QUANTICOL - A framework for hybrid limits under uncertainty

This deliverable reports on the development of a theoretical framework to study the effect of multiple scales and imprecision in the emergent behaviour of collective adaptive systems (CAS). We show how to construct suitable mean-field approximations for such systems. It constitutes the main achievement of Task 1.1 and a rst step towards Task 1.3 and the linking of language specification and mean-field techniques. This document is structured in two main sections, the rst one presents results related to multiple scales, both in terms of time and of population levels. The second part focuses on mean eld results in presence of uncertainty. As for multiple scales, we discuss the following results in detail. We rst present a general frame- work for mean eld limits for systems with heterogeneous population size, following [Bor15]. This framework considers a very general class of population processes, allowing both immediate and stochas- tic transitions, guarded by Boolean predicates (to encode for example control actions), and obtaining limits in terms of stochastic hybrid systems, which are usually faster to simulate. This is discussed in detail in Section 2.2. Computing the transition rates of some immediate transitions requires the computation of stochastic hitting times, i.e., the time for a stochastic system to hit a given domain. We show how to use a uid approximation to compute this time in Section 2.3. Next, in Section 2.4, we present a general framework to combine mean eld limits with reduction of multiple time scales, with conditions providing guarantees on the correctness of exchanging these two operations [BP14]. This framework leads to a new simulation algorithm for Markov models with multiple time scales, leveraging powerful statistical abstraction tools [BMS15]. Finally, an integration of hybrid conditional moment techniques [Has+14] within the stochastic process algebra PEPA [Pou15] is discussed in Section 2.5. The second part of the document is devoted to the analysis of CAS models in the presence of uncertainty. We distinguish an uncertain model { for which a parameter exists but is not known { and an imprecise model { for which some parameters may vary. We show how mean eld limits can greatly simplify the study of uncertain and imprecise population models [BG15]. This setting encompasses the imprecise and the uncertain scenario, but also more classic models like Markov Decision Processes. It uses a differential inclusion to represent the limit. We discuss it in Section 3.1. We develop some numerical methods to analyse the class of limit models for uncertain and imprecise population models. In particular, we discuss in Section 3.2 a method based on statistical emulation for the uncertain case [BS14], and two methods, one based on differential hulls [TT15] and one based on the Pontraygin optimal control principle [BG15].