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A head normal form is a term that does not contain a beta redex in head position, i.e. that cannot be further reduced by a head reduction. When considering the simple lambda calculus (viz. without the addition of constant or function symbols, meant to be reduced by additional delta rule), head normal forms are the terms of the following shape:
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.
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.
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]
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.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
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 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: