Game Development Reference
In-Depth Information
Section 2.5 we give an outlook beyond finite automata, and in Section 2.6
we conclude.
This chapter is written as a gentle introduction to the subject of infinite
games and automata theory. Many concepts, techniques, and proofs are
explained using examples without being formally rigorous. For the interested
reader who wants to learn more about the subject we give some pointers to
the literature in the conclusion.
2.2 Basic notations and definitions
For a set X we denote by X the set of finite sequences and by X ω
the set
X ω
of infinite sequences over X .For α
let
Inf( α )= {x ∈ X | x occurs infinitely often in α}.
We can view an infinite sequence α ∈ X ω as a mapping α : N → X .
Consequently, we write α ( i ) to denote the element at position i in α , i.e.,
α = α (0) α (1) α (2)
···
Infinite sequences are also called infinite words or ω -words.
A game graph (also called arena ) is a tuple G =( V E ,V A ,E,c ) with
V E : vertices of Eva (player 1, circle),
V A : vertices of Adam (player 2, box),
• E ⊆ V × V : edges with V = V E ∪ V A ,
• c : V → C with a finite set of colours C .
The sets V E and V A are disjoint (each vertex belongs to exactly one of the
players). We are interested in plays of infinite duration played on such graphs.
Therefore, we assume that each vertex has at least one successor. Most of
the time we consider games on finite arenas but in Section 2.4 we also use
games on infinite graphs.
Example 2.1
Figure 2.1 shows a simple game graph with V E =
{
x 1 ,x 2
}
and V A =
. We assume that C = V and that the colouring is the
identity function. Whenever we consider game graphs without specifying the
colouring, then we implicitly identify the vertices with the colours.
{
y 1 ,y 2
}
A play in G is an infinite sequence α = v 0 v 1 v 2 ···
of vertices such that
( v i ,v i +1 )
E for all i
0. By c ( α ) we denote the corresponding sequence of
colours c ( v 0 ) c ( v 1 ) c ( v 2 )
···
A game is a pair of a game graph and a winning condition G
=( G, Win )
C ω . Eva wins a play α if c ( α )
with Win
Win . Otherwise Adam wins.
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