Its Syntax and Semantics“ (Barendregt, 1981). Yet another way to answer this question is to provide the following quotation from “Lambda Calculus. ![]() For example, the combinators $(\lambda y.\lambda x.y x)$ and $(\lambda y.y)$ are observationally equivalent - but how can we show that? Simply noting that they are $\eta$-equivalent packs a somewhat subtle and long-winded argument into an easily reusable concept. The availability of eta-reduction does not change whether a term has a normal form, for example.Įta is, however, sometimes useful as an auxiliary reasoning concept. ![]() So in that sense, eta equivalence does not add anything fundamentally new to the calculus. The eta conversion is intuitively justified by the fact that in the end the only thing you can really do with a lambda term is to apply it to something, so at that point it's not going to matter whether you've already eta reduced or not. $$ (\lambda x.M x) N \rightsquigarrow M N $$Ĭan be justified either as a beta or an eta reduction, but the eta reduction could happen before there's even an $N$ there. You could say that it anticipates the beta redex that will arise if and when the left-hand side is applied to something. Beta conversion is the familiar $(\lambda x.M)N \rightsquigarrow M$.Įta conversion is $\lambda x.M x \rightsquigarrow M$ when $M$ does not contain $x$ free.Įta is not a special case of beta, because there is no beta redex anywhere yet.
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