  We present a method for solving implicit (factored) Markov decision processes
(MDPs) with very large state spaces. We introduce a property of state space
partitions which we call epsilon-homogeneity. Intuitively, an
epsilon-homogeneous partition groups together states that behave approximately
the same under all or some subset of policies. Borrowing from recent work on
model minimization in computer-aided software verification, we present an
algorithm that takes a factored representation of an MDP and an 0<=epsilon<=1
and computes a factored epsilon-homogeneous partition of the state space. This
partition defines a family of related MDPs - those MDPs with state space equal
to the blocks of the partition, and transition probabilities "approximately"
like those of any (original MDP) state in the source block. To formally study
such families of MDPs, we introduce the new notion of a "bounded parameter MDP"
(BMDP), which is a family of (traditional) MDPs defined by specifying upper and
lower bounds on the transition probabilities and rewards. We describe
algorithms that operate on BMDPs to find policies that are approximately
optimal with respect to the original MDP. In combination, our method for
reducing a large implicit MDP to a possibly much smaller BMDP using an
epsilon-homogeneous partition, and our methods for selecting actions in BMDPs
constitute a new approach for analyzing large implicit MDPs. Among its
advantages, this new approach provides insight into existing algorithms to
solving implicit MDPs, provides useful connections to work in automata theory
and model minimization, and suggests methods, which involve varying epsilon, to
trade time and space (specifically in terms of the size of the corresponding
state space) for solution quality.
