| dc.description.abstract | Though I used Prolog, the Ray prototype knowledge system is not just a backward chaining system. Qualitative reasoning is modeled by a rule language with an interpreter for expressing outcomes in terms of values from discrete ordinal scales. The rule interpreter proceeds forwards from conditions to conclusions, in contrast with Prolog. In the earlier chapters, quantitative analysis is done using Prolog to solve discrete optimization problems.
From the efficiency point of view, the structure of the rules is important. Even the ordering of the facts in the algorithmic knowledge base is important. A consistent vocabulary has to be maintained. In my prototype knowledge system, there is no way to check the consistency of newly added rules. The knowledge engineer who maintains the system has to assume the responsibility of maintaining the consistency of newly added rules.
In spite of these limitations, a working knowledge system with a couple of hundred rules and facts is easily realizable on the basis of this prototype. Powerful Prolog compilers are currently available even on personal computers. Walker (1987) is a testimony to the power of Prolog for building knowledge systems.
Chapter 4: Conclusions 4.0 Logic of Multilevel Clustering
To recapitulate, preanalysis for two-level clustering results in a discrete optimization problem with a single unknown, i.e., the number of blocks the dataset is to be divided into at the first level. For a two-level hierarchical agglomerative clustering algorithm, solution of a cubic equation followed by a simple rounding procedure will ensure optimal choice to minimize the number of similarity measures to be calculated. In the case of a hybrid clustering algorithm, the objective function is proved to be monotonic, and thus extreme point solutions are obtained as optimal solutions. It is also proved that two-level clustering is optimal if and only if N is a product of two prime numbers.
I have no hesitation in concluding that Horn clause logic and Prolog have proved to be suitable for axiomatizing the logic of two-level clustering. But traditional mathematics is also necessary to derive new knowledge which can be formalized in Horn clause logic and Prolog.
4.1 Optimal Number of Levels
Preanalysis for multilevel clustering results in multidimensional discrete optimization problems. The optimal number of levels is also to be decided. A straightforward Horn clause specification of the logic of multilevel clustering together with the logic of optimization results in an almost working Prolog program for the case of blocks with equal number of elements. From the results of the program, it is noticed that the recursive part of the multilevel objective function can be handled by a simple scheme.
I would like to conclude that the problem of optimal number of levels for a multilevel clustering method is solved by the Horn clause specification transformed into Prolog. As pointed out earlier, I feel that in the case of unequal size blocks, some constraints are to be imposed if the whole exercise of finding the optimal number of levels is not to degenerate into a triviality. The equal size block case has the inherent constraint of equality, which makes the optimization exercise interesting. And what’s more, the equal size block case may arise naturally because of the data producing mechanism. Anyway, further research is needed to settle the practical relevance of these two models.
4.2 Possible Further Research
Preanalysis for multilevel clustering should also include reasoning about possible proximity measures and their relevance to cluster algorithm selection. Also, my work deals mostly with hierarchical agglomerative clustering algorithms because of the versatility these algorithms supposedly possess in recognizing clusters of different shapes and sizes. I use the assumption that this versatility is invariant with the selection of parameters of multilevel clustering, providing rationale for the optimization problems solved in this thesis. This crucial assumption needs to be verified by further research in various domains like remote sensing. The verification may be by conducting a lot of empirical experiments or by proving theorems similar to the ones developed in Negi (1989). If further research points out the necessity of imposing additional restrictions on the values of P or K, those restrictions can easily be added within the logic programming framework.
Prolog, though admired for its elegance, is usually decried for lack of efficiency. If preanalysis is to be done online, efficiency considerations become important. Further research can aim at producing efficient Prolog programs which can be put to use in a practical setting.
My model of cluster algorithm selection by qualitative reasoning is only a beginning. It needs further research to automate the design of clustering algorithms in a domain-independent fashion. The subarea of machine learning in artificial intelligence is relevant for this purpose. A computer system capable of making better algorithm selections after being commented upon by an expert about the merits or demerits of the earlier selections is said to be exhibiting learning behaviour. The slogan ALGORITHM = LOGIC + CONTROL, coined by Kowalski (1979), can be used to advantage. If in some sense the logic component is fixed, learning a new clustering algorithm can be reduced to that of learning control. Minton (1989) implements a domain-independent algorithm based on explanation-based learning for the purpose of learning search control.
A more ambitious research effort would be to build a knowledge-based clustering environment including preanalysis of the clustering problem, algorithm selection, algorithm execution, and cluster validation followed by algorithm modification. Jain and Dubes (1989) in their book on algorithms for clustering data emphasize the need for cluster validation. Negi (1989) proposes a design based on an object-oriented scheme to represent different clustering algorithms. Such proposals can be attempted in the logic programming framework, and Prolog programs can be built. | |
