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

Accelerating Parametric Probabilistic Verification

We present a novel method for computing reachability probabilities of parametric discrete-time Markov chains whose transition probabilities are fractions of polynomials over a set of parameters. Our algorithm is based on two key ingredients: a graph decomposition into strongly connected subgraphs combined with a novel factorization strategy for polynomials. Experimental evaluations show that these approaches can lead to a speed-up of up to several orders of magnitude in comparison to existing approache

Qualitative Reachability for Open Interval Markov Chains

Interval Markov chains extend classical Markov chains with the possibility to describe transition probabilities using intervals, rather than exact values. While the standard formulation of interval Markov chains features closed intervals, previous work has considered also open interval Markov chains, in which the intervals can also be open or half-open. In this paper we focus on qualitative reachability problems for open interval Markov chains, which consider whether the optimal (maximum or minimum) probability with which a certain set of states can be reached is equal to 0 or 1. We present polynomial-time algorithms for these problems for both of the standard semantics of interval Markov chains. Our methods do not rely on the closure of open intervals, in contrast to previous approaches for open interval Markov chains, and can characterise situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely.Comment: Full version of a paper published at RP 201

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

Parameter Synthesis for Parametric Interval Markov Chains

AELOS_HCERES2020, STR_HCERES2020Interval Markov Chains (IMCs) are the base of a classic probabilistic specification theory introduced by Larsen and Jonsson in 1991. They are also a popular abstraction for probabilistic systems. In this paper we study parameter synthesis for a parametric extension of Interval Markov Chains in which the endpoints of intervals may be replaced with parameters. In particular, we propose constructions for the synthesis of all parameter values ensuring several properties such as consistency and consistent reachability in both the existential and universal settings with respect to implementations. We also discuss how our constructions can be modified in order to synthesise all parameter values ensuring other typical properties

Parameter Synthesis in Markov Models: A Gentle Survey

This paper surveys the analysis of parametric Markov models whose transitions are labelled with functions over a finite set of parameters. These models are symbolic representations of uncountable many concrete probabilistic models, each obtained by instantiating the parameters. We consider various analysis problems for a given logical specification $\varphi$: do all parameter instantiations within a given region of parameter values satisfy $\varphi$?, which instantiations satisfy $\varphi$ and which ones do not?, and how can all such instantiations be characterised, either exactly or approximately? We address theoretical complexity results and describe the main ideas underlying state-of-the-art algorithms that established an impressive leap over the last decade enabling the fully automated analysis of models with millions of states and thousands of parameters

Parametric LTL on Markov Chains

This paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the until-modality is equipped with a bound that contains variables; e.g., $\Diamond_{\le x}\ \varphi$ asserts that $\varphi$ holds within $x$ time steps, where $x$ is a variable on natural numbers. The central problem studied in this paper is to determine the set of parameter valuations $V_{\prec p} (\varphi)$ for which the probability to satisfy pLTL-formula $\varphi$ in a Markov chain meets a given threshold $\prec p$, where $\prec$ is a comparison on reals and $p$ a probability. As for pLTL determining the emptiness of $V_{> 0}(\varphi)$ is undecidable, we consider several logic fragments. We consider parametric reachability properties, a sub-logic of pLTL restricted to next and $\Diamond_{\le x}$, parametric B\"uchi properties and finally, a maximal subclass of pLTL for which emptiness of $V_{> 0}(\varphi)$ is decidable.Comment: TCS Track B 201

Parameter-Independent Strategies for pMDPs via POMDPs

Markov Decision Processes (MDPs) are a popular class of models suitable for solving control decision problems in probabilistic reactive systems. We consider parametric MDPs (pMDPs) that include parameters in some of the transition probabilities to account for stochastic uncertainties of the environment such as noise or input disturbances. We study pMDPs with reachability objectives where the parameter values are unknown and impossible to measure directly during execution, but there is a probability distribution known over the parameter values. We study for the first time computing parameter-independent strategies that are expectation optimal, i.e., optimize the expected reachability probability under the probability distribution over the parameters. We present an encoding of our problem to partially observable MDPs (POMDPs), i.e., a reduction of our problem to computing optimal strategies in POMDPs. We evaluate our method experimentally on several benchmarks: a motivating (repeated) learner model; a series of benchmarks of varying configurations of a robot moving on a grid; and a consensus protocol.Comment: Extended version of a QEST 2018 pape