854 research outputs found
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 : do all parameter
instantiations within a given region of parameter values satisfy ?,
which instantiations satisfy 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., asserts that
holds within time steps, where is a variable on natural
numbers. The central problem studied in this paper is to determine the set of
parameter valuations for which the probability to
satisfy pLTL-formula in a Markov chain meets a given threshold , where is a comparison on reals and a probability. As for pLTL
determining the emptiness of is undecidable, we consider
several logic fragments. We consider parametric reachability properties, a
sub-logic of pLTL restricted to next and , parametric B\"uchi
properties and finally, a maximal subclass of pLTL for which emptiness of 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
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