Inheritance Is Subtyping

Since Luca Cardelli wrote a seminal paper [3] on the semantics of inheritance in 1984, program-ming language researchers have constructed a variety of structural models of object-oriented programming (OOP) founded on Cardelli’s work. Since Cardelli approached OOP from the perspective of functional programming, he identified inheritance with record subtyping—an el-egant choice in this context. Although Cardelli did not formally define inheritance, he equated it with record extension and proved that for a small functional language with records, variants, and function types–but no recursive record types–that syntactic and semantic record subtyping were equivalent. William Cook et al [4] subsequently added ThisType and recursive record types, narrowing the typing of this in inherited methods, and reached a profoundly different conclusion: inheritance is not subtyping. Meanwhile, object-oriented (OO) program design emerged as an active area of research within software engineering, spawning class-based OO languages like C++, Java, and C#, which strictly define inheritance in terms of class hierarchies. In these languages, subtyping is identified with inheritance. In contrast to Cardelli’s expansive formulation of inheritance base