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Q1. Erik Sandewall:

A fairly large part of your work is devoted to the time structure. Then, under "future work" you mention addressing the topic of continuous change. Won't you then have to use the real numbers (or something similar) as time structure?

A1. Jixin Ma (26.1):

Yes, though it is not necessary to directly use the real numbers as the time structure. In fact, a general time theory based on both points and intervals is proposed previously (The Computer Journal 37(2), 1994, pp.114-123). With the corresponding density axioms, it seems appropriate for addressing continuous change.

C1-1. Erik Sandewall:

In this case, why don't you introduce them at once, even for the present work? Is there some essential aspect of the present results that you are going to lose when you proceed to continuous change?

C1-2. Jixin Ma (26.1):

First of all, the approch addressing both moments and points as primitive is proposed in the paper in order to overcome the so-called Dividing Instant Problem, i.e., the problem in specifying if intervals is "closed"/"open" at their ending points in the case intervals are constructed out of points. It is argued again and again by many researchers that no solution is satisfactory to the question of what preconception about the closed/open nature of intervals should be taken:

• If all intervals include their ending-points, then adjacent intervals would have ending-points in common. Hence, if two adjacent intervals correspond to a given fluent and its negation, respectively, there will be a point at which the fluent is both true and false.

• Similarly, if all intervals don't include their ending-points, there will be points at which the truth or falsity of some fluents are undefined.

• Another approach is to take point-based intervals as semi-open (e.g., all intervals include their left ending-points, and exclude their right ones) so that they may sit conveniently next to one another. However, on the one hand, since this approach insists that every interval contains only a single ending-point, the choice of which ending-point of intervals should be included/excluded seems arbitrary, and hence unjustifiable and artificial. On the other hand, although the approach may offer a solution to some practical questions, there are some other critical questions which remain problematical. The fundamental reason is that in a system where time intervals are all taken as semi-open, it will be difficult to represent time points in a consistent structure so that they can stand between intervals conveniently.

There are three reasons of taking a discrete time structure:

1. It is simple and adequate for many practical applications.

2. It is noted that, if time intervals are addressed as temporal objects which can be associated with fluents by meta-predicates such as "HOLDS" or "TRUE", one shall face the possibility that a fluent might be neither true nor false throughout some interval. In the worst case, the so-called intermingling problem may arise, that is the possibility of a fluent changing its truth value infinitely often in an interval with a finite duration. This will lead to some difficulties in characterising the relationships between the negation of a fluent and the negation of involved sentances.

The approach that addresses intervals as a finite ordered union of points/moments sucessfully overcomes (or at least bypasses) such a problem. For the purpose of modelling continuous change, it seems necessary (this maybe arguable) to extend such a discrete time model to a dense one. However, in this case, some careful treatments have to be taken for dealing with the Intermingling Problem.

3. Since times are explicitly addressed in the formalism, the corresponding Frame Problem would become more difficult to be handled if the time structure is modelled as dense in the first place. Hence, the strategy is to tackle the simple case (that is, the discrete model) first, then to extend the idea to the general case.

This on-line debate page is part of a discussion at recent workshop; similar pages are set up for each of the workshop articles. The discussion is organized by the area Reasoning about Actions and Change within the Electronic Transactions on Artificial Intelligence (ETAI).

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