Austin Melton

Proximity relations in the fuzzy relational database model

Abstract The fuzzy relational model of Buckles and Petry is a rigorous scheme for incorporating non-ideal or fuzzy information in a relational database. In addition to providing a consistent scheme for representing fuzzy information in the relational structure, the model possesses two critical properties that hold for classical relational databases. These properties are that no two tuples have identical interpretations and each relational operation has a unique result. The fuzzy relational model relies on similarity relations for each scalar domain in the fuzzy database. These relations are reflexive, symmetric, and max-min transitive. In addition to introducing fuzziness into the relational model, each similarity relation induces equivalence classes in its domain. It is the existence of these equivalence classes that provides the model with the important properties possessed by classical relational databases. In this paper, we extend the fuzzy relational database model of Buckles and Petry to deal with proximity relations for scalar domains. Since reflexivity and symmetry are the only constraints placed on proximity relations, they generalize the notion of similarity relations. We show that it is possible to induce equivalence classes from proximity relations; thus, the ‘nice’ properties of the fuzzy relational model of Buckles and Petry are preserved. Furthermore, the removal of the max-min transitivity restriction also provides database users with more freedom to express their value structures.

Deriving structurally based software measures

Abstract Many software engineering methods place internal structural constraints on the documents (including specifications, designs, and code) that are produced. Examples of such structural constraints are low coupling, high cohesion, reuse in designs and code, and control structuredness and data-abstraction in code. The use of these methods is supposed to increase the likelihood that the resulting software will have desirable external attributes, like reliability and maintainability. For this reason, we believe that the software engineering community needs to know how to measure internal attributes and needs to understand the relationships between internal and external software attributes. This can only be done if we have rigorous measures of the supposedly key internal attributes. We believe that measurement theory provides an appropriate basis for defining such measures. By way of example, we show how it is used to define a measure of coupling.

A philosophy for software measurement

Abstract We as a group—called the Grubstake Group—are convinced that software measures are essential for “controlling” software. Thus, we are dedicated to producing an environment in which software measures can be confidently used by software managers and programmers. However, we are also convinced that such an environment can only be created if there exists a formal and rigorous foundation for software measurement. This foundation will not have to be understood by the users of the software measures, but it will have to be understood by those who define, validate, and provide tool support for the measures. It is this foundation that we are introducing in this article.

Galois connections and computer science applications

We have presented an existence theorem and some important properties of Galois connections. We have also shown how data structures problems can be simplified and better understood when Galois insertions are used. In particular, the proof of correctness of an implementation follows simply from the construction of a Galois insertion. We plan further applications of Galois connections theory to computing-related problems.

Functional dependencies and normal forms in the fuzzy relational database model

Abstract The fuzzy relational database model as defined by Buckles and Petry employs sets in place of atomic values for components of tuples in database relations. This technique for dealing with imprecision in relational databases is intuitively appealing. Moreover, the model preserves several important properties of classical relational databases. In recent works we have demonstrated that the existence of partitions on scalar domains is the key to ensuring conformity with the classical relational model. Specifically, by restricting the components of fuzzy tuples to be nonempty subsets of equivalence classes from domain partitions, it is possible to define the notions of redundant tuples and consistent database relations and to specify a well-defined fuzzy relational algebra. Since these properties are obtained by working only with equivalence classes, the fuzzy relational model of Buckles and Petry is generalized to an equivalence classes model of relational databases. In this work, additional properties of the fuzzy relational database model are presented. By employing equivalence classes from domain partitions, we define functional dependencies and normal forms for the fuzzy relational model. These definitions extend the corresponding classical definitions. Moreover, our definitions of functional dependencies and normal forms provide valuable guidelines for designing fuzzy relational databases.

An extended version of the fuzzy relational database model

Abstract The fuzzy relational database model proposed by Buckles and Petry is a formal method for organizing and using fuzzy information in relational databases. The model possesses two key properties that hold for classical relational databases: no two tuples have identical interpretations and each relational algebra operation has a unique result. The original fuzzy relational database model was based on similarity relations which were defined on finite scalar domains. Buckles and Petry later extended the model to incorporate fuzzy number domains; this extension was done without loss of consistency with the representation or the relational algebra. In a recent paper we have extended the original model to proximity relations which generalize similarity relations; the key properties of Buckles and Petrys model are preserved in the extension. In this work we demonstrate that the existence of partitions on finite scalar domains in the fuzzy relational model is the key to preserving the important properties of the classical relational model. We explain how proximity relations defined on finite scalar domains are used to partition the domains. Further, we show that normal fuzzy sets can be employed as domain values. These fuzzy sets become members of fi nite scalar domains and their characteristic functions are employed in generating the proximity relations of the fuzzy relational database model.

An equivalence classes model of fuzzy relational databases

Abstract The technique of employing sets of values for tuple components to express imprecision in relational databases was proposed by Buckles and Petry in their classic works on fuzzy relational databases. In addition to providing an intuitively appealing scheme for representing fuzzy information, the model of Buckles and Petry possesses several key properties of the classical relational model. By employing finite scalar domains with similarity relations and special fuzzy number domains, Buckles and Petry have demonstrated that the classical properties of uniqueness of tuple interpretations and well-definedness of the relational algebra can be retained in the fuzzy relational database model. The key to the preservation of these properties is the fact that scalar domains with similarity relations and the fuzzy number domains can be partitioned into equivalence classes. However, since equivalence classes can be constructed without assuming the existence of similarity relations or special fuzzy number domains, it is desirable to generalize the fuzzy relational database model to one based only on equivalence classes. In this work we show that the important properties of classical relational databases (and of fuzzy relational databases) are preserved in a generalized model built on equivalence relations on finite database domains. Further, we generalize the notion of a functional dependency to the fuzzy relational model.

A synthesis of software science measures and the cyclomatic number

A solution is obtained to the problem of defining a software measure or a family of measures which simultaneously detect those aspects of software complexity that are detected by the software science measures and the cyclomatic number. The authors present a family of measures, called weighted measures that is built on the software science measures by adding weights to certain operators and operands; the size of the weights is determined by a theorem which relates nesting levels and the cyclomatic number. Thus, by construction the weighted measures synthesize the software science measures and the cyclomatic number. Further, by applying the weighted measures, the software science measures, and the cyclomatic number to sample programs, it is shown that the weighted measures also synthesize in practice the software science measures and the cyclomatic number. >

A standard representation of imperative language programs for data collection and software measures specification

Abstract Software measures and software tools are often defined in terms of a particular, limited programming language. For example, a number of software measures are defined only for structured programs. Several approaches to program testing and debugging are defined using a specific simple language. As a result, implementing tools and measures so that they can be applied to “real” programs in “real” programming languages is difficult. Further, independent evaluation and comparison of measures and tools is difficult. We propose a standard representation of imperative language programs that is independent of the syntax of any particular programming language, and that supports the definition of a wide range of tools and measures. Additionally, the standard representation masks the actual program semantics. Thus the standard representation provides a vehicle by which large volumes of industrial software can be made available to researchers while protecting the proprietary nature of the programs.

A Category of Galois Connections

We study Galois connections by examining the properties of three categories. The objects in each category are Galois connections. The categories differ in their hom-sets; in the most general category the morphisms are pairs of functions which commute with the maps of the domain and codomain Galois connections. One of our main results is that one of the categories—the one which is the most closely related to the closed and open elements of the Galois connections—is Cartesian-closed.