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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.
When combined with the idea of weak reduction, the resulting call-by-value reduction is indeed a faithful approximation. [19] Unfortunately, weak reduction is not confluent, [17] and the traditional reduction equations of the lambda calculus are useless, because they suggest relationships that violate the weak evaluation regime. [19]
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 lambda calculus, a beta redex is a term of the form: [3] [4] (.). A redex is in head position in a term , if has the following shape (note that application has higher priority than abstraction, and that the formula below is meant to be a lambda-abstraction, not an application):
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]
Java's lambda expressions are just syntactic sugar. Anything that can be written with a lambda expression can be rewritten as a call to construct an instance of an anonymous inner class implementing the interface, [ a ] and any use of an anonymous inner class can be rewritten using a named inner class, and any named inner class can be moved to ...
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 ...