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We propose a compositional approach for constructing abstractions of generalMarkov decision processes using approximate probabilistic relations. Theabstraction framework is based on the notion of $\delta$-lifted relations,using which one can quantify the distance in probability between theinterconnected gMDPs and that of their abstractions. This new approximaterelation unifies compositionality results in the literature by incorporatingthe dependencies between state transitions explicitly and by allowing abstractmodels to have either finite or infinite state spaces. Accordingly, one canleverage the proposed results to perform analysis and synthesis over abstractmodels, and then carry the results over concrete ones. To this end, we firstpropose our compositionality results using the new approximate probabilisticrelation which is based on lifting. We then focus on a class of stochasticnonlinear dynamical systems and construct their abstractions using both modelorder reduction and space discretization in a unified framework. We provideconditions for simultaneous existence of relations incorporating the structureof the network. Finally, we demonstrate the effectiveness of the proposedresults by considering a network of four nonlinear dynamical subsystems(together 12 dimensions) and constructing finite abstractions from theirreduced-order versions (together 4 dimensions) in a unified compositionalframework. We benchmark our results against the compositional abstractiontechniques that construct both infinite abstractions (reduced-order models) andfinite MDPs in two consecutive steps. We show that our approach is much lessconservative than the ones available in the literature.

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