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I want to reduce this to normal form. I think it's already in normal form, but that one can get the same expression infinitely.

But that necessarily means the first $x$ does not bind any of the inner $x$ variables? I am struggling to get rules for binding clear when its the same variable

A. Lindberg
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  • Maybe relevant: https://math.stackexchange.com/questions/118298/lambda-calc-bound-and-free-variables. – Martín-Blas Pérez Pinilla Mar 12 '19 at 09:40
  • I'm not very well versed in $\lambda$-calculus conventions, but $\lambda x.(\lambda x.xx)$ looks very ambiguous to me. Does the outer $\lambda$ expression take in an $x$ and feed back the function which applies its argument to itself, the function which applies $x$ to the argument, the function which applies the argument to $x$, or the function that applies $x$ to itself? I don't know. – Arthur Mar 12 '19 at 09:45
  • You're correct: the first bounded $x$ is not binding the inner $x$'s. – Berci Mar 12 '19 at 10:02

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The term is not in normal form, because it contains a redex: $(\lambda x.xx)(\lambda x.xx)$. However, beta-reducing this redex yields just the same term again: $$(\lambda x. (\lambda x.xx)(\lambda x.xx)) \to_\beta (\lambda x. (\lambda x.xx)(\lambda x.xx))$$ Since this is the only beta-reduction possible, $(\lambda x. (\lambda x.xx)(\lambda x.xx))$ is not normalizable (= does not have a $\beta$ normal form).

You are partially right in your assumption that the outermost abstraction $\lambda x$ does, in a way, not affect the variables in the inner subterms: Just like in predicate logic with quantifiers binding variables, an outermost binding of variables will get "overwritten" by a new abstraction in a subterm.
This is due to the definition of substitution and free and bound variables: $\lambda x.M$ binds any free occurrence of $x$ in $M$. But since every occurrence $x$ is already bound by another abstraction in $M$, there are no occurrences of variables that are bound by the outermost abstraction. To find the free variables in $(\lambda x. (\lambda x.xx)(\lambda x.xx))$, you would recursively go down the structure of the lambda term and look for the free occurences of $x$ in each of the subterms $(\lambda x.xx)$ and $(\lambda x.xx)$. Since all occurrences $x$ belong to a subterm of the inner $\lambda x.M$ respectively and are therefore already bound by other abstractions, there are no free occurrences of $x$ in any of the subterms and thus none in the term as a whole, so the outer $\lambda x$ does not effectively bind anything.
If you were to apply the term $(\lambda x. (\lambda x.xx)(\lambda x.xx))$ to some term $P$ and performed beta reduction, the operational semantics would tell you to substitute every free occurrence of $x$ by $P$. The recursive definition of substitution first passes down the substitution $[P/x]$ to each of the subterms in the application $(\lambda x.xx)(\lambda x.xx)$, but then for each of the two subterms $(\lambda x.xx)$ would state that nothing happens, because there are no free occurrences of $x$ in these terms. So the inner abstractions take operational precedence over the outermost abstraction due to the way free vs. bound variables and substitution are defined; they are "new" abstractions that are independent of the first $\lambda x$.
This also means that the $x$'s in the two subterms of the form $(\lambda x.xx)$ are independent: The $x$'s belong to different bindings and therefore behave as different varialbes, in the same way that the outermost $\lambda x$ is a different abstraction from the ones in the respective subterms.
Nevertheless, I sand only "partially right" because the outermost abstraction techincally does have scope over the entire subterm $((\lambda x.xx)(\lambda x.xx))$. If there were free occurences of $x$ in any of the subterms, then these variables would be bound. It just so happens that all of none variables that are in the scope of the outer $\lambda x$ are free, so the abstraction is vacuous and doesn't have an actual effect. Still, the scope of $\lambda x$ is the entire suberm $((\lambda x.xx)(\lambda x.xx))$.

Natalie Clarius
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