Learning and Designing Stochastic Processes from Logical Constraints
It addresses the problem of learning stochastic process parameters from logical constraints, which is relevant for systems where only qualitative properties are observable, but the results are demonstrated only on simple examples.
This paper develops a method to learn parameters of stochastic processes from qualitative observations (satisfaction of linear temporal logic formulae) rather than quantitative state observations, unifying system identification and design. The method is demonstrated on examples like rumor spreading and gene regulation.
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.