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The lambda calculus provides simple semantics for computation which are useful for formally studying properties of computation. The lambda calculus incorporates two simplifications that make its semantics simple. The first simplification is that the lambda calculus treats functions "anonymously;" it does not give them explicit names.
In programming language semantics, normalisation by evaluation (NBE) is a method of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics. A term is first interpreted into a denotational model of the λ-term structure, and then a canonical (β-normal and η-long) representative is extracted by ...
Hendrik Pieter (Henk) Barendregt (born 18 December 1947, Amsterdam) [1] is a Dutch logician, known for his work in lambda calculus and type theory. Life and work [ edit ]
Based on the operators within lambda calculus, application and abstraction, it is possible to develop an algebra whose group structure uses application and abstraction as binary operators. Application is defined as an operation between lambda terms producing a λ-term, e.g. the application of λ onto the lambda term a produces the lambda term λa.
In the 1930s Alonzo Church sought to use the logistic method: [a] his lambda calculus, as a formal language based on symbolic expressions, consisted of a denumerably infinite series of axioms and variables, [b] but also a finite set of primitive symbols, [c] denoting abstraction and scope, as well as four constants: negation, disjunction, universal quantification, and selection respectively ...
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.