Game Development Reference
In-Depth Information
For infinite-state games, the reduction to ω -regular payoffs described above
can be problematic also because optimal maximising/minimising strategies
in infinite-state games with ω -regular payoffs (even reachability payoffs)
do not necessarily exist. For example, even if we somehow check that the
value of ( v, q 0 )in G
×
M is 1, this does yet mean that player
has a
( P =1 M )-winning strategy in v .
For finite-state games, the two problems discussed above usually disappear.
However, there are still some issues related to complexity. In particular,
the results about the type of optimal strategies in G
M do not carry over
to G . For example, assume that we are given a linear-time objective P =1 M
where M is a deterministic Rabin-chain automaton. If G has finitely many
states, then G×M is also finite-state and hence we can rely on the results
presented by McIver and Morgan [2002] and Chatterjee et al. [2004b] and
conclude that the value of ( v, q 0 ) is computable in time polynomial in
the size of G
×
M and there is an optimal maximising MD strategy σ
computable in polynomial time. From this we can deduce that the existence
of a ( P =1 M )-winning strategy for player
×
in v is decidable in polynomial
time. However, since the optimal MD strategy σ may depend both on the
current vertex of G and the current state of M , we cannot conclude that
if player
has some ( P =1 M )-winning strategy in v , then he also has an
MD ( P =1 M )-winning strategy in v (still, the strategy σ can be translated
into a FD ( P =1 M )-winning strategy which simulates the execution of M
on the history of a play).
To sum up, linear-time objectives are closely related to ω -regular payoffs,
but the associated problems cannot be seen as 'equivalent' in general.
Branching-time logics
Branching-time logics such as CTL, CTL , or ECTL (see, e.g., Emerson
[1991]) allow explicit existential/universal quantification over runs. Thus,
one can express that a given path formula holds for some/all runs initiated
in a given state.
In the probabilistic setting, the existential/universal path quantifiers are
replaced with the probabilistic operator P introduced in the previous
section. In this way, every (non-probabilistic) branching-time logic determines
its probabilistic counterpart. The probabilistic variants of CTL, CTL , and
ECTL are denoted by PCTL, PCTL , and PECTL , respectively (see
Hansson and Jonsson [1994]).
The syntax of PCTL path and state formulae is defined by the following
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