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One step reduction

Recall that we introduced the $\beta $ rule


\begin{displaymath}
(\lambda x.M) M' \rightarrow _\beta M\{x \leftarrow M'\}
\end{displaymath}

and said, informally, that we would permit this rule to be used in all ``contexts''. The $\beta $ rule is not the only basic rule possible. For instance, observe that the two terms $\lambda x.(M x)$ and $M$ are equivalent with respect to $\beta $ reduction--for any term $M'$, $(\lambda x.(M x)) M' \rightarrow _\beta M M'$. This can be formalized by a rule (normally called $\eta$) which says


\begin{displaymath}
\lambda x.(M x) \rightarrow _\eta M
\end{displaymath}

Given a set of basic rules such as $\beta $, $\eta$, ..., we can inductively define a one step reduction that permits any of these basic rules to be used in any context. Let us denote one step reduction by $\rightarrow $. We define $\rightarrow $ through inference rules such as the following:

if $M \rightarrow _x M'$ for some basic rule $x$ such as $\beta $, $\eta$, ..., then $M \rightarrow M'$

Following the conventional notation used in logic, we present such a rule in the following form


\begin{displaymath}
\frac{M \rightarrow _x M'}{M \rightarrow M'}
\end{displaymath}

Here is the complete set of rules defining $\rightarrow $:


\begin{displaymath}
\begin{array}{cccc}
\displaystyle\frac{M \rightarrow _x M'}...
...&
\displaystyle\frac{N \to N'}{MN \rightarrow MN'}
\end{array}\end{displaymath}

Notice that all that we have done is to formalize the fact that we permit the basic reductions $\rightarrow _x$ within any subterm. In the ``calculations'' we have seen so far, we have already informally used this form of applying the $\beta $ rule.

In the discussion that follows, we shall dispense with the $\eta$ rule and assume that the only basic rule is the rule $\beta $.


next up previous contents
Next: Normal forms Up: The (untyped) lambda calculus Previous: Encoding arithmetic   Contents
Madhavan Mukund 2004-04-29