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Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
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:
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.
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 ...
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.