Lambda calculus is a mathematical system focused on function abstraction and application, using variables and substitution. The simplicity of lambda calculus revolves around three rules: variables, lambda abstractions, and function applications. Lambda terms in the system follow rules for validity based on variables, abstractions, and applications. Reduction operations in lambda calculus involve alpha conversion and beta reduction, with eta reduction emphasizing extensionality. The Y combinator, a fixed-point combinator, is a key concept allowing for recursive function calls without named references. Implementation in Scheme requires delaying evaluation to avoid infinite recursion. The Y combinator has practical applications in recursive function definitions, providing a unique approach to handling functions in languages like Scheme.
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