Compositional Reasoning for Markov Decision Processes

Markov decision processes (MDPs) have long been used to model qualitative aspects of systems in the presence of uncertainty. However, much of the literature on MDPs takes a monolithic approach, by modelling a system as a particular MDP; properties of the system are then inferred by analysis of that particular MDP. In this paper we develop compositional methods for reasoning about the qualitative behaviour of MDPs. We consider a class of labelled MDPs called weighted MDPs from a process algebraic point of view. For these we define a coinductive simulation-based behavioural preorder which is compositional in the sense that it is preserved by structural operators for constructing MDPs from components. For finitary convergent processes, which are finite-state and finitely branching systems without divergence, we provide two characterisations of the behavioural preorder. The first uses a novel qualitative probabilistic logic, while the second is in terms of a novel form of testing, in which benefits are accrued during the execution of tests.