Search results
Results From The WOW.Com Content Network
Typed lambda calculi are strictly weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can. On the other hand, typed lambda calculi allow more things to be proven.
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.
The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus. The fixed-point combinator may be used in a number of different areas:
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution.This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness" conditions in such implicit calculi are a notorious ...
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.
For example, one can write the meaning of sleeps as the following lambda expression, which is a function from an individual x to the proposition that x sleeps. λ x . s l e e p ′ ( x ) {\displaystyle \lambda x.\mathrm {sleep} '(x)} Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing ...
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.
A redex [1] (one says also β-redex) is a term of the lambda calculus of the form (λ x. t) u. If a term has the shape (λ x. t) u 1... u n it is said to be a head redex. A head normal form is a term of the lambda calculus which is not a head redex. [a] A head reduction is a (non empty) sequence of contractions of a term which contracts head ...