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

Wu and Huang (Advances in Computer Games, pp. 180-194, 2006) presented a new family of k-in-a-row games, among which Connect6 (a kind of six-in-a-row) attracted much attention. For Connect6 as well as the family of k -in-a-row games, this paper proposes a new threat-based proof search method, named relevance-zone-oriented proof (RZOP) search, developed from the lambda search proposed by Thomsen (Int. Comput. Games Assoc. J., vol. 23, no. 4, pp. 203-217, 2000). The proposed RZOP search is a novel, general, and elegant method of constructing and promoting relevance zones. Using this method together with a proof number search, this paper solved effectively and successfully many new Connect6 game positions, including several Connect6 openings, especially the Mickey Mouse opening, which used to be one of the popular openings before we solved it.

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Relevance-Zone-Oriented Proof Search for Connect6

Recently, Monte Carlo tree search (MCTS) has become a well-known game search method, and has been successfully applied to many games. This method performs well in solving search trees with numerous branches, such as Go, Havannah, etc. Connect6 is a game involving a search tree with numerous branches, and it is also one of the sudden-death games. This paper thus proposes a new MCTS variant related to Connect6, called two-stage MCTS. The first stage focuses on threat space search (TSS), which is designed to solve the sudden-death problem. For the double-threat TSS in Connect6, this study proposes an algorithm called iterative threat space search (ITSS) which combines normal TSS with conservative threat space search (CTSS). The second stage uses MCTS to estimate the game-theoretic value of the initial position. This stage aims at finding the most promising move. The experimental result shows that two-stage MCTS is considerably more efficient than traditional MCTS on those positions with TSS solution in Connect6. Furthermore, according to Connect6 heuristic knowledge, this paper uses relevance-zone search to accelerate identifying winning and losing moves.

Two-Stage Monte Carlo Tree Search for Connect6

This paper proposes a new approach for proof number (PN) search, named job-level PN (JL-PN) search, where each search tree node is evaluated or expanded by a heavy-weight job, which takes normally over tens of seconds. Such JL-PN search is well suited for parallel processing, since these jobs are allowed to be performed by remote processors independently. This paper applies JL-PN search to solving automatically several Connect6 positions including openings on desktop grids. For some of these openings, so far no human expert had been able to find a winning strategy. Our experiments also show that the speedups for solving the test positions are roughly linear, fluctuating from sublinear to superlinear. Hence, JL-PN search appears to be a quite promising approach to solving games.

Job-level proof-number search for connect6

This paper introduces an approach, called generic job-level search, to leverage the game-playing programs which are already written and encapsulated as jobs. Such an approach is well suited to a distributed computing environment, since these jobs are allowed to be run by remote processors independently. In this paper, we present and focus on a job-level proof number search (JL-PNS), a kind of generic job-level search for solving computer game search problems, and apply JL-PNS to solving automatically several Connect6 positions, including some difficult openings. This paper also proposes a method of postponed sibling generation to generate nodes smoothly, and some policies, such as virtual win, virtual loss, virtual equivalence, flagging, or hybrids of the above, to expand the nodes. Our experiment compared these policies, and the results showed that the virtual-equivalence policy, together with flagging, performed the best against other policies. In addition, the results also showed that the speedups for solving these positions are 8.58 on average on 16 cores.

https://ir.nctu.edu.tw:443/bitstream/11536/21399/1/000316479500004.pdf

Job-Level Proof Number Search

This paper presents a volunteer-computing-based grid environment or called a desktop grid environment for Connect6 applications The Connect6 application described in this paper is to let professional Connect6 players to develop or solve openings, based on two programs, NCTU6 and Verifier NCTU6 is to make Connect6 moves, written by the team led by Wu [19][21] NCTU6 Verifier (abbr Verifier), modified from NCTU6, is to verify whether one player wins in a given game position, or to generate the defensive moves if not winning in the position Since both NCTU6 and Verifier consume huge amount of computation resources and requires on-demand responses, we design a desktop grid environment that provides players with on-demand computing through dynamic resource provisioning The underlying desktop grid achieves high throughput computing by harvesting the idle CPU times on desktop computers connected to the Internet

A Volunteer-Computing-Based Grid Environment for Connect6 Applications

A nonogram puzzle is played on a rectangular grid of pixels with clues given in the form of row and column constraints. The aim of solving a nonogram puzzle, an NP-complete problem, is to paint all the pixels of the grid in black and white while satisfying these constraints. This paper proposes an efficient approach to solving nonogram puzzles. We propose a fast dynamic programming (DP) method for line solving, whose time complexity in the worst case is O(kl) only, where the grid size is l×l and k is the average number of integers in one constraint, always smaller than l. In contrast, the time complexity for the best line-solving method in the past is O(kl2). We also propose some fully probing (FP) methods to solve more pixels before running backtracking. Our FP methods can solve more pixels than the method proposed by Batenburg and Kosters (before backtracking), while having a time complexity that is smaller than theirs by a factor of O(l). Most importantly, these FP methods provide useful guidance in choosing the next promising pixel to guess during backtracking. The proposed methods are incorporated into a fast nonogram solver, named LalaFrogKK. The program outperformed all the programs collected in webpbn.com, and also won both nonogram tournaments that were held at the 2011 Conference on Technologies and Applications of Artificial Intelligence (TAAI 2011, Taiwan). We expect that the proposed FP methods can also be applied to solving other puzzles efficiently.

https://ir.nctu.edu.tw:443/bitstream/11536/22772/1/000324586300005.pdf

Lung-Ping Chen

Papers

A Volunteer-Computing-Based Grid Environment for Connect6 Applications

An Efficient Approach to Solving Nonograms