Just as the $\lambda$-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 $\lambda$-calculus, since B\"ohm 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 $\lambda$-calculi sharing the same $\lambda$-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 $\lambda$-calculus in both common sense and Felleisen's sense~\cite{ felleisen1991-expressive-power}.
Categories: cs.PL cs.SE