OWL Cardinality
After posting my problems with cardinality in an open world, I've been giving it some further thought. Coincidentally, Andrew made a comment which closely reflected the opinion that I have come to.
If minimum cardinality means nothing in an open world, then there was very little point in including it in the OWL language. Whoever did so probably forgot about the open world assumption at the time. I can't blame them, as I find it very easy to forget. So I think I should assume that the intent was for this to operate in a closed world assumption. Similarly, I have to assume the same about the intent of maximum cardinality.
I'm still a little uncomfortable about switching from an open world assumption over to a closed world assumption without warning. I'm wondering if there should be some kind of user-settable flag which would allow cardinality to be determined in a closed world. Whether I do this or not, I should at least look at the iTQL structure for both types of queries. Once I have them both it would not be too much work to switch between the two paradigms.
Unfortunately, owl:sameAs makes the closed world assumption difficult as well.
Calculating
For either type of cardinality, any statements which are declared owl:sameAs will need to be considered to be a single statement. In this case, we need to count all objects in the relevant statements (relevant here means that the predicate is the one that the cardinality constraint applies to), which are also the owl:sameAs other objects in the relevant statements. If the number of these objects is n, then we need to reduce the count of the total relevant statements by (n-1).
An alternate method of counting, would be to find and count all relevant statements whose objects are not declared the owl:sameAs anything. Then count all the groups of owl:sameAs objects which appear in relevant statements, and add this to the total.
Unfortunately, both methods here require arithmetic operations (addition and subtraction). iTQL does not support this, though one day it may have to. While I need it here, I'm not sure that I want to add it just yet. I may have to consider a non-iTQL approach. I'll give it further thought before I commit to anything.
I should note here, that because of inferencing we can be sure that all owl:sameAs statements will be fully symmetric by the time constraints are being checked. Similarly, inferencing means that if one object appears in a statement, then all other objects which are the owl:sameAs it will also appear in similar statements. At least this allows for a certain consistency when dealing with these statements.
3 comments:
Some of your comments bothered me, since they did not match my understanding of OWL's lineage. I checked with Jim Hendler and Bijan Parsia at the University of Maryland and they assure me that the cardinality restricitions in OWL Full and DL are Open World and were intended to be so. That does mean that you can't validate using them. To a logician, consistency checking is a different concept from validation and I'm pretty sure you meant validation.
Bijan particularly points you to the DL Handbook for further reading.
Other people working on OWL reasoning using secondary storage include InstanceStore from the University of Manchester and DLDB by Jeff Heflin at Lehigh University. You might want to speak with them.
OWL was designed to be Open World. Validation was not intended by OWLs designers.
My suggestion is to (ab)use the OWL cardinality constraints for validation purposes and assume a closed world, since validation seems to be more frequently required.
Stefan
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