Central Limit Model Checking
release_vl2f23usa5bkfdargq2l3ujlmu
by
Luca Bortolussi, Luca Cardelli, Marta Kwiatkowska, Luca Laurenti
2018
Abstract
We consider probabilistic model checking for continuous-time Markov chains
(CTMCs) induced from Stochastic Reaction Networks (SRNs) against a fragment of
Continuous Stochastic Logic (CSL) extended with reward operators. Classical
numerical algorithms for CSL model checking based on uniformisation are limited
to finite CTMCs and suffer from the state sapce explosion problem. On the other
hand, approximate techniques such as mean-field approximations and simulations
combined with statistical inference are more scalable, but can be time
consuming and do not support the full expressiveness of CSL. In this paper we
employ a continuous-space approximation of the CTMC in terms of a Gaussian
process based on the Central Limit Approximation (CLA), also known as the
Linear Noise Approximation (LNA), whose solution requires solving a number of
differential equations that is quadratic in the number of species and
independent of the population size. We then develop efficient and scalable
approximate model checking algorithms on the resulting Gaussian process, where
we restrict the target regions for probabilistic reachability to convex
polytopes. This allows us to derive an abstraction in terms of a
time-inhomogeneous discrete-time Markov chain (DTMC), whose dimension is
independent of the number of species, on which model checking is performed.
Using results from probability theory, we prove the convergence in distribution
of our algorithms to the corresponding measures on the original CTMC. We
implement the techniques and, on a set of examples, demonstrate that they allow
us to overcome the state space explosion problem, while still correctly
characterizing the stochastic behaviour of the system. Our methods can be used
for formal analysis of a wide range of distributed stochastic systems,
including biochemical systems, sensor networks and population protocols.
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