Agent-in-the-Loop Simulation

data structures used in practical reasoning algorithms.

Adjacency structures also restrict the form of representations in ways that eliminate pathological cases that would case dfficulties in practical reasoning.

For example, spaces represented using adjacency structures must have a consis-tent dimension and cannot include stray pieces of lower dimension. Adjacency structures can only describe situations that end abruptly as if they were pieces of some larger situation wth no edges. This fits nicely with the intuitive belief that the universe does not have edges.

either of these spaces are not connected.

The cells used to represent space are not restricted n shape, arrangement, or dimensionality. Previous formalisms have been confined to regular cell arrange-ments (e.g. Pavlidis 1977) or low dimensions. Many representations handle only rectangular arrays. It is not materially easier to define the topological structure of these restricted classes of cell complexes and non-regular cell arrangements are occasionally useful. For example, biological systems, such as the human retina, do not have perfectly regular cell arrangements. Non-regular tessellations are useful in creating compact variable-resolution representations for situations (Brooks Lozano-Perez 1985, Rom and Peleg 1988, Funt 1980). Also, we see in Chapter 4 that it is convenient to be able to use non-regular cell shapes for prov-ing algorithms correct, even when these algorithms only manipulate regular cell arrangements.

A ellular representation also cannot use more than finitely many cells to represent a bounded region of N-space. It is possible to create cellular repre-sentations in which infinitely many cells touch at a point or along a face, but these representations cannot have the topology of I'V or an N-manifold. Be-cause boundaries are placed on or between cells, this restriction also makes it impossible to represent infinitely dense sets of boundaries directly. Cellular rep-resentations can branch, as shown 'in Figure 26. In later chapters we will see a few applications 'in which researchers have proposed such models for time. However, the branches must occur at cell boundaries and thus infinitely dense branches cannot be directly represented.

A second, and closely-related, limitation of cellular representations is that digitized functions cannot distinguish functions that approach a limiting value asymptotically, without ever reaching it, from functions that actually reach the

Figure 26 A branching time line.

limiting value. When the dfferences from the limit value become small enough, the values must be represented using the same cell in the value space as the limit value. Since it cannot distinguish the two cases, an algorithm using digitized data must treat the asymptotic function as though it actually reached the limit point.

We can cast this observationinto a second form which is more directly relevant to practical applications:

If a property is changing in value with a slope of constant sign and it is moving towards a limiting value, the property either becomes ndistinguishable from the limiting value after some finite amount of time or else the slope becomes indistinguishable from zero after some finite amount of time.

Suppose, for example, that you are shovelling snow out of a driveway.8 After some finite period of time, it must either be the case that you have removed all but negligibly much of the snow, or else your rate of shovelling has become negligible. This generalization will prove useful 'in explaining data from both linguistic semantics and high-level reasoning.

In the applications discussed in this thesis, boundaries are always induced

8 Of finite extent!

by contrasts in cell labelling. Because of this, they always satisfy the subset condition. This condition states that an adjacency set must be in the boundaries if any subset of it is in the boundaries. For example, if the edge between two cells belongs to the boundaries, 'its endpoints are also part of the boundaries.

Similarly, if an entire cell belongs to the boundaries, so do all of the edges and vertices that it touches.

Aside from the subset condition, boundaries can be any collection of ad-jacency sets. Boundaries can intersect one another and a boundary can end abruptly in the middle of a region. Figure 27 shows examples of real situations in which boundaries end abruptly. In the applications presented in this thesis, it 'is typically best to place boundaries between cells. However, it is occasionally helpful to place boundaries on cells and even to create boundaries more than one cell wide. The formalism allows all of these options.

Figure 27. Boundaries can end abruptly in the middle of regions. Left: a bent finger. Right: a torus seen in 2D projection.

Thus cellular topology imposes a number of restrictions on the form of rep-resentations for situations. However, these restrictions seem to eliminate only pathological situations that are of little use in practical reasoning. Later chap-ters discuss how some of these restrictions apply to various application domains and confirm that they are not prohibiting useful types of representations.