The theory of classification part 5: axioms, assertions and subtyping

The algebraic calculus for reasoning about the complete behavior of object types and the effects of axioms upon subtyping were analyzed. The translation of pure algebra into a piecemeal treatment in terms of variants, pre-, and post conditions was studied. The existing object subtyping rules were applied to derive subtyping rules governing the strengthening, or weakening of the assertions as there was a direct relationship between axiom strengthening, and subtyping. It was found that weaker preconditions co-existed with stronger invariants, and the same system satisfied the stronger of the two

NOOP: A mathematical model of object-oriented programming

Computer software is ubiquitous. More than 35 × 10 18 computer instructions are executed around the globe each second. As computers dominate more aspects of our lives, there is a growing need to reason more accurately about computer software. Most contemporary computer software is written using object-oriented (OO) programming languages, such as J AVA, C#, and C++. How should we mathematically characterize object-oriented software? This is the question this thesis addresses by presenting an accurate domain-theoretic model of mainstream object-oriented programming. Mainstream object-oriented languages are class-based. In such languages, the name of a class is part of the meaning of an object, a property often called "nominality". Most mainstream OO languages also conform to a static type discipline. Hence, the focus of this thesis is the construction of an accurate model of nominal, statically-typed OO languages. In statically-typed nominal OO languages, class names are also part of the meaning of corresponding class types, and class inheritance (subclassing) is explicitly declared; one class is a subclass of another only if it is declared as such. When static type systems are formulated to describe sets of objects, subtyping is defined so that subclassing is consistent with subtyping. Nevertheless, some programming languages (PL) theoreticians dismiss this identification as a design error because the only published models of OO languages exclude nominal information from objects and define subtyping in a way that ignores nominality. In nominal OO languages, program behavior depends on the nominal information embedded in objects. This thesis builds a model of OO languages called NOOP that includes nominal information and defines static types in accord with mainstream OO language designs. In NOOP , the meaning of every object includes its class name. Similarly, types are defined such that objects belong to a particular class type if and only if they are members of classes that inherit from the class corresponding to the class type. To demonstrate the utility of the model, we show that in NOOP inheritance and OO subtyping coincide. This work shows that mainstream OO languages are not technically defective in identifying inheritance and subtyping. In models that include nominal information and define types that respect nominal information, this identification is mathematically correct. The folklore among OO programming language researchers that "inheritance is not subtyping" is incorrect

The theory of classification part 1: perspectives on type compatibility

Object oriented type theory was used to study the syntactic and semantic type compatibility. In syntactic compatibility the components provides all the expected operations. In semantic compatibility the component's operations behave in the expected way. Various examples of failure due to type-related software design faults were discussed