185 research outputs found
A Markov Chain Model Checker
Markov chains are widely used in the context of performance and reliability evaluation of systems of various nature. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both the discrete [17,6] and the continuous time setting [4,8]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen Twente Markov Chain Checker ), where properties are expressed in appropriate extensions of CTL. We illustrate the general bene ts of this approach and discuss the structure of the tool. Furthermore we report on first successful applications of the tool to non-trivial examples, highlighting lessons learned during development and application of )
Quantitative model checking of continuous-time Markov chains against timed automata specifications
We study the following problem: given a continuous-time Markov chain (CTMC) C, and a linear real-time property provided as a deterministic timed automaton (DTA) A, what is the probability of the set of paths of C that are\ud
accepted by A (C satisfies A)? It is shown that this set of paths is measurable and computing its probability can be reduced to computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of partial differential equations. For the special case of single-clock DTA, the system of integral equations can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations
A tool for model-checking Markov chains
Markov chains are widely used in the context of the performance and reliability modeling of various systems. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both discrete [34, 10] and continuous time settings [7, 12]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen-Twente Markov Chain Checker EÎMC2, where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore, we report on successful applications of the tool to some examples, highlighting lessons learned during the development and application of EÎMC2
Efficient CSL Model Checking Using Stratification
For continuous-time Markov chains, the model-checking problem with respect to
continuous-time stochastic logic (CSL) has been introduced and shown to be
decidable by Aziz, Sanwal, Singhal and Brayton in 1996. Their proof can be
turned into an approximation algorithm with worse than exponential complexity.
In 2000, Baier, Haverkort, Hermanns and Katoen presented an efficient
polynomial-time approximation algorithm for the sublogic in which only binary
until is allowed. In this paper, we propose such an efficient polynomial-time
approximation algorithm for full CSL. The key to our method is the notion of
stratified CTMCs with respect to the CSL property to be checked. On a
stratified CTMC, the probability to satisfy a CSL path formula can be
approximated by a transient analysis in polynomial time (using uniformization).
We present a measure-preserving, linear-time and -space transformation of any
CTMC into an equivalent, stratified one. This makes the present work the
centerpiece of a broadly applicable full CSL model checker. Recently, the
decision algorithm by Aziz et al. was shown to work only for stratified CTMCs.
As an additional contribution, our measure-preserving transformation can be
used to ensure the decidability for general CTMCs.Comment: 18 pages, preprint for LMCS. An extended abstract appeared in ICALP
201
Bisimulations and Logical Characterizations on Continuous-time Markov Decision Processes
In this paper we study strong and weak bisimulation equivalences for
continuous-time Markov decision processes (CTMDPs) and the logical
characterizations of these relations with respect to the continuous-time
stochastic logic (CSL). For strong bisimulation, it is well known that it is
strictly finer than CSL equivalence. In this paper we propose strong and weak
bisimulations for CTMDPs and show that for a subclass of CTMDPs, strong and
weak bisimulations are both sound and complete with respect to the equivalences
induced by CSL and the sub-logic of CSL without next operator respectively. We
then consider a standard extension of CSL, and show that it and its sub-logic
without X can be fully characterized by strong and weak bisimulations
respectively over arbitrary CTMDPs.Comment: The conference version of this paper was published at VMCAI 201
Learning and Designing Stochastic Processes from Logical Constraints
Stochastic processes offer a flexible mathematical formalism to model and
reason about systems. Most analysis tools, however, start from the premises
that models are fully specified, so that any parameters controlling the
system's dynamics must be known exactly. As this is seldom the case, many
methods have been devised over the last decade to infer (learn) such parameters
from observations of the state of the system. In this paper, we depart from
this approach by assuming that our observations are {\it qualitative}
properties encoded as satisfaction of linear temporal logic formulae, as
opposed to quantitative observations of the state of the system. An important
feature of this approach is that it unifies naturally the system identification
and the system design problems, where the properties, instead of observations,
represent requirements to be satisfied. We develop a principled statistical
estimation procedure based on maximising the likelihood of the system's
parameters, using recent ideas from statistical machine learning. We
demonstrate the efficacy and broad applicability of our method on a range of
simple but non-trivial examples, including rumour spreading in social networks
and hybrid models of gene regulation
Towards Light-Weight Probabilistic Model Checking
YesModel checking has been extensively used to verify various systems. However, this usually has been done by experts who have a good understanding of model checking and who are familiar with the syntax of both modelling and property specification languages. Unfortunately, this is not an easy task for nonexperts to learn description languages for modelling and formal logics/languages for property specification. In particular, property specification is very daunting and error-prone for nonexperts. In this paper, we present a methodology to facilitate probabilistic model checking for nonexperts. The methodology helps nonexpert users model their systems and express their requirements without any knowledge of the modelling and property specification languages
A Probabilistic Temporal Logic with Frequency Operators and Its Model Checking
Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic
(CSL) are often used to describe specifications of probabilistic properties for
discrete time and continuous time, respectively. In PCTL and CSL, the
possibility of executions satisfying some temporal properties can be
quantitatively represented by the probabilistic extension of the path
quantifiers in their basic Computation Tree Logic (CTL), however, path formulae
of them are expressed via the same operators in CTL. For this reason, both of
them cannot represent formulae with quantitative temporal properties, such as
those of the form "some properties hold to more than 80% of time points (in a
certain bounded interval) on the path." In this paper, we introduce a new
temporal operator which expressed the notion of frequency of events, and define
probabilistic frequency temporal logic (PFTL) based on CTL\star. As a result,
we can easily represent the temporal properties of behavior in probabilistic
systems. However, it is difficult to develop a model checker for the full PFTL,
due to rich expressiveness. Accordingly, we develop a model-checking algorithm
for the CTL-like fragment of PFTL against finite-state Markov chains, and an
approximate model-checking algorithm for the bounded Linear Temporal Logic
(LTL) -like fragment of PFTL against countable-state Markov chains.Comment: In Proceedings INFINITY 2011, arXiv:1111.267
A Sample-Driven Solving Procedure for the Repeated Reachability of Quantum CTMCs
Reachability analysis plays a central role in system design and verification.
The reachability problem, denoted , asks whether the system
will meet the property after some time in a given time interval .
Recently, it has been considered on a novel kind of real-time systems --
quantum continuous-time Markov chains (QCTMCs), and embedded into the
model-checking algorithm. In this paper, we further study the repeated
reachability problem in QCTMCs, denoted , which
concerns whether the system starting from each \emph{absolute} time in will
meet the property after some coming \emph{relative} time in . First
of all, we reduce it to the real root isolation of a class of real-valued
functions (exponential polynomials), whose solvability is conditional to
Schanuel's conjecture being true. To speed up the procedure, we employ the
strategy of sampling. The original problem is shown to be equivalent to the
existence of a finite collection of satisfying samples. We then present a
sample-driven procedure, which can effectively refine the sample space after
each time of sampling, no matter whether the sample itself is successful or
conflicting. The improvement on efficiency is validated by randomly generated
instances. Hence the proposed method would be promising to attack the repeated
reachability problems together with checking other -regular properties
in a wide scope of real-time systems
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