Approximate Probabilistic Relations for Compositional Abstractions of Stochastic Systems

In this paper, we propose a compositional approach for constructing abstractions of general Markov decision processes (gMDPs) 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 allowing abstract models to have either finite or infinite state spaces. 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. 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.