  This paper is concerned with a compositional approach for constructing both
infinite (reduced-order models) and finite abstractions (a.k.a. finite Markov
decision processes (MDPs)) of large-scale interconnected discrete-time
stochastic systems. The proposed framework is based on the notion of stochastic
simulation functions enabling us to employ an abstract system as a substitution
of the original one in the controller design process with guaranteed error
bounds. In the first part of the paper, we derive sufficient small-gain type
conditions for the compositional quantification of the probabilistic distance
between the interconnection of stochastic control subsystems and that of their
infinite abstractions. We then construct infinite abstractions together with
their corresponding stochastic simulation functions for a particular class of
discrete-time nonlinear stochastic control systems. In the second part of the
paper, we leverage small-gain type conditions for the compositional
construction of finite abstractions. We propose an approach to construct finite
MDPs as finite abstractions of concrete models or their reduced-order versions
satisfying an incremental input-to-state stability property. We demonstrate the
effectiveness of the proposed results by applying our approaches to a fully
interconnected network of 20 nonlinear subsystems (totally 100 dimensions). We
construct finite MDPs from their reduced-order versions (together 20
dimensions) with guaranteed error bounds on their output trajectories. We also
apply the proposed results to a temperature regulation in a circular building
and construct compositionally a finite abstraction of a network containing 1000
rooms. We employ the constructed finite abstractions as substitutes to
compositionally synthesize policies regulating the temperature in each room for
a bounded time horizon.
