A Calculus of Inheritance
This work addresses foundational issues in declarative programming for language designers and theorists, offering a novel theoretical framework.
The paper tackles the problem of inheritance in programming languages by introducing inheritance-calculus, which unifies various constructs under a record abstraction and models inheritance as set union, eliminating the multiple-inheritance linearization problem. It shows that this calculus is more expressive than the λ-calculus and establishes a convergence hierarchy among λ-calculi.
Just as the $λ$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, inheritance-calculus uses three primitives (record, definition, inheritance) as the foundation of declarative programming. By unifying modules, classes, objects, methods, fields, and locals under a single record abstraction, the calculus models inheritance simply as set union. Consequently, composition is inherently commutative, idempotent, and associative, structurally eliminating the multiple-inheritance linearization problem. Its semantics is first-order~\cite{vanemden1976-predicate-logic-semantics, reynolds1972-definitional-interpreters, aczel1977-inductive-definitions}, denotational, and computable by tabling~\cite{tamaki1986-tabled-resolution}, even for cyclic inheritance hierarchies. These three properties extend to the $λ$-calculus, since Böhm tree equivalence~\cite{barendregt1984-lambda-calculus} is fully abstract for the first-iteration approximation of a sublanguage of inheritance-calculus. As a corollary, this establishes a convergence hierarchy $\text{eager} \subsetneq \text{lazy} \subsetneq \text{fixpoint}$ among $λ$-calculi sharing the same $λ$-syntax. Inheritance-calculus is distilled from MIXINv2, a practical implementation in which the same code acts as different function colors~\cite{nystrom2015-function-color}; ordinary arithmetic yields the relational semantics of logic programming~\cite{vanemden1976-predicate-logic-semantics}; $\mathtt{this}$ resolves to multiple targets; and programs are immune to nonextensibility in the sense of the Expression Problem~\cite{wadler1998-expression-problem}. This makes inheritance-calculus strictly more expressive than the $λ$-calculus in both common sense and Felleisen's sense~\cite{felleisen1991-expressive-power}.