Fuzzy logic methods have been used successfully in many real-world applications, but the foundations of fuzzy logic remain under attack. Taken together, these two facts constitute a paradox. A second paradox is that almost all of the successful fuzzy logic applications are embedded controllers, while most of the theoretical papers on fuzzy methods deal with knowledge representation and reasoning. I hope here to resolve these paradoxes by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential. My conclusions are based on a mathematical result, on a survey of literature on the use of fuzzy logic in heuristic control and in expert systems, and on practical experience developing expert systems. An apparent paradoxAs is natural in a research area as active as fuzzy logic, theoreticians have investigated many formal systems, and a variety of systems have been used in applications. Nevertheless, the basic intuitions have remained relatively constant. At its simplest, fuzzy logic is a generalization of standard propositional logic from two truth values, false and true, to degrees of truth between 0 and 1.Formally, let A denote an assertion. In fuzzy logic, A is assigned a numerical value t(A), called the degree of truth of A , such that 0 5 t(A) I 1. For a sentence composed from simple assertions and the logical connectives "and" (A), "or" (v), and "not" ( 1 ) degree of truth is defined as follows: MIT Press, 1993, pp 698-703 Definition 1: Let A and B be arbitrary as- sertions. Then t ( A A B ) = min [ t(A), t(B)) t(A v B ) = max { t ( A ) , t ( B ) ] t(A) = t(B) if , either t ( B ) = t ( A ) or t(B) = 1-t(A). WA direct proof of Theorem 1 appears in the sidebar, but it can also be proved using similar results couched in more abstractProposition: Let P be a finite Boolean algebra of propositions and let z be a truthassignment function P + [0,1], supposedly truth-functional via continuous connectives. Then for all p E P, Q) E { 0, 1 ] WThe link between Theorem 1 and this proposition is that l ( A A 4) = B v (4 A -IB) is a valid equivalence of Boolean algebra. Theorem 1 is stronger in that it relies on only one particular equivalence, while the proposition is stronger because it applies to any connectives that are truth-functional and continuous (as defined in its authors'The equivalence used in Theorem 1 is rather complicated, but it is plausible intupaper).itively, and it is natural to apply it in reasoning about a set of fuzzy rules, since 7 ( A A 4 ) and B v (4 A 4 ) are both reexpressions of the classical implication 4 4 B. It was chosen for this reason, but the same result can also be proved using many other ostensibly reasonable logical aquivalences.It is important to be clear on what exactly Theorem 1 says, and what it does not say. On the one hand, the theorem applies to any more general formal system that includes the four postulates listed in Definition 1. Any extension of fuzzy logic to accommodate first-order sentences, for example, collapses to two trut...

Abstract-Mammograms are X-ray images of the breast which are used to detect breast cancer. When mammograms are analyzed by computer, the pectoral muscle should preferably be excluded from processing intended for the breast tissue. For this and other reasons, it is important to identify and segment out the pectoral muscle. In this paper, a new, adaptive algorithm is proposed to automatically extract the pectoral muscle on digitized mammograms; it uses knowledge about the position and shape of the pectoral muscle on mediolateral oblique views. The pectoral edge is first estimated by a straight line which is validated for correctness of location and orientation. This estimate is then refined using iterative "cliff detection" to delineate the pectoral margin more accurately. Finally, an enclosed region, representing the pectoral muscle, is generated as a segmentation mask. The algorithm was found to be robust to the large variations in appearance of pectoral edges, to dense overlapping glandular tissue, and to artifacts like sticky tape. The algorithm has been applied to the entire Mammographic Image Analysis Society (MIAS) database of 322 images. The segmentation results were evaluated by two expert mammographic radiologists, who rated 83.9% of the curve segmentations to be adequate or better.

Abstract-Mammograms are X-ray images of the breast which are used to detect breast cancer. When mammograms are analyzed by computer, the pectoral muscle should preferably be excluded from processing intended for the breast tissue. For this and other reasons, it is important to identify and segment out the pectoral muscle. In this paper, a new, adaptive algorithm is proposed to automatically extract the pectoral muscle on digitized mammograms; it uses knowledge about the position and shape of the pectoral muscle on mediolateral oblique views. The pectoral edge is first estimated by a straight line which is validated for correctness of location and orientation. This estimate is then refined using iterative "cliff detection" to delineate the pectoral margin more accurately. Finally, an enclosed region, representing the pectoral muscle, is generated as a segmentation mask. The algorithm was found to be robust to the large variations in appearance of pectoral edges, to dense overlapping glandular tissue, and to artifacts like sticky tape. The algorithm has been applied to the entire Mammographic Image Analysis Society (MIAS) database of 322 images. The segmentation results were evaluated by two expert mammographic radiologists, who rated 83.9% of the curve segmentations to be adequate or better.

Abstract-Five methods that generate multiple prototypes from labeled data are reviewed. Then we introduce a new sixth approach, which is a modification of Chang's method. We compare the six methods with two standard classifier designs: the 1-nearest prototype (1-np) and 1-nearest neighbor (1-nn) rules. The standard of comparison is the resubstitution error rate; the data used are the Iris data. Our modified Chang's method produces the best consistent (zero errors) design. One of the competitive learning models produces the best minimal prototypes design (five prototypes that yield three resubstitution errors).Index Terms-Competitive learning, Iris data, modified Chang's method (MCA), multiple prototypes, nearest neighbor (1-nn) rule.

We study indices for choosing the correct number of components in a mixture of normal distributions. Previous studies have been confined to indices based wholly on probabilistic models. Viewing mixture decomposition as probabilistic clustering (where the emphasis is on partitioning for geometric substructure) as opposed to parametric estimation enables us to introduce both fuzzy and crisp measures of cluster validity for this problem. We presume the underlying samples to be unlabeled, and use the expectation-maximization (EM) algorithm to find clusters in the data. We test 16 probabilistic, 3 fuzzy and 4 crisp indices on 12 data sets that are samples from bivariate normal mixtures having either 3 or 6 components. Over three run averages based on different initializations of EM, 10 of the 23 indices tested for choosing the right number of mixture components were correct in at least 9 of the 12 trials. Among these were the fuzzy index of Xie-Beni, the crisp Davies-Bouldin index, and two crisp indices that are recent generalizations of Dunn's index.