Making sense of terms without normal forms

Recall that for terms with normal forms, the normal form denotes the ``value'' of the term. What about terms without normal forms? Are all terms without normal forms equally ``meaningless''?

Suppose we want an equivalence $\approx$ on lambda terms such that:

1. $(\lambda x M)N \approx M\{x \leftarrow N\}$--that is, $\approx$ the equivalence induced by the $\beta$ reduction.

2. If $M$ and $N$ do not have normal forms, then $M \equiv N$.

3. Functions that are equated by $\approx$ yield equivalent results for the same arguments. That is, if $M \approx N$ then for all $R$, $MR \approx NR$.

Then, we can show that $\approx$ is the trivial relation the equates all terms--in other words, $M \approx N$ for all $M$, $N$!

To see this, consider the function $F$ defined by

$$F x b = \mathrm{~if~} b \mathrm{~then~} x \mathrm{~else~} (F x b)$$

As we have seen above, we can plug in our definition for if-then-else and then unravel this recursive definition to yield a lambda term for $F$.

It turns out that

$$F = GG, \mathrm{~where~} G = \lambda fxb. (\mathrm{~if~} b \mathrm{~then~} x \mathrm{~else~} (f f x b)$$

Now, consider $F X \langle{\texttt{tt}}\rangle$ and $F X \langle{\texttt{ff}}\rangle$, where $\langle{\texttt{tt}}\rangle$ and $\langle{\texttt{ff}}\rangle$ are the encodings of the truth values true and false.

• $F X \langle{\texttt{tt}}\rangle \rightarrow \mathrm{if~} \langle{T}\rangle \mathrm{~then~} X \mathrm{~else~} (F X \langle{\texttt{tt}}\rangle ) \rightarrow X$.

• $F X \langle{\texttt{ff}}\rangle \rightarrow \mathrm{if~} \langle{F}\rangle \mat... ...(F X \langle{\texttt{ff}}\rangle ) \rightarrow F X \langle{\texttt{ff}}\rangle$.

Now, suppose we have an equivalence $\approx$ as described above. Then

$$\begin{array}{lcl} F Z & \rightarrow & (\lambda xb.(\mathrm... ...~} Z \mathrm{~else~} G))\\ & \rightarrow & \ldots \end{array}$$

Since $F Z$ does not terminate for any $Z$, we have $FX \approx FY$ for all $X$ and $Y$. But, since $FX \approx FY$ implies that $FX M \approx FY M$ for all M (by Condition 3 in the definition of $\approx$). But then we have $FX \langle{\texttt{tt}}\rangle \approx FY \langle{\texttt{tt}}\rangle$. Since $F Z \langle{\texttt{tt}}\rangle \rightarrow Z$ for all $Z$, this means that $X \approx Y$ for all $X$ and $Y$!