Church's formal system used in computatability, programming languages and proof theory to represent effective functions, programs and their computation, and proofs.
Its dynamics is a rewriting system representing computation by mean of the beta-reduction, that substitutes each occurrence of the variable with the argument term. From this operational point of view, it represents the ideal formal model of a functional programming language and it can be enriched with constant values and datatypes. Such a typed restriction constitutes the bridge across logic's proof-theory and computer science's theory of computability known as the Curry-Howard correspondence: a term is a proof of its type formula.
- Henk Barendregt, Lambda Calculus. Its Syntax and Semantics, 1981. The monograph of the subject.