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Lambda calculus is a type of formal system from mathematical logic used in computer science for function definition, application and recursion.
Functional computer programming arose out of this type of calculus, and acts in turn as a model for software development via providing simple semantics for computation with functions. Unlike with imperative functions, which can change program status because they don't have referential transparency, functional programming doesn't change program status.
Lambda calculus makes language code threads result in exactly the same values, regardless of the time or machine the code thread was run on. This is not the case with imperative function programming, which typically has unpredictable program responses.
Functional programming can be realized in any programming language, even those that were not designed with default functional programming structures.