As with regular MDPs, -stationary MDPs can also be generalized with general environment and agent operators. The resulting model inherits the advantages of both approaches of generalization: a broad scale of decision problems can be discussed simultaneously, while the underlying environment is allowed to change over time as well. This family of MDPs will be called generalized -stationary MDPs or -MDPs for short.
Given a prescribed , a generalized -MDP is defined by the tuple , with and , , if there exists a generalized MDP such that . Note that the last assumption requires that the asymptotic distance of the corresponding dynamic-programming operator sequence and is small.
Note also, that the given definition is indeed a generalization of both concepts: setting , and for all yields a generalized MDP, while setting and for all simplifies to an -stationary MDP.