Causal Space and Restricted Situations

Suppose a miner has been started digging in an upper level. If there is nothing that blocks digging, it will break through and fall to the next level down provided nothing happens to interrupt this. I want to use a formalism like that proposed in the draft [McCarthy 94]

The most straightforward way of saying that the miner will eventually fall through is to say he will if no events occur. This is much too strong a condition. Here's an idea.

1. Introduce a concept of causal space. In many of the approximate theories we will want to use, causal space will not correspond to real space.
2. Causal space has points. Events occur at points. Points persist from situation to situation. Thus we can talk about the same point in related situations.
3. Distance in causal space is defined by the time of propagation of effects of events. Events cannot affect fluents associated with far away points in a short time.
4. Regions are sets of causal points, normally they will be built up from something like open sets in causal space, i.e. if a point is in the ``middle'' of a region, sufficiently nearby points will also be in the region.
5. The fewer points there are in causal space close to a given point, the more conveniently local computations can be done. We will want to work with spaces without very unlikely causal connections. For example, President Clinton could conceivably wake up at 5am with a sudden desire to telephone John McCarthy, but it is better to ignore the possibility and put him at a much larger causal distance than this possibility allows.

   This suggests regarding causal space as a metric space. Indeed it may be a metric space, but we want to reserve judgment on what if any topology we shall want to use.