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Lambda Calculus as a Workflow Model by Peter Kelly, Paul Coddington, and Andrew Wendelborn; mentions graph reduction as a common means of evaluating lambda expressions and discusses the applicability of lambda calculus for distributed computing (due to the Church–Rosser property, which enables parallel graph reduction for lambda expressions).
In the context of the lambda calculus, normal-order reduction refers to leftmost-outermost reduction in the sense given above. [10] Normal-order reduction is normalizing, in the sense that if a term has a normal form, then normal‐order reduction will eventually reach it, hence the name normal. This is known as the standardization theorem. [11 ...
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:
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.
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 redexes. A head reduction of a term t (which is supposed not to be in head normal form) is a head reduction which starts from a term t and ends on a head normal form. From ...
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.
The name "normal order" comes from the lambda calculus, where normal order reduction will find a normal form if there is one (it is a "normalizing" reduction strategy). [31] Lazy evaluation is classified in this article as a binding technique rather than an evaluation order.
Such an essentially semantic, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms. NBE was first described for the simply typed lambda calculus. [1]