Generalization of alpha-beta and SSS1 search procedures

Search procedures, such as alpha-beta and SSS1, are used to solve minimax game trees. With a notable exception of B1, most of these procedures assume the static model, i.e., the computation is done solely on the basis of static values given to terminal nodes. The first goal of this paper is to generalize these to the informed model, which permits the usage of heuristic information pertaining to nonterminal nodes, such as upper and lower bounds, and estimates, of the exact values realizable from the corresponding game positions. We provide a general framework, within which various conventional procedures including alpha-beta and SSS1 can be naturally generalized to the informed model.For the static model, it is known that SSS1 surpasses alpha-beta in the sense that it explores only a subset of the nodes which are explored by alpha-beta. The second goal of this paper is, assuming the informed model, to develop a precise characterization of the class of search procedures that surpass alpha-beta. It turns out that the class contains many search procedures other than SSS1 (even for the static model). Finally some computational comparison among these search procedures is made by solving the 4 × 4 Othello game.     

博弈树启发式搜索的α-β剪枝技术研究

博弈是启发式搜索的一个重要应用领域,博弈的过程可以用一棵博弈搜索树表示,通过对博弈树进行搜索求取问题的解,搜索策略常采用α-β剪枝技术。在深入研究α-β剪枝技术的基础上,提出在扩展未达到规定深度节点时,对扩展出的子节点按照估价函数大小顺序插入到搜索树中,从而在α-β剪枝过程中剪掉更多的分枝,提高搜索效率。     

Improved speedup bounds for parallel alpha-Beta search

In this paper we investigate the ``mandatory-work-first' approach to parallel alpha-beta search first proposed by Akl, Barnard, and Doran. This approach is based on a version of alpha-beta search without deep cutoffs and a two-stage evaluation process, the second stage of which is often pruned. Our analysis shows that for best-first ordering on the lookahead tree, this approach provides greater speedup than the Palphabeta tree-splitting technique, and that for worst-first ordering, mandatory work first provides only slightly worse speedup than Palphabeta.     

基于概率的剪枝算法

以alpha-beta剪枝算法为研究对象,提出一种基于alpha-beta剪枝和概率剪枝因素相结合的概率剪枝算法,来解决博弈树搜索问题。利用概率剪枝算法,可减少博弈树搜索深度,从而加快搜索进程。     

基于概率的剪枝算法

以alpha—beta剪枝算法为研究对象，提出一种基于alpha—beta剪枝和概率剪枝因素相结合的概率剪枝算法．来解决博弈树搜索问题。利用概率剪枝算法，可减少博弈树搜索深度，从而加快搜索进程。     

Minimax search algorithms with and without aspiration windows

Investigation of several algorithms for computing exact minimax values of game trees (utilizing backward pruning) are discussed. The focus is on trees with an ordering similar to that actually found in game playing practice. The authors compare the algorithms using two different distributions of the static values, the uniform distribution and a distribution estimated from practical data. A systematic comparison of using aspiration windows for all of the usual minimax algorithms is presented. The effects of aspiration windows of varying size and position are analyzed. Increasing the ordering of moves to near the optimum results in unexpectedly high savings. Algorithms with linear space complexity benefit most. Although the ordering of the first move is of predominant importance, that of the remainder has only second-order effects. The use of an aspiration window not only makes alpha-beta search competitive, but there also exist dependencies of its effects on certain properties of the trees     

The history heuristic and alpha-beta search enhancements inpractice

Many enhancements to the alpha-beta algorithm have been proposed to help reduce the size of minimax trees. A recent enhancement, the history heuristic, which improves the order in which branches are considered at interior nodes is described. A comprehensive set of experiments is reported which tries all combinations of enhancements to determine which one yields the best performance. In contrast, previous work on assessing their performance has concentrated on the benefits of individual enhancements or a few combinations. The aim is to find the combination that provides the greatest reduction in tree size. Results indicate that the history heuristic combined with transposition tables significantly outperforms other alpha-beta enhancements in application-generated game trees. For trees up to depth 8, this combination accounts for 99% of the possible reductions in tree size, with the other enhancements yielding insignificant gains     

Depth-limited search for real-time problem solving

We propose depth-limited heuristic search as a general paradigm for real-time problem solving in a dynamic environment. When combined with iterative-deepening, it provides the ability to commit to an action almost instantaneously, but allows the quality of that decision to improve as long as time is available. Once a deadline is reached, the best decision arrived at is executed. We illustrate the paradigm in three different settings, corresponding to single-agent search, two-player games, and multi-agent problem solving. First we review two-player minimax search with alpha-beta pruning. Minimax can be extended to themaxn algorithm for more than two players, which admits a much weaker form of alpha-beta pruning. Finally, we explore real-time search algorithms for single-agent problems. Minimax is specialized tominimin, which allows a very powerfulalpha pruning algorithm. In addition,real-time-A ^* allows backtracking while still guaranteeing a solution and making locally optimal decisions.This research was supported by an NSF Presidential Young Investigator Award, NSF Grant IRI-8801939, and and equipment grant from Hewlett-Packard. Thanks to Valerie Aylett for drawing the figures.     

Unsynchronized iteratively deepening parallel alpha-beta search

A parallel alpha-beta search algorithm called unsynchronized iteratively deepening parallel alpha-beta search is described. The algorithm's simple control strategy and strong performance in complicated positions make it a viable alternative to the principal variation splitting algorithm (PVSA). Processors independently carry out iteratively deepening searches on separate subsets of moves. The iterative deepening is unsynchronized, e.g. one processor may be in the middle of the fifth iteration while another is in the middle of the sixth. Narrow windows diminish the importance of backing-up a score to the root of the tree as quickly as possible (one of the principal objectives of the PVSA). Speedups measured on one, two, four, and eight chess-playing computers are reported     

Challenging state-of-the-art move ordering with Adaptive Data Structures

The field of game playing is a particularly well-studied area within the context of AI, leading to the development of powerful techniques, such as the alpha-beta search, capable of achieving competitive game play against an intelligent opponent. It is well-known that tree pruning strategies, such as alpha-beta, benefit strongly from proper move ordering, i.e., by searching the best element first. A wide range of techniques have been developed over the years to achieve good move ordering, and improved tree pruning, in the field, in general and in particular, in the alpha-beta search, have been extensively studied. Inspired by the formerly unrelated field of Adaptive Data Structures (ADSs), we had previously introduced the History-ADS technique, which employs an adaptive list to achieve effective and dynamic move ordering, in a domain independent fashion. Our previous results confirmed that it performs well in a very wide range of cases, and in varied types of board games. However, our previous work did not compare the performance of the History-ADS heuristic to any established move ordering strategy. In an attempt to address this problem, we present here a comparison to two well-known, acclaimed strategies, which operate on a similar philosophy to the History-ADS, namely, the History Heuristic, and the Killer Moves technique. We also introduce, in this work, a mechanism by which these established move ordering strategies can be approximated, or directly implemented, in terms of ADSs. We confirm that, in a wide range of two-player and multi-player games, at various points in the game’s progression, the History-ADS performs at least as well as these strategies, and, in fact, outperforms them in the majority of cases.     

On the branching factor of the alpha-beta pruning algorithm

An analysis of the alpha-beta pruning algorithm is presented which takes into account both shallow and deep cut-offs. A formula is first developed to measure the average number of terminal nodes examined by the algorithm in a uniform tree of degree n and depth d when ties are allowed among the bottom positions: specifically, all bottom values are assumed to be independent identically distributed random variables drawn from a discrete probability distribution. A worst case analysis over all possible probability distributions is then presented by considering the limiting case when the discrete probability distribution tends to a continuous probability distribution. The branching factor of the alpha-beta pruning algorithm is shown to grow with n as Θ(n/lnn), therefore confirming a claim by Knuth and Moore that deep cut-offs only have a second order effect on the behavior of the algorithm.     

Robust nonlinear task space control for 6 DOF parallel manipulator

This paper presents a robust nonlinear controller equipped with a friction estimator for a 6 degree of freedom (DOF) parallel manipulator in the task space coordinates. The requisite 6 DOF system states for the task space control are acquired by an alpha-beta tracker and a numerical forward kinematic solution. The Friedland-Park friction observer in the joint space coordinates provides friction estimates that help to improve the control performance. Finally, the RNTC turns out to outper form a nonlinear task space control and a popularly adopted PID control with the friction estimator in the joint space coordinates.     

Design, analysis, and implementation of a parallel tree search algorithm

The alpha-beta algorithm for searching decision trees is adapted to allow parallel activity in different parts of a tree during the search. The algorithm has been implemented in a procedural simulation language (GASP IV). The simulation environment provides the illusion of multiple software processes and multiple hardware processors. A number of preliminary experiments have been done to gather statistics on run time, nodes scored, and nodes visited. Results indicate that a substantial reduction in time of search occurs because of the use of parallelism. An analytic expression for the storage requirements of the algorithm is derived. The analysis provides an example of the classical tradeoff between time and storage.     

Parallel game-tree search

The design issues affecting a parallel implementation of the alpha-beta search algorithm are discussed with emphasis on a tree decomposition scheme that is intended for use on well ordered trees. In particular, the principal variation splitting method has been implemented, and experimental results are presented which show how such refinements as progressive deepening, narrow window searching, and the use of memory tables affect the performance of multiprocessor based chess playing programs. When dealing with parallel processing systems, communication delays are perhaps the greatest source of lost time. Therefore, an implementation of our tree decomposition based algorithm is presented, one that operates with a modest amount of message passing within a network of processors. Since our system has low search overhead, the principal basis for comparison is the communication overhead, which in turn is shown to have two components.     

Low overhead alternatives to SSS1

Of the many minimax algorithms, sss1 is noteworthy because it usually searches the smallest game trees. Its success can be attributed to the accumulation and use of information acquired while traversing the tree. The main disadvantages of sss1 are its high storage needs and management costs. This paper describes a class of methods, based on the popular alpha-beta algorithm, that acquire and use information to guide a tree search. They retain a given search direction and yet are as good as sss1, even while searching random trees. Further, although some of these new algorithms also require substantial storage, they are more flexible and can be programmed to use only the space available, at the cost of some degradation in performance.     

A comparison of two paradigms for distributed shared memory

Two paradigms for distributed shared memory on loosely-coupled computing systems are compared: the shared data-object model as used in Orca, a programming language specially designed for loosely-coupled computing systems, and the shared virtual memory model. For both paradigms two systems are described, one using only point-to-point messages, the other using broadcasting as well. The two paradigms and their implementations are described briefly. Their performances are compared on four applications: the travelling-salesman problem, alpha-beta search, matrix multiplication and the all-pairs shortest-paths problem. Measurements were obtained on a system consisting of 10 MC68020 processors connected by an Ethernet. For comparison purposes, the applications have also been run on a system with physical shared memory. In addition, the paper gives measurements for the first two applications above when remote procedure call is used as the communication mechanism. The measurements show that both paradigms can be used efficiently for programming large-grain parallel applications, with significant speed-ups. The structured shared data-object model achieves the highest speed-ups and is easiest to program and to debug.     

博弈树置换表启发式算法研究

博弈树搜索对于计算机博弈至关重要。优秀的搜索算法通过搜索较少的节点就可以获得最佳路径,从而提高计算机的博弈水平。论文以中国象棋计算机博弈作为背景,在alpha-beta基本搜索算法上,详细阐述了置换表启发算法的原理和哈希冲突,引进了双层置换表的概念及其替换策略,增强了引擎的搜索效率。实验结果表明了该算法的有效性。     

Evolutionary-based heuristic generators for checkers and give-away checkers

Abstract: Two methods of genetic evolution of linear and non-linear heuristic evaluation functions for the game of checkers and give-away checkers are presented in the paper. The first method is based on the simplistic assumption that a relation 'close' to partial order can be defined over the set of evaluation functions. Hence an explicit fitness function is not necessary in this case and direct comparison between heuristics (a tournament) can be used instead. In the other approach a heuristic is developed step-by-step based on the set of training games. First, the end-game positions are considered and then the method gradually moves 'backwards' in the game tree up to the starting position and at each step the best fitted specimen from the previous step (previous game tree depth) is used as the heuristic evaluation function in the alpha-beta search for the current step. Experimental results confirm that both approaches lead to quite strong heuristics and give hope that a more sophisticated and more problem-oriented evolutionary process might ultimately provide heuristics of quality comparable to those of commercial programs.     

Heuristic search in PARLOG using replicated worker style parallelism

Most concurrent logic programming languages hide the distribution of processes among physical processors from the programmer. For parallel applications based on heuristic search, however, it is important for the programmer to accurately control this distribution. With such applications, an inferior distribution strategy easily leads to enormous search overheads, thus decreasing speedup on parallel hardware.

To solve this problem, various language extensions for concurrent logic languages have been proposed, such as mapping notations and priorities. We present an alternative approach that does not require any new language features. Our solution is to use the replicated workers paradigm in a concurrent logic language (PARLOG). This paradigm has thus far mainly been used in parallel procedural languages, such as Linda and Orca. We show that it is just as useful for logic languages. We have implemented two parallel applications, the Traveling Salesman Problem and alpha-beta search, using this approach. Also, we have done some performance measurements of these programs on a multiprocessor. These experiments show that significant speedups can be obtained in this way.     

Quantitative deduction and its fixpoint theory

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.