Compositional Abstraction-based Synthesis of General MDPs via Approximate Probabilistic Relations

We propose a compositional approach for constructing abstractions of general Markov decision processes using approximate probabilistic relations. The abstraction framework is based on the notion of $\delta$-lifted relations, using which one can quantify the distance in probability between the interconnected gMDPs and that of their abstractions. This new approximate relation unifies compositionality results in the literature by incorporating the dependencies between state transitions explicitly and by allowing abstract models to have either finite or infinite state spaces. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract models, and then carry the results over concrete ones. To this end, we first propose our compositionality results using the new approximate probabilistic relation which is based on lifting. We then focus on a class of stochastic nonlinear dynamical systems and construct their abstractions using both model order reduction and space discretization in a unified framework. We provide conditions for simultaneous existence of relations incorporating the structure of the network. Finally, we demonstrate the effectiveness of the proposed results by considering a network of four nonlinear dynamical subsystems (together 12 dimensions) and constructing finite abstractions from their reduced-order versions (together 4 dimensions) in a unified compositional framework. We benchmark our results against the compositional abstraction techniques that construct both infinite abstractions (reduced-order models) and finite MDPs in two consecutive steps. We show that our approach is much less conservative than the ones available in the literature.