Modeling the Game of Go by Ising Hamiltonian, Deep Belief Networks and Common Fate Graphs

Abstract

Three different models of the game of Go are developed by establishing an analogy between this game and physical systems susceptible to analysis under the well-known Ising model in two dimensions. The Ising Hamiltonian is adapted to measure the ‘energy’ of the Go boards generated by the interaction of the game pieces (stones) as players make their moves in an attempt to control the board or to capture rival stones. The proposed models are increasingly complex. The first or Atomic-Go model, consists of the straightforward measurement of local energy employing the adapted Ising Hamiltonian. The second or Generative Atomic-Go model, employs a Deep Belief Network (a generative graphical model popular in machine learning) to generate board configurations and compensate for the lack of information in mostly-empty boards. The third or Molecular-Go model, incorporates Common Fate Graphs, which are an alternative representation of the Go board that offers advantages in pattern analysis. Simulated games between different Go playing systems were used to test whether the models are able to capture the energy changes produced by moves between players of different skills. The results indicate that the latter two models reflect said energy differences correctly. These positive results encourage further development of analysis tools based on the techniques discussed.