Suppose a bounded region *R* is given, containing an *Evader*
whose position at any time *t* is unknown, and the
movement of the evader (i.e. *e*) is continuous. The goal
is finding a continuous path (a *
solution*) that guarantees that for some
*t*_{d} *e*(*t*_{d}) is visible from . It has been shown in
a paper by Steven LaValle and John Hinrichsen that such a path can be found by searching an information state
graph induced by the cell decomposition of *R*, where the cells are
bounded by inflections and bitangents of (the boundary of *R*).

To understand this method, note that from any
position in *R*, the pursuer is able to see some parts of *R*, but not
others. Also note that this correspods to seeing some intervals or ,nut not seeing others. These *gaps* in visibility of correspond to subregions of *R* that may contain an unseen evader. A
gap is called *contaminated* if there is a possibility of an
evader hiding in the gap's corresponding subregion. If we know that an
evader can't be in an unseen region, we label its corresponding gap
*cleared*. One suggested method of detecting
the presence and position of the gaps for an actual robot
is performing a 360 degree sweep of distances to the
wall, and finding discontinuities in the measurements.

The cells are partitioned by inflections and bitangents because the
relevant pursuit *events*, such as cleared/contaminated
gaps appearing, dispearring, merging, or splitting, occur only when
the pursuer crosses an inflection or a bitangent. We therefore use an
information state determined by the cell location of the pursuer and
the status of the gaps (whether they are cleared or contaminated), and
can build an information state graph whose nodes are a set all
possible information states, with edges between them determined by
information state changes possible by crossing cell boundaries. This
graph can then be searched to find what sequence of cells must be
visited to end up in an information state where all gaps have been cleared.

In this project, we will build on these results to show that a solution appropriate for a very limited robot can be exctracted from the cell decomposition. We will also speculate about the possibility of the topology of the cell decomposition graph being inferred by the same limited robot without a priori knowledge of the map, and without the need to build the map using robot sensor data. These two results can lead to a complete algorithm for such a robot in an unknown enviroment.