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Let's look at GDL in the context of a
specific game.

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namely Tic-tac-toe.

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As fundamental entities, we include white
and black, the roles of the game.

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we also include 1, 2, and 3, which are
the.

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We will use this indices for rows and
columns of the Tic-tac-toe board.

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And we'll use x, o and b, meaning blank,
as

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marks that go in the cells of the
Tic-tac-toe board.

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we'll use the ternary function constant.

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Cell together with a row index and a
column index and a mark to designate the

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proposition that the cell in the specified
row

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and column contains the specified mark or
blank.

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For example the data cell 2, 3, o, says
that there's

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an o in the cell in row 2 and column 3.

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We use the unary

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function constant control to say whose
turn it is to mark a cell.

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For example, the proposition control of
white denotes

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the proposition that it's white's turn to
play.

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In Tic-tac-toe, there are only two types
of actions a player can perform.

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It can either mark a cell or it can do
nothing, which

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is what a player does when it's not his
turn to play.

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The binary relation con, binary function
constant

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mark, together with a row and a column,

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and designates the action of playing,
placing a mark in row m and column n.

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Mark place there depends on who does the
action.

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And fi,

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going back to the object constants, we
have noop

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which refers to the act of doing nothing
at all.

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Finally, we have some helper vocabulary.

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row, column, diagonal, line, open.
whose purpose will become clear soon.

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The state of the game of Tic-tac-toe was
an arbitrary subset of the propositions.

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Propositions in a state are assumed to be
true whenever the game

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is in that state and all others are
assumed to be false.

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For example, we can just describe the
Tic-tac-toe state shown here

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on the left with the set of propositions
shown on the right.

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cell 1, 1, contains an x, cell 2, 2,
contains an o, cell 3, 3, contains an x.

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The other cells are all blank.

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And it's black's turn to play.

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Using this conceptualization of states, we
can define the game of

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Tic-tac-toe with a small set of logical
sentences, as shown here.

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The game has thousands of states.

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And it can be described by just one page
of rules.

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A similar parsimony is possible for other
games.

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For example, chess is more than 10 to the
30

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[INAUDIBLE]

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states.

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And yet, it can be described in about four
pages of rules with this search on here.

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let's look at each of these groups of
rules in more detail.

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we first of, identified the two roles in
the

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game, namely white and black, using the
role relation.

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Next, we define the propositions in the
game.

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Since there are only 29 propositions, we
could

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do this by writing out 29 ground atoms,
however.

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we can do this more economically by
writing just

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two rules, as shown here, together with
some ground atoms.

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Now, the first rule says that an
expression

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of the form cell of x,y,w is true.

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if x is an index, and

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y is an index, and w is a filler, that is
we'll

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see x over b.
And an index there is 1, 2, or 3.

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The second rule says that a player, every
player

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[INAUDIBLE],

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that there is a proposition of the form
control of

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w for each of the two roles in the game.

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Now we can do the same for actions and
expression mark of x, y is an action

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for w if w is a role and x is an index and
y is an index.

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And noop is a possible action for either
of the two players.

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Okay, here we characterize the initial
state by writing

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all relevant propositions that are true in
the initial state.

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In this case all cells are blank and the x
player has control.

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Next, we then have to find legality.

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Player may mark a cell if that cell is
blank, and it has control.

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Otherwise, the only legal action is noop.

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Next, we look at the update rules for the
game.

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In other words it's physics.

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It's dynamics.

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First rule here says that a cell is marked
with an

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x or an o if the appropriate player marks
that cell.

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And the second rule handles the other
player.

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If a cell is blank and is not, not marked
on that step then it remains blank.

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The cell contains a mark.

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It retains that mark on the subsequent
state.

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Finally, control alternates on each play.

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Before we get to rewards and termination,
here's some supporting concepts.

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A row of marks means that there are

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[UNKNOWN].

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three marks all with the same first
coordinate.

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The column and diagonal relations are
defined analgously.

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A line is a row of marks of the same type
or column or a diagonal.

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Finally a game is open provided there is
some cell containing a blank.

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Alright, here we have the definition of
goals.

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A white player gets 100 points, since
there's a

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line of x marks, and no line of o marks.

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If there are no lines of either sort,
white gets 50 points.

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There's a line of o marks and no line of x
marks then white gets zero, zero points.

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The rewards for black are analogous.

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The final is termination.

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The game terminates whenever either player
has a line of marks

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of the appropriate type or if the board is
not open.

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That is, there are no cells containing
blanks.