A computer analysis tool for structural decomposition using entropy metrics

Abstract

The decomposition of a metric space into successive subregions exhibiting distinctive characteristics is a problem of broad application. In pattern classification, the object is to partition the space such that pattern classes are easily separable; that is, so that each subregion of the partition contains predominantly samples of only one class. In piece-wise-constant approximation the decompositions produced contain samples whose values are sufficiently close to allow approximation with a specified degree of accuracy. In defining software it is quite often necessary to derive a structural model of a computer program which contains modules, i.e., partitions exhibiting the flow relations or connectivities among the elements (statements) in a program. The subsequent analysis and manipulation of the structural model produces useful design alternatives that enhance the operational qualities of the software generated in terms of program control, logic paths, data transfer and other relevant software issues. The basic feasibility of this approach has been demonstrated by numerous investigators. 1 - 5 However, the analytical and diagnostic tools for performing structural decompositions require further refinement and development. For example, the metrics usually used 6 , 7 for defining the topology of a given software structure are primarily single attribute measures. Although the entropy metric proposed in this paper is metrizable in terms of its hypergraph representation, 8 the extension to a multi-attribute unique formulation is, as yet, elusive. This is because an all-purpose problemindependent metric space places unrealizable constraints on the structure it proposes to define. Thus, as Koontz et al. 9 point out, even when a metric is given and a structure well known, the notion of neighboring points can not be rigorously defined for finite point sets from a computational point of view, since the simplest Euclidean distance measure must be scaled by a factor indicating its own respective distance to the nearest neighbor in order to avoid overlapping and ambiguous regions. Although conceptually, the construction of a neighborhood and the determination of the limit point of a sequence of real numbers is a widely used idea, a more fundamental requirement for metrizable hyper-spaces is that of specifying the existence of a limit point of a set. The resultant necessary and sufficient conditions for identifying metrizable spaces is given by Hausdorff. 10 However, equivalent normalizations and the use of discrete semi-metrics over a restricted space have precluded some of these inherent problems in the quest for such a unique, multi-attribute metric. Thus, the primary emphasis is to obtain realizable decompositions using readily-implementable metrics, as well as focus upon suitable partitioning alternatives in terms of identifying mathematically consistent criteria for structural decompositions.