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Despite the variety of games treated in general game playing, all games share a common abstract structure.

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Each game takes place in an environment with finitely many states.

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The one distinguished initial state and one or more terminal states.

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In addition each game has a fixed finite number of players.

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Each player has finitely many possible actions in any given game state

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and each state has an associated goal value for each player.

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The dynamic model for general games is synchronous update:

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All players move on all steps, although some moves could be no-ops.

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And the environment updates only in response to the moves taken by the players.

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Given this common structure we can think of a game as a state graph like the one shown here.

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In this case you have a game with one player with 8 states names s1 through s8.

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There's one initial state, namely s1, and two terminal states, that's s4 and s8.

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The numbers associated with each state indicate the values of those states

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and the arcs on the graph capture the transition function for the game.

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For example, if the game is in state s1 and the player does action a the game will move to state s2.

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if the player does action b the game will move to state s5, and so forth.

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In the case of multiple players with simultaneous moves, the arcs become multi-arcs with one arc for each combination of players actions.

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Here's an example of a simultaneous move game with 2 players.

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it its state is s1 and both players perform action a, we follow the arc labeled "a / a".

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If the first player does action b and the second player does action a we follow the arc "b / a".

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We also have different goals for the different players: for example in state s4 player 1 gets 100 points while player 2 gets 0 points

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and in state s8 the situation is reversed.

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State graph captures the essential information about a game, and since all games that we're considering are finite it's possible at least in principle to describe such games in the form of state graphs.

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Unfortunately such explicit representation are not practical in all cases: even though the number of states and actions a finite, these sets can be extremly large and of course the corresponding graphs can be larger still.

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For example in chess there are thousands of possible moves and more than 10^30 states.

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Fortunately we can solve this problem exploiting regularities in the game to produce compact encodings.

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In practice we rarely think of states as monolithic entities and more frequently characterize states in terms of propositions that are true in those states.

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Similarly we often think of actions as having some structure as well.

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This leads to a notion of a structured state machine like the one shown here.

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The overall structure is the same as in the simple state machine shown earlier but in this case we've revealed the substructure of states.

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Now by itself this does not help us since the graph is the same size as before.

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However actions frequently effect only some of the propositions that are true in states and leave others unchanged.

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By exploiting these limitations on the effects of actions we can encode state graphs compactly by writing logical rules in place of explicit graphs.

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GDL or Game Description Language is a formal language for encoding games in this way.

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As we'll see it's based on symbolic logic.

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It's widely used in the litterature and is used in virtually all general game playing competitions.

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Moreover its form is the basis for some more expressive variance that have significant value in real-world applications such as enterprise management, computational law.

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Here's an example of GDL, in this case the rules for the game of tic-tac-toe.

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We discuss the specifics of GDL in the next lesson, for now the details are unimportant.

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The one thing to note is that this one page of rules fully describes a game with thousands of states.

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It's a significant savings over the state graph.

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The improvement in more complex games can be even more dramatic, for example it's possible to describe the rules of chess with many, many more states in just about 4 pages of rules like these.