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Computational and Algebraic Structure of Board Games

Chun-Kai Hwang, John Reuben Gilbert, Tsung-Ren Huang, Chen-An Tsai, Yen-Jen Oyang

TL;DR

This work investigates the algebraic and computational structure of board games, focusing on Xi Gua Qi and two-player games like Go and chess. It develops a graph-based, matrix-operator framework to model game states and movements, and introduces a PBCR approach to extract predictive rules from neural networks. The paper proves that two-player movement operators inhabit non-ring, non-group subsets of matrix spaces, with transition matrices having entries from $\{-1,0,1\}$ or rational numbers, and derives a closed-form transition representation for such games. It demonstrates strong predictive performance for neural models and discusses future links to quantum game theory and spin networks for a holistic analysis of game complexity.

Abstract

We provide two methodologies in the area of computation theory to solve optimal strategies for board games such as Xi Gua Qi and Go. From experimental results, we find relevance to graph theory, matrix representation, and mathematical consciousness. We prove that the decision strategy of movement for Xi Gua Qi and Chinese checker games belongs to a subset that is neither a ring nor a group over set Y={-1,0,1}. Additionally, the movement for any board game with two players belongs to a subset that is neither a ring nor a group from the razor of Occam. We derive the closed form of the transition matrix for any board game with two players such as chess and Chinese chess. We discover that the element of the transition matrix belongs to a rational number. We propose a different methodology based on algebra theory to analyze the complexity of board games in their entirety, instead of being limited solely to endgame results. It is probable that similar decision processes of people may also belong to a matrix representation that is neither a ring nor a group.

Computational and Algebraic Structure of Board Games

TL;DR

This work investigates the algebraic and computational structure of board games, focusing on Xi Gua Qi and two-player games like Go and chess. It develops a graph-based, matrix-operator framework to model game states and movements, and introduces a PBCR approach to extract predictive rules from neural networks. The paper proves that two-player movement operators inhabit non-ring, non-group subsets of matrix spaces, with transition matrices having entries from or rational numbers, and derives a closed-form transition representation for such games. It demonstrates strong predictive performance for neural models and discusses future links to quantum game theory and spin networks for a holistic analysis of game complexity.

Abstract

We provide two methodologies in the area of computation theory to solve optimal strategies for board games such as Xi Gua Qi and Go. From experimental results, we find relevance to graph theory, matrix representation, and mathematical consciousness. We prove that the decision strategy of movement for Xi Gua Qi and Chinese checker games belongs to a subset that is neither a ring nor a group over set Y={-1,0,1}. Additionally, the movement for any board game with two players belongs to a subset that is neither a ring nor a group from the razor of Occam. We derive the closed form of the transition matrix for any board game with two players such as chess and Chinese chess. We discover that the element of the transition matrix belongs to a rational number. We propose a different methodology based on algebra theory to analyze the complexity of board games in their entirety, instead of being limited solely to endgame results. It is probable that similar decision processes of people may also belong to a matrix representation that is neither a ring nor a group.

Paper Structure

This paper contains 8 sections, 3 theorems, 2 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Simulation: Xi Gua Qi
  4. Method 1: Game Tree
  5. Method 2: Probabilistic Neural Network Rule Models
  6. Results
  7. Discussion
  8. Conclusions and Future Works

Key Result

Theorem 1

If we view the snapshot of the board game as a vertex of a graph, the edge of this graph represents a piece movement, and the edge is an operator (a matrix) or a function mapping f, $f:A \mapsto A$, where $A=X^{21}, X=\{1,2,3\}$, 1, and 2 are the different colors of the chess pieces, and 3 represent

Figures (3)

  • Figure 1: The initial playing state of Xi Gua Qi
  • Figure 2: The Graph of the board during playing state
  • Figure 3: The mutable tree for the game pieces of red color blocked by the opponent’s yellow color pieces that correspond to Figure \ref{['fig2']}

Theorems & Definitions (6)

  • Theorem 1: Neither A Ring nor A Group
  • proof
  • Theorem 2: Neither A Ring nor A Group for Chinese Checker Game
  • proof
  • Theorem 3: Neither A Ring nor A Group for Any Board Game with Two Players such as Chess and Chinese Chess
  • proof