph is a tree, this relationship is especially easy to analyze.Several examples are studied in detail.

An algorithmic solution of N-person games

Abstract: Two-person, perfect information, constant sum games have been studied in Artificial Intelligence This paper opens up the issue of playing n-person games and proposes a procedure for constant sum or non-constant sum games It is proved that a procedure, maxn, locates an equilibrium point given the entire game tree The minimax procedure for 2-person games using look ahead finds a saddle point of approximations, while maxn finds an equilibrium point of the values of the evaluation function for n-person games using look ahead Maxn is further analyzed with respect to some pruning schemes

A review of game-tree pruning†

Abstract: Chess programs have three major components: move generation, search, and evaluation. All components are important, although evaluation with its quiescence analysis is the part which makes each program’s play unique. The speed of a chess program is a function of its m ove generation cost, the complexity of the position under study and the brevity of its evaluation. More important, however, is the quality of the mechanisms used to discontinue (prune) search of unprofitable continuations. The most reliable pruning method in popular use is the robust alpha-beta algorithm, and its many supporting aids. These essential parts of game-tree searching and pruning are reviewed here, and the performance of refinements, such as aspiration and principal variation search, and aids like transposition and history tables are compared. † Much of this article is a revision of material condensed from an entry entitled ‘‘Computer Chess Methods,’’ prepared for the Encyclopedia of Artificial Intelligence , S. Shapiro (editor), to be published by John Wiley & Sons in 1987. The transposition table pseudo code of Figure 7 is similar to that in another paper: ‘‘Parallel Search of Strongly Ordered Game Trees,’’ T.A. Marsland and M. Campbell, ACM Computing Surveys, Vol 14, No. 4, copyright 1982, Association for Computing Machinery Inc., and is reprinted by permission.

Solving tree problems on a Mesh-connected processor Array

Abstract: In this paper we present techniques that result in O ( n ) time algorithms for computing many properties and functions of an n -node forest stored in an n × n mesh of processors. Our algorithms include computing simple properties like the depth, the height, the number of descendents, the preorder (resp. postorder, inorder) number of every node, and a solution to the more complex problem of computing the Minimax value of a game tree. Our algorithms are asymptotically optimal since any nontrivial computation will require Ω ( n ) time on the mesh. All of our algorithms generalize to higher dimensional meshes.

Ambush strategies in search games on graphs

Abstract: A blind searcher and a blind hider move at below unit speed along a finite length graph Q known to both, until the first time T when they meet. A two person zero-sum game arises if the searcher pays the hider T units. We consider circumstances under which it may be optimal for the searcher to “lie in wait” at a node of Q, hoping the hider will come to him. We also explicitly define a notion of “equilibrium in distribution” for such games, which has been implicit in the literature. We show that for the graph consisting of two nodes connected by three arcs of equal length there are optimal ambush strategies but there is no equilibrium in distribution.

War game with variable game board

Abstract: A decoy board game is provided including a gameboard with a grid pattern divided into four segments defining discrete squares of different representations, such as water, land, mountain, island, starting, headquarter and airstrip in which four players can play the game The players are provided with four sets of game pieces adapted for movement along the grid pattern formed by the squares Each set of playing pieces includes six different types of pieces having predetermined movements and capable of removing an opponents pieces in accordance with rules of the game and by result of a random number generator

An Explanation of and Cure for Minimax Pathology

Abstract: The traditional approach to choosing moves in game-playing programs is the minimax procedure. The general belief underlying its use is that increasing search depth improves play. Recent research has shown that given certain simplifying assumptions about a game tree's structure, this belief is erroneous: searching deeper decreases the probability of making a correct move. This phenomenon is called game tree pathology. Among these simplifying assumptions is uniform depth of win/loss (terminal) nodes, a condition which is not true for most real games. Analytic studies in [10] have shown that if every node in a pathological game tree is made terminal with probability exceeding a certain threshold, the resulting tree is nonpathological. This paper considers a new evaluation function which recognizes increasing densities of forced wins at deeper levels in the tree. This property raises two points that strengthen the hypothesis that uniform win depth causes pathology. First, it proves mathematically that as search deepens, an evaluation function that does not explicitly check for certain forced win patterns becomes decreasingly likely to force wins. This failing predicts the pathological behavior of the original evaluation function. Second, it shows empirically that despite recognizing fewer mid-game wins than the theoretically predicted minimum, the new function is nonpathological.

Making best use of available memory when searching game trees

Abstract: When searching game trees, Algorithm SSS* examines fewer terminal nodes than the alphabeta procedure, but has the disadvantage that the storage space required by it is much greater. ITERSSS* is a modified version of SSS* that does not suffer from this limitation. The memory M that is available for use by the OPEN list can be fed as a parameter to ITERSSS* at run time. For successful operation M must lie above a threshold value MO. But MO is small in magnitude and is of the same order as the memory requirement of the alphabeta procedure. The number of terminal nodes of the game tree examined by ITERSSS* is a function of M, but is never greater than the number of terminals examined by the alphabeta procedure. For large enough M, ITERSSS* is identical in operation to SSS*.

Phased state space search

Abstract: PS*, a new sequential tree searching algorithm based on the State Space Search (SSS*), is presented. PS*(k) divides each MAX node of a game tree into k partitions, which are then searched in sequence. By this means two major disadvantages of SSS*, storage demand and maintenance overhead, are significantly reduced, and yet the corresponding increase in nodes visited is not so great even in the random tree case. The performance and requirements of PS* are compared on both theoretical and experimental grounds to the well known αβ and SSS* algorithms. The basis of the comparison is the storage needs and the average count of the bottom positions visited. To appear in the Procs. of the ACM/IEEE Fall Joint Computer Conference, Dallas, Nov. 1986.

Predicting the performance of minimax and product in game-tree searching

Abstract: The discovery that the minimax decision rule performs poorly in some games has sparked interest in possible alternatives to minimax. Until recently, the only games in which minimax was known to perform poorly were games which were mainly of theoretical interest. However, this paper reports results showing poor performance of minimax in a more common game called kalah. For the kalah games tested, a non-minimax decision rule called the product rule performs significantly better than minimax. This paper also discusses a possible way to predict whether or not minimax will perform well in a game when compared to product. A parameter called the rate of heuristic flaw (rhf) has been found to correlate positively with the. performance of product against minimax. Both analytical and experimental results are given that appear to support the predictive power of rhf.

A note on a core catcher of a cooperative game

Abstract: In Driessen (1986) it is shown that for games satisfying a certain condition the core of the game is included in the convex hull of the set of certain marginal worth vectors of the game, while it is conjectured that the inclusion holds without any condition on the game. In this note it is proved that the inclusion holds for all games.

Estimation of minimax values

Abstract: In estimating minimax values, an important topic in the study of heuristic game tree searches, a pathological phenomenon sometimes results when the conventional minimax procedure is used as a back-up process. In this paper exact methods are derived for two different games, one using product-propagation rules as a back-up process and another not using any back-up process. The method of estimating minimax values in a heuristic game tree search should depend on both the static evaluation function and the structure of the game tree; it can be very different for different games.

Polynomial evaluations of bi-valued game trees

Abstract: ABSTF~CT. Let T be a bi-valued game tree with probability p that a given leaf has been labeled with a i and probability l-p that it has been labeled with a 0. If A is an algorithm for solving T, then let V(A) denote the cost of solving T when using A. The cost V(A) will depend upon the value of p so that we may regard V(A) as a function of p. For straight algorithms A we develop a method for computing the function V(A) as a polynomial in p. The polynomial representation of V(A) is then used in the study of even Fibonacci trees.

State Space Algorithms for Searching Game Trees

Abstract: Modifying SSS*’s node expansion strategy yields different state space algorithms for searching game trees. The dual node expansion employed by DUAL*, for example, performs most often superior to SSS*. Introducing directional search characteristics to SSS* and DUAL* gives insight in the utility of global node information. As a result, the totally directional αβ search is shown to be just a restricted case of SSS* and D UAL*.