Lambda calculus

The lambda calculus (or λ-calculus) is a mathematical formalism developed by the logician Alonzo Church (q.v.) in the 1930s to model the mathematical notion of substitution of values for bound variables. Consider the definition f(x) = x + 1, which defines f to be the successor function. The variable x in this definition is a bound variable in the sense that replacement of all instances of x by some other variable (say, y) yields a definition f(y) = y + 1, which is semantically equivalent. In the λ-calculus, the successor function f may be defined by the λ-expression λx(x + 1). The subexpression (x + 1) is referred to as the body of the λ-expression. The subexpression λx is referred to as the bound variable part and specifies that x is to be regarded as a bound variable in the body with which λx is associated.