The primary use of an I-map is to simplify the description of a plan. In Section 11.1.3, a plan was defined as a function on the history I-space, . Suppose that an I-map, , is introduced that maps from to . A feedback plan on is defined as . To execute a plan defined on , the derived I-state is computed at each stage by applying to to obtain . The action selected by is .

To understand the effect of using instead of as the domain of , consider the set of possible plans that can be represented over . Let and be the sets of all plans over and , respectively. Any can be converted into an equivalent plan, , as follows: For each , define .

It is not always possible, however, to construct a plan, , from some . The problem is that there may exist some for which and . In words, this means that the plan in requires that two histories cause different actions, but in the derived I-space the histories cannot be distinguished. For a plan in , both histories must yield the same action.

An I-map has the potential to collapse down to a smaller I-space by inducing a partition of . For each , let the preimage be defined as

This yields the set of history I-states that map to . The induced partition can intuitively be considered as the ``resolution'' at which the history I-space is characterized. If the sets in (11.26) are large, then the I-space is substantially reduced. The goal is to select to make the sets in the partition as large as possible; however, one must be careful to avoid collapsing the I-space so much that the problem can no longer be solved.

Consider the partition of
that is induced by
.
For each , the set
, as defined in
(11.26), is the set of all histories that lead to the same
state estimate. A plan on can no longer distinguish between
various histories that led to the same state estimate. One
implication is that the ability to encode the amount of uncertainty in
the state estimate has been lost. For example, it might be wise to
make the action depend on the covariance in the estimate of ;
however, this is not possible because decisions are based only on the
estimate itself.

Steven M LaValle 2020-08-14