Today, we advertise the acceptance of an additional contribution: the article by Antonis Kakas and Rob Miller has been accepted to the ETAI following our tough discussion and refereeing procedure. As the readers recognize, we have had an extensive open discussion about the paper, which has now been followed by confidential refereeing by three referees. All three referees recommended unconditional acceptance of the article. Two of them also made additional suggestions for further improvement of the article. These suggestions have been added to the open discussion about the article, only the identity of the reviewers being kept confidential. We congratulate the authors to having passed this test!

Also today, Jixin Ma and Sergio Brandano pursue the discussion about standard vs. non-standard ontologies of time.

Antonis Kakas and Rob Miller
Reasoning about Actions, Narratives and Ramification

Jixin Ma:

To Sergio,

First of all, what do you mean "the classical one"? (the classical continuous time structure)? Does it refer to the classical physical model of time, where the structure is a set of points which is isomorphic to the real line?

I can just quote myself ...

Here in Pisa, we write ``continuity'' and we read ``axiom of completeness'', which is what everyone commonly means when speaking about (the founding notion of) continuity.

Concerning the core theory that you and Jixin are willing to obtain, I already developed a Basic Time Structure which may be of interest. It is as simple as I managed to design it, without un-useful complications. The structure works well in my case. you are welcome to read and comment my contribution, which may be found in my ETAI's reference.

So, you didn't refer "the classical one" to "the Basic Time Structure" you developed, did you? If No, why did you develop it? What is your convincing argument(s) on the need of such a structure? Is it also an alternative to the classical one? (Sorry, I am here using the similar question raised by youself to ask you, though I don't have to). If Yes, I shouldn't ask this question.

At the ontological level, the notion of continuous time vi discrete time is closely related to questions "Is the set of time elements dense or not?", and " Are there really time atoms?".

The word "continuity", even at the ontological level, can not be read as "continuous with some exception".

What I actually said is very clear as you quoted above. Does it imply that "the word continuity can be read as continuous with some exception"? In fact, even when Pat talked about "continuous with some exception", he didn't really mean that it is as same as the word "continuity". What he means, as I understand, is just that, with the exception of time moments, each time intervals can be decomposed into (at least two) sub-intervals.

The axiom of completeness states: Let be  A  and  B  non empty subsets of  S  such that  a < b  for all  a in A  and  b in B . Then exists  xi in S  such that  a < xi < b  for all  a in A  and  b in B . Now, the set  S , that is your domain, may consists as well either of time-points or time-intervals;  S  holds real numbers on the former case, intervals from the real line on the latter case.

Firstly, you said here, "the former case" and "the latter case". Can these two cases mixed together? In other words, can the domain contains both time-points and time-intervals. I suppose it should. Otherwise, you will meet some problem in satisfying the so-called completeness axiom (see below).

Secondly, you take time-points as real numbers, and intervals "from" the real line. Is your intervals are sets of real numbers limited by their end-points (real numbers)? If no, what are they? If yes, have you consider the dividing instant problem? This problem is more obvious with your time structure when you try to impose the axiom of completeness (see below).

Thirdly, if the domain S consists time-intervals, you need to re-define (or revise, or, at least, explain) the mean of the relation   <   between elements of the domain S. After you have done this properly (You didn't show how, if you can, to do it. You just claim that the domain "may" contains either time-points or time-intervals), you have to show, for the case that interval  a  in  A  is immediately before interval  b  in  B  (that is there is no other time elements standing between  a  and  b ), what is the required  xi  such that  a < xi < b . Obviously,  xi  cannot be an interval (non-pointlike), otherwise, it will overlap with  a  and  b . Therefore, if you can define what is it, it has to be a point (This is why I said earlier in the above that is your domain contains intervals, it needs to contain points as well). Now, you meet the dividing instant problem, as I expected.

As for general treatments, the Basic Time Structure does not have to impose the axiom of density or discreteness (Similar arquements apply to issues such as linear/non-linear, bounded/un-bounded). Therefore, the time structure as a whole may be continuous or discrete, or neither continuous nor discrete.

I agree with your premise: the Basic Time Structure does not have to impose the choice, in fact it leaves you free in that sense. As soon as you make the choice, then you obtain either a continuous structure or a discrete structure, just depending on this choice. I do not agree, instead, with your conclusion. If I leave you the freedom to choose, it does not mean the Structure is neither continuous nor discrete; it simply means you still have to make the choice. Sorry, I can not draw nothing different out of it.

If you don't impose the continuous axiom (!!! as argued by Pat, it does Not has to be the so-called axiom of completeness !!!) or discrete axiom, the structure can be neither continuous nor discrete. I think it is very easy to form a structure which satifies the basic axiomatisation, but does not satisfy the continuous requirement, and does not satisfy the discrete requirement. In fact, you can wirte down any extra constraint as long as it is consistent with the basic theory.

Sergio Brandano:

Pat Hayes wrote (ENRAC 21.4.1998):

Why cannot time be continuous in some places but discontinuous at others? There is no mathematical objection to such a structure, and it has been argued that a continuum punctuated by a sparse collection of points of discontinuity might be a plausible mathematical picture of time which seems to 'flow smoothly' except when things happen suddenly. (Similar arguments can be made for describing spatial boundaries, by the way; and elementary physics makes similar assumptions, where velocity is supposed to change smoothly except when 'impact' occurs.)

If a given temporal structure includes the solution to the problem of representing ``perceived smooth'' flux and ``perceived fast'' flux of time, then that temporal structure is necessarily agent-centric, since different agents may have a different perception of the world. In being agent-centric, this structure can not aim at generality. In fact, if we design an agent-centric temporal structure and the world is inhabited by more than one agent, then we must design a more general structure that reconciles the different views from the different agents. I say ``must'' because, otherwise, we pre-destine agents to never interact with each other, which would be a major restriction.

Sergio