What Is an Alternative Beta?

In the lambda calculus, one term is the beta paradigm (normative), and it is possible without a "beta reduction". One term is the beta-eta paradigm , and it is possible to have neither a beta reduction nor an "eta reduction". One term is the head paradigm if there is no "beta-reducible formula at the head position". ^[1]

In the lambda calculus, the beta reducer (redex) is a term of the form

here

Yes (may) involve variables

Item.

"Beta reduction at the head position" is to apply the following rewrite rule to a beta reducer

here

Is the item

Medium variable

Replace with item

the result of.

A beta reduction is in the head position if it has the following form:

Not any reduction in this form is an internal beta reduction.

In general, there are many different possible beta reductions for a given term. Normal order reduction is an evaluation strategy that always applies the "beta reduction of the head position" rule until no more such reductions are possible. At this point, the term of the result is the "head paradigm".

In contrast, in applying order reduction , internal reduction is applied first, and head reduction is applied only when no more internal reductions are possible.

Normal order reduction is complete, in the sense that if an item has a head normal form, normal order reduction can always reach it. Conversely, application order reduction may not end, even when the term has a canonical form. For example, using application order reduction, the following reduction sequences are possible:

Using normal order reduction, the same starting point quickly reduces to the normal form: