I did, should you (3)
As you may remember, my goal for reading this paper is a better understanding of the theories behing QSR and thus be able to make a better decision regarding what theory to use to represent my problem. I know, I haven’t talked about what “the problem” is yet, but suffice it to say I want to reason about some objects in space.
So far, I’ve concluded that I needed spatial qualitative reasoning (instead of qualitative reasoning); I have decided on my spatial entities (regions) and topology (to describe the relationships among my entities) and I have defined my basic relationships between my entities (binary relations). So let’s go on, what other factors do I need to consider?
It has been shown, that there are three primitives necessary to define a theory, how they are defined will depend on the notions for “connection” used. We need to define Connection, Parthood and Fusion. All of these use a notion of “connected”; are these three notions the same or different? Actually, the first question should be: Is there a difference between connection and parthood? If you think of a region as being a set of “things”, these “things” are held together and connected; is that the same kind of “connection” that happens when to regions touch? Should you define connection and parthood based on the same connection primitive or not? Is everything within a region connected the same way as regions that connect?
There are three definitions each for Connection (C), Parthood (P) and Fusion(\sigma). We can describe what theory we are going to use by the triple <i,j,k>. For exaple, <3,1,2> means that uses C3 to define topological connection, C1 to define Parthood and C2 to define Fusion. C1 means that to sets are connected iff they share an element. C2 means that two sets are connected iff one set shares an element with the closure of another. C3 means that tow sets are connected iff the sets closures share an element. We can now define Connection, Parthood and Fusion, independent from, dependent on or anything in between each other. <3,1,2> is on of six theories that uses three independent primitives. There are three uniform theories and six theories where Parthood is defined in terms of topological connection, but fusion is an indpendent primitive.Depending on the type chosen, a few more axioms may be necessary, check page 11 of the paper for details.
You will also need to decide if boundaries matter or not. When two objects touch and you zoom in really, really, really close, what does it look like: Object A ends, Object B starts; or Object A ends, “the boundary”, Object B starts?
As it turns out <i,i,i> type theories don’t work well with boundaries. So if you wanted a <2,2,2> theory and boundaries you will need to go back through the paper and figure out where your requirements collide. Additionally, it actually seems that <2,1,1> is the only viable theory which will work with boundaries.
I believe that boundary-free theories are the way to go. I need a theory to solve a real-world problem. So I want a theory that makes sense for my problem. I also believe that a workable theory should not be hard to find, hence boundary-tolerant doesn’t seem as promising as boundary-free.
On the other hand, most existing boundary-free theories are uniformely typed. My believe is that there is a difference between connection and parthood. For our problem I think we should be looking at <3,1,k> or possible <1,3,k> theories. Unfortunately, there are not many others; on the bright side, fewer papers for me to read.
Thus ends the discussion about mereotopological representation for qualitative spatial reasoning. The paper contineues, however, and covers other theories such as topology via “n-interections”, as well as theories between mereotopology and fully metric spatial representation.
QSR, Research, bioinformatics | julia | 
March 23, 2009 10:58
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