connect4 solution (1988)

connect4 solution (1988) - A Knowledge-based Approach of...

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A Knowledge-based Approach of Connect-Four The Game is Solved: White Wins Victor Allis Department of Mathematics and Computer Science Vrije Universiteit Amsterdam, The Netherlands Masters Thesis, October 1988 † ABSTRACT A Shannon C-type strategy program, VICTOR, is written for Connect-Four, based on nine strategic rules. Each of these rules is proven to be correct, implying that conclusions made by VICTOR are correct. Using VICTOR, strategic rules where found which can be used by Black to at least draw the game, on each 7 × (2n) board, provided that White does not start at the middle column, as well as on any 6 × (2n) board. In combination with conspiracy-number search, search tables and depth-first search, VICTOR was able to show that White can win on the standard 7 × 6 board. Using a database of approximately half a million positions, VICTOR can play real time against opponents on the 7 × 6 board, always winning with White. † Published in 1988 as Report IR-163 by the Faculty of Mathematics and Computer Science at the Vrije Universiteit Amsterdam, The Netherlands. Also published in 1992 as Report CS 92-04 by the Faculty of General Sciences at the University of Limburg, Maastricht, The Netherlands.

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Contents Introduction 5 1. Introduction into Connect-Four 7 1.1. The Rules of the Game 7 1.2. Nomenclature 8 2. Different Approaches 10 2.1. Complexity of the Game 10 2.2. Knowledge-Based Approach 12 2.3. Correctness of Strategic Rules 14 3. Some Strategic Rules for Connect-Four 16 3.1. Useless Threats 16 3.2. Another Useless Threat 18 3.3. Odd and Even Threats 19 3.4. Tactics 21 4. Control of Zugzwang 25 4.1. Definition of Control of Zugzwang 25 4.2. Control in Initial Position 26 4.3. Control for White 27 4.4. Some Special Control Positions 28 4.5. Benefits of Controlling the Zugzwang 30 5. More Strategic Rules 32 5.1. Baseinverse 32 5.2. Claimeven 33 5.3. Combination of Rules 33 5.4. General Usage of Rules 34 6. Formal Definitions of Rules 36 6.1. Claimeven 36 6.2. Baseinverse 37 6.3. Vertical 38
6.4. Aftereven 39 6.5. Lowinverse 40 6.6. Highinverse 41 6.7. Baseclaim 42 6.8. Before 43 6.9. Specialbefore 45 6.10. Creation of Strategic Rules 46 7. Interaction of Strategic Rules 47 7.1. Dependence of Zugzwang 47 7.2. Zugzwang-Independent Rules 49 7.3. Combinations of Zugzwang-Dependent Rules 49 7.4. Collection of Possible Combinations 50 8. How to apply the Strategic Rules 51 8.1. Strategic Rules for Black 51 8.2. Strategic Rules for White 51 8.3. An Example Threat Combination for White 53 8.4. Threat Combinations in General 55 9. Implementation 58 9.1. Purpose of VICTOR 58 9.2. Evaluation of a Position 58 9.3. Bottleneck of Algorithm 59 9.4. Some Data on the Program 60 10. First Results 61 10.1. Small Boards 61 10.2. The 7 x 4 Board 62 10.3. The Standard 7 x 6 Board 62 11. Automated Analysis 63 11.1. The Oracle 63 11.2. Conspiracy-Number Search 64

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11.3. How to use Conspiracy-Number Search 65 11.4. Performance of Conspiracy-Number Search 65 11.5. Conspiracy Numbers as Oracle 67 11.6. Three-Level Search 67 12. Implementation of Conspiracy-Number Search and Search Tables 68 12.1. Search Tree in Memory during Search 68 12.2. Transpositions 68 12.3. Combination of Conspiracy-Number Search and Depth-First Search 69 12.4. Search Tables
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connect4 solution (1988) - A Knowledge-based Approach of...

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