Let's look at GDL in the context of a
specific game.
namely Tic-tac-toe.
As fundamental entities, we include white
and black, the roles of the game.
we also include 1, 2, and 3, which are
the.
We will use this indices for rows and
columns of the Tic-tac-toe board.
And we'll use x, o and b, meaning blank,
as
marks that go in the cells of the
Tic-tac-toe board.
we'll use the ternary function constant.
Cell together with a row index and a
column index and a mark to designate the
proposition that the cell in the specified
row
and column contains the specified mark or
blank.
For example the data cell 2, 3, o, says
that there's
an o in the cell in row 2 and column 3.
We use the unary
function constant control to say whose
turn it is to mark a cell.
For example, the proposition control of
white denotes
the proposition that it's white's turn to
play.
In Tic-tac-toe, there are only two types
of actions a player can perform.
It can either mark a cell or it can do
nothing, which
is what a player does when it's not his
turn to play.
The binary relation con, binary function
constant
mark, together with a row and a column,
and designates the action of playing,
placing a mark in row m and column n.
Mark place there depends on who does the
action.
And fi,
going back to the object constants, we
have noop
which refers to the act of doing nothing
at all.
Finally, we have some helper vocabulary.
row, column, diagonal, line, open.
whose purpose will become clear soon.
The state of the game of Tic-tac-toe was
an arbitrary subset of the propositions.
Propositions in a state are assumed to be
true whenever the game
is in that state and all others are
assumed to be false.
For example, we can just describe the
Tic-tac-toe state shown here
on the left with the set of propositions
shown on the right.
cell 1, 1, contains an x, cell 2, 2,
contains an o, cell 3, 3, contains an x.
The other cells are all blank.
And it's black's turn to play.
Using this conceptualization of states, we
can define the game of
Tic-tac-toe with a small set of logical
sentences, as shown here.
The game has thousands of states.
And it can be described by just one page
of rules.
A similar parsimony is possible for other
games.
For example, chess is more than 10 to the
30
[INAUDIBLE]
states.
And yet, it can be described in about four
pages of rules with this search on here.
let's look at each of these groups of
rules in more detail.
we first of, identified the two roles in
the
game, namely white and black, using the
role relation.
Next, we define the propositions in the
game.
Since there are only 29 propositions, we
could
do this by writing out 29 ground atoms,
however.
we can do this more economically by
writing just
two rules, as shown here, together with
some ground atoms.
Now, the first rule says that an
expression
of the form cell of x,y,w is true.
if x is an index, and
y is an index, and w is a filler, that is
we'll
see x over b.
And an index there is 1, 2, or 3.
The second rule says that a player, every
player
[INAUDIBLE],
that there is a proposition of the form
control of
w for each of the two roles in the game.
Now we can do the same for actions and
expression mark of x, y is an action
for w if w is a role and x is an index and
y is an index.
And noop is a possible action for either
of the two players.
Okay, here we characterize the initial
state by writing
all relevant propositions that are true in
the initial state.
In this case all cells are blank and the x
player has control.
Next, we then have to find legality.
Player may mark a cell if that cell is
blank, and it has control.
Otherwise, the only legal action is noop.
Next, we look at the update rules for the
game.
In other words it's physics.
It's dynamics.
First rule here says that a cell is marked
with an
x or an o if the appropriate player marks
that cell.
And the second rule handles the other
player.
If a cell is blank and is not, not marked
on that step then it remains blank.
The cell contains a mark.
It retains that mark on the subsequent
state.
Finally, control alternates on each play.
Before we get to rewards and termination,
here's some supporting concepts.
A row of marks means that there are
[UNKNOWN].
three marks all with the same first
coordinate.
The column and diagonal relations are
defined analgously.
A line is a row of marks of the same type
or column or a diagonal.
Finally a game is open provided there is
some cell containing a blank.
Alright, here we have the definition of
goals.
A white player gets 100 points, since
there's a
line of x marks, and no line of o marks.
If there are no lines of either sort,
white gets 50 points.
There's a line of o marks and no line of x
marks then white gets zero, zero points.
The rewards for black are analogous.
The final is termination.
The game terminates whenever either player
has a line of marks
of the appropriate type or if the board is
not open.
That is, there are no cells containing
blanks.