Just as the $λ$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, Overlay-Calculus uses three primitives (record, definition, inheritance) as the foundation of declarative programming. It trivially embeds the $λ$-calculus, although the entire semantics builds on only naive set theory; as a consequence, all constructs including inheritance are inherently commutative, idempotent, and associative; the linearization problem of multiple inheritance simply does not arise. This induces a fully abstract semantics of the lazy $λ$-calculus with respect to Böhm tree equivalence. Overlay-Calculus is distilled from the Overlay language, a practical implementation in which we observed further emergent phenomena: the Expression Problem dissolves, programs are CPS-agnostic, records natively encode random-access memory, and self-reference resolves to multiple targets. These properties suggest applications to configuration languages, dependency injection, object-oriented programming, composable effect systems, modular software architectures, file-system-as-compiler, general-purpose programming, and no-code development.

翻译：正如$λ$-演算使用三个基本要素（抽象、应用、变量）作为函数式编程的基础，覆盖演算使用三个基本要素（记录、定义、继承）作为声明式编程的基础。它平凡地嵌入了$λ$-演算，尽管整个语义仅建立在朴素集合论之上；因此，包括继承在内的所有构造本质上都是可交换、幂等且可结合的；多重继承的线性化问题根本不会出现。这导出了关于Böhm树等价的惰性$λ$-演算的完全抽象语义。覆盖演算提炼自Overlay语言，这是一种实际实现，我们在其中观察到更多涌现现象：表达式问题消解，程序对CPS无感知，记录原生编码随机存取存储器，自引用可解析为多个目标。这些特性表明其在配置语言、依赖注入、面向对象编程、可组合效应系统、模块化软件架构、文件系统即编译器、通用编程以及无代码开发等领域具有应用潜力。