Cayley Graph

In part 1 of this series, the Tic-tac-toe reinforcement learning task was expressed as a Combinatorial Group with the hypothesis that the expansion of the group into a Cayley Graph could be used to learn its associated game tree. In this instalment, the expansion of the group into a Caley Graph will be examined in a bit more detail. Initially, the Tic-tac-toe group will be set aside in favour of a simpler domain which will offer a more compact and pedagogical representation. However, the expansion of the Tic-tac-toe group should follow the same process, this article will circle back to the Tic-tac-toe domain to highlight the equivalences which should ensure that this is so.

Tic-tac-toe, (or noughts and crosses or Xs and Ox), is a turn-based game for two players who alternately tag the spaces of a $3 \times 3$ grid with their respective marker: an X or an O. The object of the game is to place three markers in a row, either horizontally, vertically, or diagonally. Given only the mechanics of Tic-tac-toe, the game can be expressed as Combinatorial Group by defining a set $A$ of generators $\{a_i\}$ which describe the actions that can be taken by either player. The Cayley Graph of this group can be constructed which will express all the possible ways the game can be played. Using the Cayley Graph as a model, it should be possible to learn the Tic-tac-toe game tree using dynamic programming techniques (hint: the game tree is a sub-graph of the Cayley Graph).