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Ontologies for time
To Pat
As an addition to my response (ENRAC 15.4 (98035)) to Pat's
suggestion of "simply map Allen and Hayes' moments to Ma and
Knight's points":
The constraint that "moments cannot meet each" will lead to the
conclusion that we can have neither a completely discrete nor a
completely dense system which contains both moments and
decomposable intervals. However, if we revise Allen and Hayes'
system to include both points and intervals (including moments), and
impose the "not-meet-each-other" constraint on points only, rather
than on moments, this objection does not apply.
To Sergio
| | 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. I really find it difficult to
believe that you like to make an exception in this sense, also because the
hat here is ``formal (temporal) reasoning''. It also seems to me that any
temporal structure must necessarily fail to be persuasive if on one hand
it includes the notion of continuity and on the other it refuses it;
how can time be continuous ... with some exception? Either it is
continuous, or it is not! That is, either the Basic Time Structure
assumes the axiom of completeness, or it does not!!
In fact, in this discussion I have not yet seen any explanation why
an alternative notion of continuous structure is needed at all? I am not
asking you to argue about your own notion, I just ask you to give a
convincing argument on the need of a notion which is an alternative to
the classical one, such as: ``the problem P of temporal reasoning about
actions and change can not be solved adopting the axiom of completeness'',
or ``the axiom of completeness is too strong an assumption for our purposes;
axiom A is better suited, because...''.
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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?
| | 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.
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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?".
For a point-based model, the continuity is usually characterised as
"Between any two points, there is a third"; while for an
interval-based model (like that of Allen), it is characterised as
"Every interval can be decomposed into two adjacent sub-intervals".
In addition, as for a model which takes both intervals and
points as primitive, one may characterise two levels of density. At
the weak level, it is only required that each interval can be divided
into two adjacent sub-intervals. At the strong level, it is required
that there is always a point within any interval. It is easy to
infer that the latter can imply the former.
As for general treatments, the Basic Time Structure does not
have to impose the axiom of density or discreteness (Similar
arguments 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.
Now, "why an alternative notion of continuous structure is needed at
all"? It has been noted that, temporal knowledge in the domain of
artifical intelligence, including "temporal reasoning about actions
and change", is usually imcomplete, and using time intervals in
many cases is more convenient and more in-keeping with common
sense of temporal concepts than to use the classical abstraction of
points. In fact, the notion of time intervals (or periods) has been
introduced for a long time in the literature. In addition, in order
to overcome/bypass the annoying question of whether various
intervals are open or
closed, various approached have been proposed. An example is Allen's
interval-based time theory. As for these time theories, the old
(classical?) notion of continuity no longer simply applies.
Jixin
Sergio Brandano wrote:
| | The following are some fragments from the current discussion:
From Pat Hayes - ENRAC 14.3.1998
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| | instantaneous intervals completely. It is also quite consistent to have
arbitrary amounts of density, discreteness, etc.; for example, one can
say that time is continuous except in a certain class of 'momentary'
intervals whose ends are distinct but have no interior points.
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From Jixin Ma - ENRAC 15.4.1998
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| | time element is a decomposable interval. In fact, generally speaking,
the basic time structure may be neither dense nor discrete anywhere,
or may be continuous over some parts and discrete over other parts.
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Pat and Jixin, what do you mean when you write ``continuous''?
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. I really find it difficult to
believe that you like to make an exception in this sense, also because the
hat here is ``formal (temporal) reasoning''. It also seems to me that any
temporal structure must necessarily fail to be persuasive if on one hand
it includes the notion of continuity and on the other it refuses it;
how can time be continuous ... with some exception? Either it is
continuous, or it is not! That is, either the Basic Time Structure
assumes the axiom of completeness, or it does not!!
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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.)
You talk about a 'founding notion' of continuity as being that captured by
the axiom of completeness. Here, in my view, you commit a philosophical
error (especially in Pisa!) There are intuitions about continuity which
one can try to capture in various formal ways, but there is no 'founding
notion' of continuity other than those intuitions. In the late 19th
century, famous mathematicians objected strongly to the view of the
continuum as consisting of a set of points, for example. This modern
perspective, now taught in high schools, is a modern invention, not a
'founding' notion. It is more recent than the gasoline engine, yet people
have had intuitions about smoothness, instantaneity and continuity for
eons. (Whether or not one agrees with me on this admittedly controversial
point, it seems unwise to identify a mathematical property such as
continuity with any kind of axiom until one has verified that no other
axiom will do as well; and as I am sure Sergio knows, there are many
alternative ways to axiomatize continuity.)
In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
be appropriate for different purposes or to capture different intuitions.
(I agree with Jixin that it is useful to seek a common 'core' theory which
can be extended in various ways to describe various possible more complex
temporal structures; and that this theory will have to be rather weak.)
| | In fact, in this discussion I have not yet seen any explanation why
an alternative notion of continuous structure is needed at all? I am not
asking you to argue about your own notion, I just ask you to give a
convincing argument on the need of a notion which is an alternative to
the classical one, such as: ``the problem P of temporal reasoning about
actions and change can not be solved adopting the axiom of completeness'',
or ``the axiom of completeness is too strong an assumption for our purposes;
axiom A is better suited, because...''.
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The 'dividing point' problem which gave rise to this discussion would do.
According to the modern account of the continuum, this point must exist,
and since all intervals consist of points, the light is therefore either on
or off at it. But it seems more natural, as well as formally simpler, to
just say that the question is meaningless; perhaps (though this is no
longer my own preference) because that point doesnt exist.
A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete: for example, they routinely
describe times as integers representing the number of milliseconds since
the birth of Christ. (Of course, one can always insist that these are to be
understood as being embedded in a continuum, but then what use is an axiom
whose sole purpose is to insist that times exist which have no name and
about which nothing can be asserted, other than that they exist?)
Pat Hayes
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