3,229 research outputs found
Model Checking Games for the Quantitative mu-Calculus
We investigate quantitative extensions of modal logic and the modal
mu-calculus, and study the question whether the tight connection between logic
and games can be lifted from the qualitative logics to their quantitative
counterparts. It turns out that, if the quantitative mu-calculus is defined in
an appropriate way respecting the duality properties between the logical
operators, then its model checking problem can indeed be characterised by a
quantitative variant of parity games. However, these quantitative games have
quite different properties than their classical counterparts, in particular
they are, in general, not positionally determined. The correspondence between
the logic and the games goes both ways: the value of a formula on a
quantitative transition system coincides with the value of the associated
quantitative game, and conversely, the values of quantitative parity games are
definable in the quantitative mu-calculus
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
Probabilistic modal {\mu}-calculus with independent product
The probabilistic modal {\mu}-calculus is a fixed-point logic designed for
expressing properties of probabilistic labeled transition systems (PLTS's). Two
equivalent semantics have been studied for this logic, both assigning to each
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 parity games. A shortcoming of the probabilistic modal
{\mu}-calculus is the lack of expressiveness required to encode other important
temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL).
To address this limitation we extend the logic with a new pair of operators:
independent product and coproduct. The resulting logic, called probabilistic
modal {\mu}-calculus with independent product, can encode many properties of
interest and subsumes the qualitative fragment of PCTL. The main contribution
of this paper is the definition of an appropriate game semantics for this
extended probabilistic {\mu}-calculus. This relies on the definition of a new
class of games which generalize standard two-player stochastic (parity) games
by allowing a play to be split into concurrent subplays, each continuing their
evolution independently. Our main technical result is the equivalence of the
two semantics. The proof is carried out in ZFC set theory extended with
Martin's Axiom at an uncountable cardinal
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