When the work on this thesis started, the relationship between FOPCL knowledge bases and conditional structures was to a certain extend not clear. There were speculations that the PU transformation rules could be applied in a similar way on the CS-level and that there are one-to-one relationships between FOPCL knowledge bases and their related CSs.
We have shown in chapter 3 that these assumptions don’t hold, which we expressed in assumption 1. This likely to be the most important result of this thesis, but also a very unfortunate one. First of all it destroys the possibility of switching freely between the processing of knowledge bases and CSs, so that different processing steps can be done on the level wherever they are performed easiest. Second it also forced us to look for other objectives for this thesis.
We have tried to break down the different factors which influence the relationship between FOPCL knowledge bases and their CSs. In order to do so in the given time we restricted ourselves to atomic conditionals only, left out negated predicates and in several cases only allowed for certain types of instantiation restrictions. We saw that different types of instantiation restrictions (local and non-local) have different impact on the CS. It also became clear that cases of imbalanced usage are not possible to be detected or handled on CS-level whilst cases of imbalanced sharing seem to be visible on CS-level.
At the early stages of our work it still seemed possible that there are straight forward ways to calculate the CS of a knowledge base simply out of the number of ground atoms and their relationships (e.g. how many of them overlap between two conditionals). We have shown in chapter 6 that this assumption was a bit naive, as the calculation of conditional contributions (not even conditional structures) works only in very restricted cases.
Finally the idea of c-segment reduction turned out to be an approach for easier generation of conditional contributions. We have shown the RCS method in detail in chapters 7 and 8 and we believe that the findings there, especially proposition 19 and assumption 2 are the major positive findings within this thesis.
But unfortunately once we want to put together the conditional contributions (gained by RCS) to a CS we see that all the advantage gained by RCS is eaten up by the complex and resource hungry process of combining the conditional contributions, as we discussed in chapter 9.
We found by chance that the patterns in the CA-tables can be interpreted in certain ways. It is not clear at this moment whether this "patter reading" in the end turns out to be only an esoteric exercise or if it can be developed into a useful mechanism, which might reveal even more information about the relationship between the two conditionals. The findings shown in chapter 10 can only be regarded as a starting point.
Finally we looked again whether the PU-transformations could at least partially be performed on CS-level. We saw in chapter 11 that this is only possible for a very simple case and that it is likely that all other cases will not work. Still, we found out that using the maximum ordered sums is of advantage at least in the simple case for which PU-transformation on CS-level works.
We preceded this thesis with a cite from Stanislaw Lems novel Golem XIV ([12]). Although the work on CS and PU is not comparable with questions which engage singularities and star-made objects, the results that showed up on the way sometimes caused the feeling that things are wobbling. Nevertheless the work was very interesting and exciting – maybe this thesis helps to gain a better understanding of the reasons why the PU-CS table sometimes wobbles.