A Monte-Carlo evaluation consists in estimating a position by averaging the outcome of several random continuations. The method can serve as an evaluation function at the leaves of a min-max tree. This paper presents a new framework to combine tree search with Monte-Carlo evaluation, that does not separate between a min-max phase and a Monte-Carlo phase. Instead of backing-up the min-max value close to the root, and the average value at some depth, a more general backup operator is defined that progressively changes from averaging to minmax as the number of simulations grows. This approach provides a finegrained control of the tree growth, at the level of individual simulations, and allows efficient selectivity. The resulting algorithm was implemented in a 9 × 9 Go-playing program, Crazy Stone, that won the 10th KGS computer-Go tournament.

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Efficient selectivity and backup operators in Monte-Carlo tree search

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Searching for solutions in games and artificial intelligence

Abstract Proof-number search (pn-search) is designed for finding the game-theoretical value in game trees. It is based on ideas derived from conspiracy-number search and its variants, such as applied cn-search and αβ-cn search. While in cn-search the purpose is to continue searching until it is unlikely that the minimax value of the root will change, pn-search aims at proving the true value of the root. Therefore, pn-search does not consider interim minimax values. Pn-search selects the next node to be expanded using two criteria: the potential range of subtree values and the number of nodes which must conspire to prove or disprove that range of potential values. These two criteria enable pn-search to treat efficiently game trees with a non-uniform branching factor. It is shown that in non-uniform trees pn-search outperforms other types of search, such as α-β iterative-deepening search, even when enhanced with transposition tables, move ordering for the full principal variation, etc. Pn-search has been used to establish the game-theoretical values of Connect-Four, Qubic, and Go-Moku. There pn-search was able to find a forced win for the player to move first. The experiments described here are in the domain of Awari, a game which has not yet been solved. The experiments are repeatable for other games with a non-uniform branching factor. This article describes the underlying principles of pn-search, presents an appropriate implementation, and provides an analysis of its strengths and weaknesses.

Proof-Number Search

Proof-number search

We want to express digital communication with a human touch. So we pay attention to the properties of a door because it links one place to another. We think the linking of different places and sharing of places is one of the substantial qualities of network technology. Another reason that we focus on a door is because it bears great meanings. For example, a door is the first spot for a meeting with someone or for kissing someone goodbye. It could be your children, spouse, or your friends. These days, we talk and say hello to friends by Messenger or mobiles. Accustomed to digital devices which neglect time or space, we expect that the devices can carry our emotions to others. However, these have limits in expression. We want to show through the our work "The Door" that network technology should have a little more humanity and love in the near future.

The Door

: In this thesis we investigate two issues relating to heuristic search algorithms. The first and most important issue addressed is the technique used to represent knowledge within a search tree. Previous techniques have used either single values or ranges. We demonstrate that probability distributions, using a modified B*-type search algorithm, can successfully be used as a knowledge representation technique. Furthermore we show that the use of probability distributions is superior to the use of either of the previous techniques. The former conclusion is based on experiments that show that the probability-based algorithm is able to solve a wide variety of tactical chess problems. The latter conclusion is based on both analytical examples as well as experimental results. In analyzing search algorithms that use single-valued or range-based state descriptions, several important problems arise. For each problem we show how it is solved by the use of probability-based state descriptions. Experimentally we show that the probability-based algorithm solves over one-third more problems than the comparable range-based algorithm and expands approximately one-tenth the nodes on problems that both algorithms solve. Keywords: Reports, Military publications, Periodicals.

Searching with probabilities

The B∗ tree search algorithm—New results☆

Communication is a basic problem when using VLSI technology to implement large parallel circuits. Valuable chip area must be used to run wires connecting components on a chip and current packaging technology restricts the amount of communication that can cross chip boundaries. This paper presents a large parallel architecture for generating moves in chess and shows how it can be restructured to reduce communication and permit a straightforward VLSI implementation without any performance loss. The result is a move generator comprising 64 identical custom chips performing at a rate of 500,000 moves per second, performance that is comparable to the best existing move generator.The success of the architecture of a component module like a chess move generator depends not only on its performance but on how well it meshes with the rest of the system. We discuss the requirements of a chess move generator in the context of a chess-playing system and describe how each of these are met by our design. Details of the chip design are presented along with a description of how the move generator is built using identical chips.

The design and implementation of a VLSI chess move generator

: A set of selection rules is presented for guiding the B* search procedure. The selection rules are based on a simple probability model. Using those selection procedures, the B* algorithm will expand approximately one-third as many nodes as a best first search and approximately two-thirds as many nodes as the previous best version of the B* algorithm. The B* algorithm, using the selection procedures, will expand 30% more nodes than optimum, for small trees, and up to 170% for larger trees. The issues of when the B* paradigm is useful is examined with the effect of invalid bounds on the algorithm.

Andrew J. Palay

Papers

Searching with probabilities

The B∗ tree search algorithm—New results☆

The design and implementation of a VLSI chess move generator

An experimental analysis of the B* tree search algorithm