Title: "Simple stochastic parity games"
Authors: Krishnendu Chatterjee, Marcin Jurdzi{\'n}ski, and
Thomas A. Henzinger
Many verification, planning, and control problems can be modeled as
games played on state-transition graphs by one or two players whose
conflicting goals are to form a path in the graph.
The focus here is on simple stochastic parity games, that is,
two-player games with turn-based probabilistic transitions and
$\omega$-regular objectives formalized as parity (Rabin chain)
winning conditions.
An efficient translation from simple stochastic parity games to
nonstochastic parity games is given.
As many algorithms are known for solving the latter, the translation
yields efficient algorithms for computing the states of a simple
stochastic parity game from which a player can win with
probability~1.
An important special case of simple stochastic parity games are the
Markov decision processes with B\"uchi objectives.
For this special case a first provably subquadratic algorithm is
given for computing the states from which the single player has
a strategy to achieve a B\"uchi objective with probability~1.
For game graphs with $m$ edges the algorithm works in time
$O(m \sqrt{m})$.
Interestingly, a similar technique sheds light on the
question of the computational complexity of solving simple B\"uchi
games and yields the first provably subquadratic algorithm, with a
running time of $O(n^2/ \log n)$ for game graphs with $n$ vertices
and $O(n)$ edges.