481 research outputs found
Up-To Techniques for Generalized Bisimulation Metrics
Bisimulation metrics allow us to compute distances between the behaviors of probabilistic systems. In this paper we present enhancements of the proof method based on bisimulation metrics, by extending the theory of up-to techniques to (pre)metrics on discrete probabilistic concurrent processes.
Up-to techniques have proved to be a powerful proof method for showing that two systems are bisimilar, since they make it possible to build (and thereby check) smaller relations in bisimulation proofs. We define soundness conditions for up-to techniques on metrics, and study compatibility properties that allow us to safely compose up-to techniques with each other. As an example, we derive the soundness of the up-to-bisimilarity-metric-and-context technique.
The study is carried out for a generalized version of the bisimulation metrics, in which the Kantorovich lifting is parametrized with respect to a distance function. The standard bisimulation metrics, as well as metrics aimed at capturing multiplicative properties such as differential privacy, are specific instances of this general definition
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
Process algebra for performance evaluation
This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems – like large-scale computers, client–server architectures, networks – can accurately be described using such stochastic specification formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions
Coalgebraic Behavioral Metrics
We study different behavioral metrics, such as those arising from both
branching and linear-time semantics, in a coalgebraic setting. Given a
coalgebra for a functor , we define a framework for deriving pseudometrics on which
measure the behavioral distance of states.
A crucial step is the lifting of the functor on to a
functor on the category of pseudometric spaces.
We present two different approaches which can be viewed as generalizations of
the Kantorovich and Wasserstein pseudometrics for probability measures. We show
that the pseudometrics provided by the two approaches coincide on several
natural examples, but in general they differ.
If has a final coalgebra, every lifting yields in a
canonical way a behavioral distance which is usually branching-time, i.e., it
generalizes bisimilarity. In order to model linear-time metrics (generalizing
trace equivalences), we show sufficient conditions for lifting distributive
laws and monads. These results enable us to employ the generalized powerset
construction
Approximating a Behavioural Pseudometric without Discount for<br> Probabilistic Systems
Desharnais, Gupta, Jagadeesan and Panangaden introduced a family of
behavioural pseudometrics for probabilistic transition systems. These
pseudometrics are a quantitative analogue of probabilistic bisimilarity.
Distance zero captures probabilistic bisimilarity. Each pseudometric has a
discount factor, a real number in the interval (0, 1]. The smaller the discount
factor, the more the future is discounted. If the discount factor is one, then
the future is not discounted at all. Desharnais et al. showed that the
behavioural distances can be calculated up to any desired degree of accuracy if
the discount factor is smaller than one. In this paper, we show that the
distances can also be approximated if the future is not discounted. A key
ingredient of our algorithm is Tarski's decision procedure for the first order
theory over real closed fields. By exploiting the Kantorovich-Rubinstein
duality theorem we can restrict to the existential fragment for which more
efficient decision procedures exist
Linear and Branching System Metrics
We extend the classical system relations of trace\ud
inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as elements of arbitrary metric spaces.\ud
\ud
Trace inclusion and equivalence give rise to asymmetrical and symmetrical linear distances, while simulation and bisimulation give rise to asymmetrical and symmetrical branching distances. We study the relationships among these distances, and we provide a full logical characterization of the distances in terms of quantitative versions of LTL and μ-calculus. We show that, while trace inclusion (resp. equivalence) coincides with simulation (resp. bisimulation) for deterministic boolean transition systems, linear\ud
and branching distances do not coincide for deterministic metric transition systems. Finally, we provide algorithms for computing the distances over finite systems, together with a matching lower complexity bound
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