Theoretical Computer Science
An introduction to symbolic dynamics and coding
An introduction to symbolic dynamics and coding
Symbolic Model Checking for Probabilistic Processes
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
A Logic of Probability with Decidable Model-Checking
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
Probabilistic Linear-Time Model Checking: An Overview of the Automata-Theoretic Approach
ARTS '99 Proceedings of the 5th International AMAST Workshop on Formal Methods for Real-Time and Probabilistic Systems
LICS '96 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science
Quantitative Analysis and Model Checking
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
Model checking meets performance evaluation
ACM SIGMETRICS Performance Evaluation Review
Principles of Model Checking (Representation and Mind Series)
Principles of Model Checking (Representation and Mind Series)
Reasoning about MDPs as Transformers of Probability Distributions
QEST '10 Proceedings of the 2010 Seventh International Conference on the Quantitative Evaluation of Systems
PRISM 4.0: verification of probabilistic real-time systems
CAV'11 Proceedings of the 23rd international conference on Computer aided verification
Model Checking MDPs with a Unique Compact Invariant Set of Distributions
QEST '11 Proceedings of the 2011 Eighth International Conference on Quantitative Evaluation of SysTems
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A finite state Markov chain M is often viewed as a probabilistic transition system. An alternative view - which we follow here - is to regard M as a linear transform operating on the space of probability distributions over its set of nodes. The novel idea here is to discretize the probability value space [0,1] into a finite set of intervals. A concrete probability distribution over the nodes is then symbolically represented as a tuple D of such intervals. The i-th component of the discretized distribution D will be the interval in which the probability of node i falls. The set of discretized distributions is a finite set and each trajectory, generated by repeated applications of M to an initial distribution, will induce a unique infinite string over this finite set of letters. Hence, given a set of initial distributions, the symbolic dynamics of M will consist of an infinite language L over the finite alphabet of discretized distributions. We investigate whether L meets a specification given as a linear time temporal logic formula whose atomic propositions will assert that the current probability of a node falls in an interval. Unfortunately, even for restricted Markov chains (for instance, irreducible and aperiodic chains), we do not know at present if and when L is an (omega)-regular language. To get around this we develop the notion of an epsilon-approximation, based on the transient and long term behaviors of M. Our main results are that, one can effectively check whether (i) for each infinite word in L, at least one of its epsilon-approximations satisfies the specification; (ii) for each infinite word in L all its epsilon approximations satisfy the specification. These verification results are strong in that they apply to all finite state Markov chains. Further, the study of the symbolic dynamics of Markov chains initiated here is of independent interest and can lead to other applications.