10,785 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
Model Checking the Quantitative mu-Calculus on Linear Hybrid Systems
We study the model-checking problem for a quantitative extension of the modal
mu-calculus on a class of hybrid systems. Qualitative model checking has been
proved decidable and implemented for several classes of systems, but this is
not the case for quantitative questions that arise naturally in this context.
Recently, quantitative formalisms that subsume classical temporal logics and
allow the measurement of interesting quantitative phenomena were introduced. We
show how a powerful quantitative logic, the quantitative mu-calculus, can be
model checked with arbitrary precision on initialised linear hybrid systems. To
this end, we develop new techniques for the discretisation of continuous state
spaces based on a special class of strategies in model-checking games and
present a reduction to a class of counter parity games.Comment: LMCS submissio
Lukasiewicz mu-Calculus
We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations
Algorithms for Game Metrics
Simulation and bisimulation metrics for stochastic systems provide a
quantitative generalization of the classical simulation and bisimulation
relations. These metrics capture the similarity of states with respect to
quantitative specifications written in the quantitative {\mu}-calculus and
related probabilistic logics. We first show that the metrics provide a bound
for the difference in long-run average and discounted average behavior across
states, indicating that the metrics can be used both in system verification,
and in performance evaluation. For turn-based games and MDPs, we provide a
polynomial-time algorithm for the computation of the one-step metric distance
between states. The algorithm is based on linear programming; it improves on
the previous known exponential-time algorithm based on a reduction to the
theory of reals. We then present PSPACE algorithms for both the decision
problem and the problem of approximating the metric distance between two
states, matching the best known algorithms for Markov chains. For the
bisimulation kernel of the metric our algorithm works in time O(n^4) for both
turn-based games and MDPs; improving the previously best known O(n^9\cdot
log(n)) time algorithm for MDPs. For a concurrent game G, we show that
computing the exact distance between states is at least as hard as computing
the value of concurrent reachability games and the square-root-sum problem in
computational geometry. We show that checking whether the metric distance is
bounded by a rational r, can be done via a reduction to the theory of real
closed fields, involving a formula with three quantifier alternations, yielding
O(|G|^O(|G|^5)) time complexity, improving the previously known reduction,
which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated
to approximate the metrics using binary search.Comment: 27 pages. Full version of the paper accepted at FSTTCS 200
Fixpoint Games on Continuous Lattices
Many analysis and verifications tasks, such as static program analyses and
model-checking for temporal logics reduce to the solution of systems of
equations over suitable lattices. Inspired by recent work on lattice-theoretic
progress measures, we develop a game-theoretical approach to the solution of
systems of monotone equations over lattices, where for each single equation
either the least or greatest solution is taken. A simple parity game, referred
to as fixpoint game, is defined that provides a correct and complete
characterisation of the solution of equation systems over continuous lattices,
a quite general class of lattices widely used in semantics. For powerset
lattices the fixpoint game is intimately connected with classical parity games
for -calculus model-checking, whose solution can exploit as a key tool
Jurdzi\'nski's small progress measures. We show how the notion of progress
measure can be naturally generalised to fixpoint games over continuous lattices
and we prove the existence of small progress measures. Our results lead to a
constructive formulation of progress measures as (least) fixpoints. We refine
this characterisation by introducing the notion of selection that allows one to
constrain the plays in the parity game, enabling an effective (and possibly
efficient) solution of the game, and thus of the associated verification
problem. We also propose a logic for specifying the moves of the existential
player that can be used to systematically derive simplified equations for
efficiently computing progress measures. We discuss potential applications to
the model-checking of latticed -calculi and to the solution of fixpoint
equations systems over the reals
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