11.4.3.2 Approximating nondeterministic and probabilistic I-spaces

Many other derived I-spaces extend directly to continuous spaces, such as the limited-memory models of Section 11.2.4 and Examples 11.11 and 11.12. In the present context, it is extremely useful to try to collapse the I-space as much as possible because it tends to be unmanageable in most practical applications. Recall that an I-map, ${\kappa}: {\cal I}_{hist}\rightarrow {\cal I}_{der}$, partitions ${\cal I}_{hist}$ into sets over which a constant action must be applied. The main concern is that restricting plans to ${\cal I}_{der}$ does not inhibit solutions.

Consider making derived I-spaces that approximate nondeterministic or probabilistic I-states. Approximations make sense because $X$ is usually a metric space in the continuous setting. The aim is to dramatically simplify the I-space while trying to avoid the loss of critical information. A trade-off occurs in which the quality of the approximation is traded against the size of the resulting derived I-space. For the case of nondeterministic I-states, conservative approximations are formulated, which are sets that are guaranteed to contain the nondeterministic I-state. For the probabilistic case, moment-based approximations are presented, which are based on general techniques from probability and statistics to approximate probability densities. To avoid unnecessary complications, the presentation will be confined to the discrete-stage model.