Appendix B: Decision Time
In this section we
compare the decision times in non-hierarchical and hierarchical
representations. For a one-dimensional environment (see Figure
B1) that contains 32 places we build a hierarchical
representation shown in Figure B2. This hierarchy contains
multiple representations of the same environment at different
resolutions. On one hand, the time to spread the activity
between place 1 and place 32 only by using the representation
with the highest resolution is proportional to the number of
connections in between place 1 and place 32, namely 31. On the
other hand, the time to spread the activity between place 1 and
place 32 when all levels of the hierarchy are used is
proportional to twice the depth of the hierarchy, namely 10 (= 2
* 5). This is the number of steps required by the spreading
activity from place 32 to reach place 1. We can extrapolate that
in general, whereas the reaction time for spreading the activity
between two places in the case of a nonhierarchical
representation of space is linear with the distance between the
two places, in the case of a hierarchical representation of
space it varies logarithmically with the distance between the
two places (see Figure B3). Therefore, by using hierarchical,
low-resolution representations of the environment the agent can
speed the spatial planning process.
Figure B1. One-dimensional environment that contains 32
Figure B2. Multiple level representation of a one-dimensional
environment that contains 32 places.
Linear and logarithmic reaction times as a function of the
distance between places. Circles represent time to spread the
activity in a hierarchical representation of space; diamonds
represent time to spread the activity in a non-hierarchical
representation of space.
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