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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.
Optimal reduction is not a reduction strategy for the lambda calculus in a narrow sense because performing β-reduction loses the information about the substituted redexes being shared. Instead it is defined for the labelled lambda calculus, an annotated lambda calculus which captures a precise notion of the work that should be shared.
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.
NBE was first described for the simply typed lambda calculus. [1] It has since been extended both to weaker type systems such as the untyped lambda calculus [2] using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory. [3] [4] [5] [6]
Viewing the lambda calculus as an abstract rewriting system, the Church–Rosser theorem states that the reduction rules of the lambda calculus are confluent. As a consequence of the theorem, a term in the lambda calculus has at most one normal form , justifying reference to " the normal form" of a given normalizable term.
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 ...
In September 2006 he posted to the Types mailing list "A very short note on homotopy lambda calculus", [14] which sketched the outlines of a type theory with dependent products, sums and universes and of a model of this type theory in Kan simplicial sets. It began by saying "The homotopy λ-calculus is a hypothetical (at the moment) type system ...
A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction.In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below).