complex expression `3 + 4` evaluates to `7`. The evaluation of the
expression computing a sum. There is a clear sense in which the
expression `7` is simpler than the expression `3 + 4`: `7` is
-syntactically simple, and `3 + 4` is syntactically complex.
+syntactically simple, and `3 + 4` is syntactically complex.
Now let's take this folk notion of computation, and put some pressure
-on it.
+on it.
##Church arithmetic##
Is the final result simpler? This time, the answer is not so straightfoward.
Compare the starting expression with the final expression:
- * 3 4
+ * 3 4
(\lrf.l(rf))(\fz.f(f(fz)))(\fz.f(f(f(fz))))
~~> 12
(\fz.f(f(f(f(f(f(f(f(f(f(f(fz))))))))))))
And if we choose different numbers, the result is even less clear:
- * 3 6
+ * 3 6
(\lrf.l(rf))(\fz.f(f(fz)))(\fz.f(f(f(f(f(fz))))))
~~> 18
(\fz.f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(fz))))))))))))))))))
In the arithmetic example, there is only one number that corresponds
to the sum of 3 and 4 (namely, 7). But there are many sums that add
-up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc.
+up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc.
So the unevaluated expression contains information that is missing
from the evaluated value: information about *how* that value was
In this example, reduction returns the exact same lambda term. There
is no simplification at all.
- (\x.xxx)(\x.xxx) ~~> ((\x.xxxx)(\x.xxxx)(\x.xxxx))
+ (\x.xxx)(\x.xxx) ~~> ((\x.xxxx)(\x.xxxx)(\x.xxxx))
Even worse, in this case, the "reduced" form is longer and more
complex by any measure.