The history I-space appears to be quite complicated. Every I-state corresponds to a history of actions and observations. Unfortunately, the length of the I-state sequence grows linearly with the number of stages. To overcome this difficultly, it is common to map history I-states to some simpler space. In many applications, the ability to perform this simplification is critical to finding a practical solution. In some cases, the simplification fully preserves the history I-space, meaning that completeness, and optimality if applicable, is not lost. In other cases, we are willing to tolerate a simplification that destroys much of the structure of the history I-space. This may be necessary to obtain a dramatic reduction in the size of the I-space.