2 research outputs found
On the equivalence of game and denotational semantics for the probabilistic mu-calculus
The probabilistic (or quantitative) modal mu-calculus is a fixed-point logic
de- signed for expressing properties of probabilistic labeled transition
systems (PLTS). Two semantics have been studied for this logic, both assigning
to every process state a value in the interval [0,1] representing the
probability that the property expressed by the formula holds at the state. One
semantics is denotational and the other is a game semantics, specified in terms
of two-player stochastic games. The two semantics have been proved to coincide
on all finite PLTS's, but the equivalence of the two semantics on arbitrary
models has been open in literature. In this paper we prove that the equivalence
indeed holds for arbitrary infinite models, and thus our result strengthens the
fruitful connection between denotational and game semantics. Our proof adapts
the unraveling or unfolding method, a general proof technique for proving
result of parity games by induction on their complexity
The Complexity of Nash Equilibria in Stochastic Multiplayer Games
We analyse the computational complexity of finding Nash equilibria in
turn-based stochastic multiplayer games with omega-regular objectives. We show
that restricting the search space to equilibria whose payoffs fall into a
certain interval may lead to undecidability. In particular, we prove that the
following problem is undecidable: Given a game G, does there exist a Nash
equilibrium of G where Player 0 wins with probability 1? Moreover, this problem
remains undecidable when restricted to pure strategies or (pure) strategies
with finite memory. One way to obtain a decidable variant of the problem is to
restrict the strategies to be positional or stationary. For the complexity of
these two problems, we obtain a common lower bound of NP and upper bounds of NP
and PSPACE respectively. Finally, we single out a special case of the general
problem that, in many cases, admits an efficient solution. In particular, we
prove that deciding the existence of an equilibrium in which each player either
wins or loses with probability 1 can be done in polynomial time for games where
the objective of each player is given by a parity condition with a bounded
number of priorities