A Calculus of Inheritance

Abstract

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, denotational, and computable by tabling, even for cyclic inheritance hierarchies. These three properties extend to the $λ$-calculus, since Böhm tree equivalence 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; ordinary arithmetic yields the relational semantics of logic programming; $\mathtt{this}$ resolves to multiple targets; and programs are immune to nonextensibility in the sense of the Expression Problem. This makes inheritance-calculus strictly more expressive than the $λ$-calculus in both common sense and Felleisen's sense.