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Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus provides a simple and powerful model for computation, serving as the foundation for many programming languages and functional programming paradigms.

The core of lambda calculus consists of three types of expressions: variables, abstractions, and applications. Variables represent inputs to functions. Abstractions, denoted by the Greek letter lambda (λ), are used to define anonymous functions. Applications are used to apply functions to arguments. The syntax of lambda calculus is minimal, consisting of only these three types of expressions.

Lambda calculus is Turing complete, meaning it can express any computable function. It has been used to model and analyze the behavior of programming languages, to prove properties of programs, and to study the theoretical limits of computation. Lambda calculus has also influenced the design of functional programming languages, such as Lisp, Haskell, and ML, which use similar concepts of functions and higher-order functions.

Despite its simplicity, lambda calculus is a rich and expressive system that has had a profound impact on the development of computer science and mathematics. Its elegant and concise notation has made it a valuable tool for reasoning about computation and for understanding the fundamental principles of programming languages.