Smoothed model checking for uncertain Continuous-Time Markov Chains

Abstract

We consider the problem of computing the satisfaction probability of a formula for stochastic models with parametric uncertainty. We show that this satisfaction probability is a smooth function of the model parameters. This enables us to devise a novel Bayesian statistical algorithm which performs model checking simultaneously for all values of the model parameters from observations of truth values of the formula over individual runs of the model at isolated parameter values. This is achieved by exploiting the smoothness of the satisfaction function: by modelling explicitly correlations through a prior distribution over a space of smooth functions (a Gaussian Process), we can condition on observations at individual parameter values to construct an analytical approximation of the function itself. Extensive experiments on non-trivial case studies show that the approach is accurate and several orders of magnitude faster than naive parameter exploration with standard statistical model checking methods.

Studying Emergent Behaviours in Morphogenesis Using Signal Spatio-Temporal Logic

“…The Smoothed Model Checking algorithm [7] relies on the characterisation of the satisfaction probability of a formula ϕ as a function of the parameters. Given a CTMC M θ , whose transition rates depend on a set of parameters θ, the satisfaction function of ϕ is defined as follows:…”

“…It has been proven in [7] that, if the transition rates of M θ depends smoothly on the parameters θ and polynomially on the state of the system, then the satisfaction function of ϕ is a smooth function of the parameters. The smoothed model checking approach leverages of the smoothness of the satisfaction function and transfers information across nearby parameter values.…”

“…In the experiments of [7], it has been possible to accurately approximate the satisfaction function over a wide range of parameters using less than 10% of the simulation runs required to obtain the same result with exhaustive parameter exploration. This resulted in a decrease of the total analysis time nearly by 90%.…”

“…The main technical contribution of the paper is the combination of SSTL within the statistical machine learning framework of [4,[7][8][9], in order to efficiently perform parameter space exploration and system design of spatio-temporal properties.…”

Mean Field Approximation of Uncertain Stochastic Models

“…not influenced by the environment, but unknown. In this case, the class of population models so obtained is considerably simpler, resulting in the so-called uncertain continuous time Markov chains, see [4].…”