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Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
The purpose of β-reduction is to calculate a value. A value in lambda calculus is a function. So β-reduction continues until the expression looks like a function abstraction. A lambda expression that cannot be reduced further, by either β-redex, or η-redex is in normal form. Note that alpha-conversion may convert functions.
In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.
a variation in the calculus of variations; the Kronecker delta function [20] the Feigenbaum constants [21] the force of interest in mathematical finance; the Dirac delta function [22] the receptor which enkephalins have the highest affinity for in pharmacology [23] the Skorokhod integral in Malliavin calculus, a subfield of stochastic analysis
In the 1930s, a new type of expression, the lambda expression, was introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. [40] [b] The lambda operators (lambda abstraction and function application) form the basis for lambda calculus, a formal system used in mathematical logic and programming language theory.
Howard's correspondence naturally extends to other extensions of natural deduction and simply typed lambda calculus. Here is a non-exhaustive list: Girard-Reynolds System F as a common language for both second-order propositional logic and polymorphic lambda calculus, higher-order logic and Girard's System F ω; inductive types as algebraic ...
There is no exact structural equivalent in lambda calculus for let expressions that have free variables that are used recursively. In this case some addition of parameters is required. Rules 8 and 10 add these parameters. Rules 8 and 10 are sufficient for two mutually recursive equations in the let expression. However they will not work for ...
List of equations; List of fundamental theorems ... Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) ... Addition theorem (algebraic ...
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