Probabilistic Bisimulations for PCTL Model Checking of Interval MDPs

Verification of PCTL properties of MDPs with convex uncertainties has been investigated recently by Puggelli et al. However, model checking algorithms typically suffer from state space explosion. In this paper, we address probabilistic bisimulation to reduce the size of such an MDPs while preserving PCTL properties it satisfies. We discuss different interpretations of uncertainty in the models which are studied in the literature and that result in two different definitions of bisimulations. We give algorithms to compute the quotients of these bisimulations in time polynomial in the size of the model and exponential in the uncertain branching. Finally, we show by a case study that large models in practice can have small branching and that a substantial state space reduction can be achieved by our approach.Comment: In Proceedings SynCoP 2014, arXiv:1403.784

Probabilistic Opacity in Refinement-Based Modeling

Given a probabilistic transition system (PTS) $\cal A$ partially observed by an attacker, and an $\omega$-regular predicate $\varphi$over the traces of $\cal A$, measuring the disclosure of the secret $\varphi$ in $\cal A$ means computing the probability that an attacker who observes a run of $\cal A$ can ascertain that its trace belongs to $\varphi$. In the context of refinement, we consider specifications given as Interval-valued Discrete Time Markov Chains (IDTMCs), which are underspecified Markov chains where probabilities on edges are only required to belong to intervals. Scheduling an IDTMC $\cal S$ produces a concrete implementation as a PTS and we define the worst case disclosure of secret $\varphi$ in ${\cal S}$ as the maximal disclosure of $\varphi$ over all PTSs thus produced. We compute this value for a subclass of IDTMCs and we prove that refinement can only improve the opacity of implementations

An inverse of the evaluation functional for typed Lambda-calculus

In any model of typed λ-calculus conianing some basic arithmetic, a functional p - * (procedure—* expression) will be defined which inverts the evaluation functional for typed X-terms, Combined with the evaluation functional, p-e yields an efficient normalization algorithm. The method is extended to X-calculi with constants and is used to normalize (the X-representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for βη-reduction, generalizing results of Friedman [1] and Statman [31: If two Xterms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are provably equal in the βη- calculus

Reachability in Parametric Interval Markov Chains using Constraints

Parametric Interval Markov Chains (pIMCs) are a specification formalism that extend Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into account imprecision in the transition probability values: transitions in pIMCs are labeled with parametric intervals of probabilities. In this work, we study the difference between pIMCs and other Markov Chain abstractions models and investigate the two usual semantics for IMCs: once-and-for-all and at-every-step. In particular, we prove that both semantics agree on the maximal/minimal reachability probabilities of a given IMC. We then investigate solutions to several parameter synthesis problems in the context of pIMCs -- consistency, qualitative reachability and quantitative reachability -- that rely on constraint encodings. Finally, we propose a prototype implementation of our constraint encodings with promising results

Computing Probabilistic Bisimilarity Distances via Policy Iteration

A transformation mapping a labelled Markov chain to a simple stochastic game is presented. In the resulting simple stochastic game, each vertex corresponds to a pair of states of the labelled Markov chain. The value of a vertex of the simple stochastic game is shown to be equal to the probabilistic bisimilarity distance, a notion due to Desharnais, Gupta, Jagadeesan and Panangaden, of the corresponding pair of states of the labelled Markov chain. Bacci, Bacci, Larsen and Mardare introduced an algorithm to compute the probabilistic bisimilarity distances for a labelled Markov chain. A modification of a basic version of their algorithm for a labelled Markov chain is shown to be the policy iteration algorithm applied to the corresponding simple stochastic game. Furthermore, it is shown that this algorithm takes exponential time in the worst case

Multi-objective Robust Strategy Synthesis for Interval Markov Decision Processes

Interval Markov decision processes (IMDPs) generalise classical MDPs by having interval-valued transition probabilities. They provide a powerful modelling tool for probabilistic systems with an additional variation or uncertainty that prevents the knowledge of the exact transition probabilities. In this paper, we consider the problem of multi-objective robust strategy synthesis for interval MDPs, where the aim is to find a robust strategy that guarantees the satisfaction of multiple properties at the same time in face of the transition probability uncertainty. We first show that this problem is PSPACE-hard. Then, we provide a value iteration-based decision algorithm to approximate the Pareto set of achievable points. We finally demonstrate the practical effectiveness of our proposed approaches by applying them on several case studies using a prototypical tool.Comment: This article is a full version of a paper accepted to the Conference on Quantitative Evaluation of SysTems (QEST) 201

Characterising Testing Preorders for Finite Probabilistic Processes

In 1992 Wang & Larsen extended the may- and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to find complete axiomatisations and alternative characterisations for these preorders. This paper solves both problems for finite processes with silent moves. It characterises the may preorder in terms of simulation, and the must preorder in terms of failure simulation. It also gives a characterisation of both preorders using a modal logic. Finally it axiomatises both preorders over a probabilistic version of CSP.Comment: 33 page