Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Sets as Memory traces.
Replies: 10   Last Post: Feb 11, 2013 12:48 AM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: Sets as Memory traces.
Posted: Feb 10, 2013 3:13 PM

On Feb 11, 5:31 am, Zuhair <zaljo...@gmail.com> wrote:
> On Feb 8, 1:02 pm, Zuhair <zaljo...@gmail.com> wrote:
> > On Feb 6, 9:17 pm, Zuhair <zaljo...@gmail.com> wrote:
> > > On Feb 6, 2:14 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > > > Suppose that we have three bricks, A,B,C, now one can understand the
> > > > Whole of those bricks to be an object that have every part of it
> > > > overlapping with brick A or B or C, lets denote that whole by W. Of
> > > > course clearly W is not a brick, W is the totality of all the three
> > > > above mentioned bricks. However here I want to capture the idea of
> > > > 'membership' of that whole, more specifically what do we mean when we
> > > > say that brick A is a 'member' of W. We know that A is a part of W,
> > > > but being a part of W is not enough by itself to qualify A as being a
> > > > member of W, one can observe that brick A itself can have many proper
> > > > parts of it and those would be parts of W of course (since part-hood
> > > > is transitive) and yet non of those is a member of W. So for a part of
> > > > W to be a member of W there must be some property that it must
> > > > satisfy. I'll add another primitive binary relation in addition to
> > > > part-hood, and that binary relation I'll call as "contact". So we'll
> > > > be working within a kind of Mereotopology. However axioms to
> > > > characterize contact relation would be different from those of
> > > > Mereotopology. Here only disjoint (non overlapping) objects are
> > > > allowed to be in contact. When we say A is in contact with B then we
> > > > mean that for some x,y where x is a part of A and y is a part of B
> > > > there do not exist a gap between x and y, of course as said above
> > > > provided that A and B are disjoint objects.

>
> > > > Now we come to stipulate the sufficient condition for membership of a
> > > > whole, this is:

>
> > > > x is a member of y iff x part of y And (there do not exist a part of y
> > > > that is in contact with x) And every proper part of x is in contact
> > > > with some part of x.

>
> > > > This supplies us with the impression that x is a 'solid' entity and
> > > > yet x is separate (not in contact) from some other parts of y should
> > > > those exist.

>
> > > > However this interpretation of membership using this kind of
> > > > mereotopology has its shortcomings, the greatest is that it is
> > > > limiting in the sense that only one level of membership is possible,
> > > > that is between solid blocks and collections of them, any collection
> > > > of several blocks would not be able to be a member of any object since
> > > > it does have separate proper parts. So this would only be enough to
> > > > interpret flat sets.

>
> > > > If we desire having a hierarchy of sets being members of other sets
> > > > and if we want also to keep the above background of thinking of
> > > > matters in terms of parts and wholes and contact etc.., then we need a
> > > > more complex approach, one of those would be to invoke TIME in the
> > > > picture!

>
> > > > This without doubt would complicate the whole picture but yet it does
> > > > supply us with some interpretation of sets and their membership.

>
> > > > Now instead of having a binary relation C to represent contact, we
> > > > upgrade that to a triple relation symbol C(x,y,t) to signify x in
> > > > contact with y at moment t.

>
> > > > This would revolutionize how matters are tackled here. So for example
> > > > if at moment t1 we have a whole apple P being a solid block i.e. it is
> > > > not in contact with any object and all its proper parts in contact
> > > > with some proper part of it, then P would be said to be a block at t1.
> > > > However this doesn't mean that apple P would also remain in this block
> > > > status, possibly at moment t2 the same apple P had been cut into two
> > > > separate halfs, so at moment t2 P is a whole of two separate blocks P1
> > > > and P2 and no longer being as a solid block, so at moment t2 P cannot
> > > > be an element of any object, while at moment t1 P could have been.

>
> > > > This development would introduce us to the concept of MEMORY traces!
> > > > and of course the introduction of a new binary primitive 'memorized
> > > > in' or 'carved in'.

>
> > > > A memory trace is supposed to be a record of objects in block status.
> > > > So for example suppose that an rock was immersed in some mud at moment
> > > > t1 and thus left its print on that piece of mud, then after a while
> > > > that rock was broken into two smaller parts, and one part also fell
> > > > down on another part of the mud and made another print on that piece
> > > > of mud, so suppose that mud remained like that for years, now this
> > > > piece of mud have the prints of the whole rock at solid status and
> > > > also of a part of that rock at solid status, this piece of mud would
> > > > be considered as a 'memory trace'.

>
> > > > Now we would coin another interpretation of 'set' as a 'memory trace'.
> > > > Membership would be of objects in solid status carved in the memory
> > > > trace.

>
> > > > so x carved in y  or x memorized in y is taken to mean that x is a
> > > > solid block at some moment t and memorized as such in y.

>
> > > > So we have the axiom.
>
> > > > x M y -> Exist t. x is solid at t.
>
> > > Even more appropriate is to stipulate 'carved in' as a three place
> > > relation symbol, so Cv(x,y,t) would mean x is carved in y at moment t.

>
> > > Now we can 'define' a binary relation M standing for 'memorized in'
> > > as:

>
> > > x M y <-> Exist t. x is solid at t & Cv(x,y,t)
>
> > > here of course what is meant by x M y is: x memorized as a solid
> > > object in y. And of course we can interpret set membership by the
> > > relation M defined above, and of course sets would be memory traces.

>
> > > Zuhair
>
> > > > So membership can be interpreted as this memorizing relation and sets
> > > > can be interpreted as 'memory traces'

>
> > > > It is natural to assume identity of memory traces after what is
> > > > memorized in them.

>
> > > > It needs to be stressed that memory traces are NOT the wholes of what
> > > > is memorized in them! since the whole of an apple and a half of it is
> > > > the apple itself and it is not different from the whole of three
> > > > thirds of it, but the memory traces of those are different!

>
> > > > However if an object do not change its solid status over time, i.e. if
> > > > we have the following property:

>
> > > > for all t. x is solid at t
>
> > > > then wholes (i.e. totalities) of such objects can be taken to be
> > > > memory traces of them since time is not having any differential effect
> > > > on those kinds of objects.

>
> > > > So all in all, sets here can be interpreted as memory traces and set
> > > > membership as memorizing objects in solid block status.

>
> > > > Zuhair
>
> > If we take the fossil example as a case of memory trace, then what is
> > imprinted in the fossil is the image of the memorized object not the
> > object itself. This gives me the idea of memory traces being of images
> > of memorized object as it occurs actually normally everyday in our
> > minds. We need to stipulate that an image preserves the contact status
> > of the imaged. However it is also plausible to state that all images
> > are disjoint.

>
> In reality what is plausible is to state that all images of objects in
> solid status
> are disjoint.
>
> Also it is desirable to have a unique image of an object at solid
> status.
>
> To make the formal workup we need to introduce the primitive three
> place
> relation symbol  Img(x',x,t) to signify x' is the image of x at moment
> t.
>
> Now the memorizing relation M would be defined as:
>
> x M y iff  Exist x',t. x is solid at t & Img(x',x,t) & x' Part of y.
>
> Of course we need y to be a whole of images of objects at solid status
> and this what memory trace would mean.
>
> Anyhow this needs further workup to be completed.
>
> So the images of an apple and its half at another moment
>

"temporal reasoning in artificial intelligence"

Time and Time Again: The Many Ways to Represent Time
James F Allen
The University of Rochester

I. Representations Based on Dating Schemes

A good representation of time for instantaneous events, if it is
possible, is using an absolute dating system. This involves time
stamping each event with an absolute real-time, say taken off the
system clock on the machine, or some other coarser-grained system such
as we use for dating in everyday life. For instance, a convenient
dating scheme could be a tuple consisting of the year

II. Constraint Propagation Approaches

There has been a considerable amount of work in Artificial
Intelligence in defining temporal reasoning systems that used the
technique of constraint propagation. These systems use a graph-based
representation where each time is linked to each other time with an
arc labeled with the possible temporal relationships between the times

III. Duration-Based Representations

With the exception of the first technique using absolute dates, we
have been ignoring the problem of representing temporal durations. In
this section we will examine some representations that operate
primarily using duration information. The basic technique for dealing
with duration information is seen in PERT networks. This
representation maintains a partial ordering of events in an acyclic
directed graph that
has both a distinguished beginning and ending event. Each node in the
graph represents an event and has an associated duration.

IV. Temporal Logics

So far we have only discussed the representation of temporal
information. For such a capability to be useful it must be embedded
within a more general representation that can encode general
assertions about the world. The simplest form of such a representation
assumes a linear time model and indexes each fact
with a time - e.g. in a logic-based representation the time could
simply be an extra argument. Thus Green(FROG1,T1) might assert that
the object FROG1 was green at time T1.

V. Temporal Knowledge Representation Systems

Given this discussion, it remains to be seen how a temporal reasoner
such as the ones described above can be embedded into an actual
reasoning system. In some cases, the representation is restricted
enough that the temporal reasoning can be incorporated directly. For
example, consider a database application where all
relations are homogeneous predicates, and time intervals are
represented by pairs of dates, or pseudodates. The temporal reasoning
is then built into the retrieval mechanism: to retrieve a relation R
between dates d1 and d2, the system uses its normal retrieval
mechanism to find candidate matches - say R between d3 and d4, and
then checks if d3 <= d1 and d4>= d2. If so the retrieval succeeeds.

Herc
--
www.BLoCKPROLOG.com

Date Subject Author
2/6/13 Zaljohar@gmail.com
2/6/13 Zaljohar@gmail.com
2/6/13 Graham Cooper
2/8/13 Zaljohar@gmail.com
2/9/13 JT
2/10/13 Zaljohar@gmail.com
2/10/13 Graham Cooper
2/10/13 Zaljohar@gmail.com
2/11/13 JT
2/8/13 Graham Cooper
2/8/13 Graham Cooper